1. No category

Combinatorial Optimization: What Is the State of the Art
Author(s): Victor Klee
Source: Mathematics of Operations Research, Vol. 5, No. 1 (Feb., 1980), pp. 1-26
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MATHEMATICS OF OPERATIONS
Vol. 5, No. 1, February 1980
Printed in U.S.A.
RESEARCH
COMBINATORIALOPTIMIZATION:WHAT IS THE
STATEOF THE ART*t
VICTOR KLEE
Universityof Washington
Dedicated to E. J. McShanefor his seventy-fifthbirthday
This surveyattempts,withoutincludingmany details of algorithmsor of the underlying
theory,to answerthe followingquestions:Whatis combinatorialoptimization?Whatare the
landmarksof the theory?What are the most excitingrecent advances?What are the most
promisingdirectionsfor research?Whatare the best sourcesof furtherinformation?
1. What is combinatorialoptimization? Many practical problems, especially those
from operations research and computer science, are concerned with optimizing a
real-valued function f over a finite set X of d-tuples of integers. Often X is not
presented explicitly but is defined implicitly in some manner. When f is linear and X
is defined by a finite number of linear inequality constraints with integer coefficients,
the problem is one of (linear) integer programming. Starting with the work of Gomory
(1958), finite algorithms and an extensive theory have been developed for such
problems. See Hu (1969), Garfinkel and Nemhauser (1972), Geoffrion and Marsten
(1972), Balinski (1974), Castning (1976), Hammer et al. (1977), Beale (1977), and
Hausmann (1978) for details and further references. Computational experience with
these algorithms has been mixed. Some fairly large problems have been solved, but
experience often demonstrates, even with problems of moderate size, that an algorithm requiring a very large finite number of steps may be useless in practice.
Most of combinatorial optimization (as the term is used here) deals with problems
that can be formulated as integer programs but have an underlying combinatorial
structure that lends itself to the development of special algorithms. One naturallly
hopes that these algorithms, when applicable, will be more efficient than general
integer programming algorithms. Many of the problems are associated with a directed
or undirected graph G = (N,A) and a function a defined on the (finite) node-set N or
the arc-set A or on N U A and taking values in ] - o, oo]. The pair (G, a) is often
called a network. It is assumed for simplicity that G is loopless, so that A is a set of
ordered or unordered (according as G is directed or not) pairs of distinct nodes. Arcs
may be written as ordered pairs (i, j) with the understanding that they will be
interpreted as unordered when G is undirected.
A path from node s to node t is a sequence of arcs of the form
(Xo,X), (XIX2), ... , (Xk-I,
)
(*)
with x0 = s and xk = t. When such a path exists, t is accessible from s. A path (*) is a
simplepath if there is no repetition of nodes other than that implied by the notation. It
is a circuit if xk = x0, and a simple circuit if there is otherwise no repetition. Paths and
* Received September 18, 1978.
AMS 1970 subject classification. Primary 90C35. Secondary 05C35.
IAOR1973subjectclassification.
Main:Networks.Crossreference:Computationalanalysis.
OR/MS Index 1978 subject classification. Primary: 481 Networks/graphs.
Key words.Combinatorialoptimization,survey,minimumpath, minimumspanningtree, maximumflow,
maximummatching,minimumtour,computationalcomplexity,good algorithm.
t Preparationof this paperwas supportedin partby the Officeof Naval Research.
0364-765X/80/0501/0001$01.25
Copyright ? 1980, The Institute of Management Sciences
2
VICTOR KLEE
circuitsthat are simple may convenientlybe regardedas sets (ratherthan sequences)
of arcs.
A graph is connected if each node t is accessible from each node s #= t. A tree is a
connected graph that contains no circuit. A spanningsubgraphis one that uses all
nodes. A matchingis a set of arcs no two of which have a common node. A touris a
spanningsimple circuit.
For purposesof illustration,we focus here on five problemsof networkoptimization, assuming for simplicitly that A is the domain of a and (except where the
contraryis stated)a > 0. The graphG shouldbe undirectedin (2) and (4), but in the
others it may be directedor undirected.In all but (3), the desired solution may be
regardedas a subset of A, the lengthor weightof such a set being the sum of the
a-values of its members.(Thus the objective function is a linear function of arc
lengths.)In (3) the values of a are regardedas flow capacities.
(1) (Minimumpath) For two given nodes s and t of G, find a shortestpath from s
to t or concludethat t is not accessiblefrom s.
(2) (Minimum spanning tree) For a connected G, find a connected spanning
subgraphof minimumweight.
(3) (Maximumflow) For two given nodes s and t of G such that t is accessible
from s, find an a-feasibleflow of maximumvalue from s to t.
(4) (Maximummatching)Among all matchingsin G, find one of maximumweight.
(5) (Minimumtour) For a completeG, find a shortesttour.
In addition to many practical optimizationproblems for which the immediate
mathematicalmodel is one of (1)-(5), a great variety of other problems can be
reducedto these or to a combinationof them.The readeris undoubtedlyfaced, each
day, with severalinstancesof (1). Problem(2) arisesin the constructionof communication networks.Problem(3) stems from an attemptto evaluatethe capacity of the
EasternEuropeanrail networkto supporta large-scaleconventionalwar (see Billera
and Lucas, 1976); it and its relative, the minimum-costflow problem,are used to
model a varietyof transportationproblems.Thoughalgorithmsfor (4) are perhapsnot
as obviouslyusefulas those for (1)-(3), matchingalgorithmshave in fact been applied
to problems of personnel assignment, pairing of machined parts, scheduling of
two-processorsystems,variousroutingproblems(see Lawler(1976)and his references
for all of these), evaluationof biomedicaldata (Tanimoto,(1976)), and finding the
rank of a matrix(Anderson(1975), Klee and van den Driessche(1977)).Problem(5)
is the famous traveling salesman problem, whose applications include computer
wiring, vehicle routing, clustering of data, job shop scheduling (see Lenstra and
Rinnooy Kan (1975)for all of these),archaeologicalseriation(Wilkinson(1971)),and
minimizingwallpaperwaste (Garfinkel(1977)).
For the many directapplicationsof (1)-(4), and for the reductionof otherproblems
to these, the best single sourceis the book of Lawler(1976), Combinatorial
Optimization: Networksand Matroids.It will be mentionedfrequentlyin the sequel.The books
or surveyarticlesof Berge and Ghouila-Houri(1965), Bradley(1975), Busackerand
Saaty (1964), Dantzig (1963), Eiselt and von Frajer(1977), Elmaghraby(1970), Ford
and Fulkerson (1962), Frank and Frisch (1972), Fulkerson(1966), Garfinkel and
Nemhauser(1972),Hu (1969),Iri (1969),Karp(1975b),Tarjan(1978)and Whitehouse
(1973) also contain much useful informationabout combinatorialoptimization.For
problems reducible to (5) the basic referencesare Cook (1970) and Karp (1972),
(1975a).Aho et al. (1974)containsa good introductionto this fascinatingsubject,and
by far the most completestudy is the book of Garey and Johnson(1979), Computers
and Intractability: A Guide to the Theoryof 9% -completeness.
Let n = INI and a = IA
1, the respectivenumbersof nodes and of arcs of G. When
convenient, we assume N = { 1,2, . . ., n). In each of (1)-(2) and (4)-H5), the desired
COMBINATORIALOPTIMIZATION: STATE OF THE ART
3
solution may be regarded as a subset of A or, relative to a given indexing of A, as an
ordered a-tuple of 0's and l's. There are 2a such subsets, but many may be quickly
excluded by paying some attention to the combinatorics of the problem. However,
even after such exclusion and even for moderate values of n, explicit investigation of
all the remaining subsets is impossible even on the fastest computer. To illustrate this,
consider the case in which a > 0 and the graph G = (N,A) is undirected and
complete, so that a = n(n - 1)/2.
(1) Shortest paths are simple. When G is complete the number of simple paths from
s to t is 'n-O(n-2)k!.
(2) Each connected spanning subgraph of minimum weight is a tree. When G is
complete the number of spanning trees is n n-2.
(4) When G is complete there are 1n/21 arcs in each maximum matching. The
number of matchings consisting of ln/2J arcs is k(2k )/2k when n = 2k and (2k + 1)
when n = 2k + 1.
k!(k )/2
When
G is complete the number of tours (considered as sets of edges) is
(5)
(n 1)!/2.
If a computer required only 10- 0 seconds (one-tenth of a nanosecond) to investigate a particular spanning tree, the time required to investigate all nn-2 spanning trees
would be about 0.01 seconds for n = 10, 54 hours for n = 15, 8.3 x 105 years for
n = 20, and 4.5 x 1014 years for n = 25. This explains why one speaks of the
"combinatorial explosion" in connection with such problems, and why an algorithm
that is merely "finite" may be useless except for very small problems. The goal of
combinatorial optimization is to find algorithms that are useful even for very large
problems.
2. What is a good algorithm? How is the speed of an algorithm to be measured?
One may compare the performance of two algorithms for the same problem on a few
instances of special interest, but how relevant is that to their performance on other
instances? One may base the comparison on a large number of instances chosen in a
regular or random fashion, but that may be very expensive and yet not provide a
reliable indication of performance on instances much larger than those tested. Here
we have used (and henceforth will use) the terms problem and instance in the sense of
Aho et al. (1974). Each instance is associated with particular numerical data and a
problem is the class of all instances of a specified form. For example, the maximum
matching problem is the class of all instances of the form (4).
It is expected, of course, that large instances will usually be solved more slowly than
small instances. For problems concerning a graph G = (N,A), the parameters n = IN I
and a = IA provide a natural measure of the size of an instance. When r is a function
of these parameters, an algorithm is said to be of (time) complexity O(r(n,a)) if there
is a constant c such that for all G = (N,A) the algorithm requires at most cr(n,a)
computational steps. The notion of "step" must be interpreted in terms of an
appropriate model of computation, and for relevance to practice the model should be
based on the random access operations of modem electronic computers. The discussion here is in terms of the RAM model of random access computation described by
Cook and Reckhow (1973) and on pp. 5-14 of Aho et al. (1974), using the uniform
cost criterion. Without too much distortion, the reader may simply interpret "step" as
a single arithmetic operation (addition, multiplication, comparison, etc.).
A good algorithm is one that is polynomially bounded-is of complexity O(nPaq)
from some p and q. And of course we'd like the exponents p and q to be as small as
possible. This notion, popularized by Edmonds (1965a) and Cobham (1965), is
meaningful in theoretical studies and also useful in practice. Since a < n2, an O(nPaq)
algorithm is also O(nP+2q). However, n and a are mentioned separately because a is
4
VICTOR KLEE
much less than n2 for most graphsarisingin practice.Note that if an algorithmis to
take advantageof the sparsenessof the input graph G = (N,A) in order to achieve
performancebetter than 0(n2), then G cannot be input as its full n x n adjacency
matrix.Instead, the input may consist of lists that tell, for each node i E N, which
other nodesj are adjacentto i, and that also give the arc lengthsa(i, j).
The above notion of complexityinvolves the worst-casebehaviorof an algorithm,
but average-casebehavioris also importantin many applications.For this notion to
be meaningful,the samplespace must be carefullydefined,and for it to be useful the
samplespace must be appropriatelyrelatedto the instancesin which the algorithmis
to be applied. Parts of ??6 and 7 deal with average-casebehavior,but otherwisewe
are concernedhere only with worst-casebehavior.
In the interestof brevity,the precedingthree paragraphshave glossed over some
importantpracticaland philosophicalquestionsconcerningthe encodingof data, the
relationshipbetween optimizationproblemsand decision problems,the choice of a
model for computation,the reasons for emphasizingpolynomialboundedness,and
the pitfallsassociatedwith average-casebehavior.These mattersare treatedby Cook
and Reckhow (1973), Aho et al. (1974), Lawler(1976), Weide (1977), Tarjan(1978)
and Garey and Johnson(1979). It should also be stressedthat if an algorithmworks
well in practice,frequentlyhandlinglarge instanceswith amazingease, then it is of
coursea good and usefulalgorithmeven thoughit may not be "good"in the technical
sense used here. That is true, for example, of the simplex algorithm of linear
programming.Its worst-casebehavioris not polynomiallybounded(Klee and Minty
(1972), Jeroslow(1973), Avis and Chvaital(1976), Goldfarband Sit (1980?))even for
minimumcost flow problems(Zadeh (1973)), and its average-casebehaviorhas not
been proved to be polynomiallybounded in any sense that is very convincingly
related to computationalpractice(but see Borgwardt(1977), (1980?)).Nevertheless,
there is strong empiricalevidence that the average-casebehavior is polynomially
bounded(Dantzig(1963,p. 160)).(On the otherhand, specialnetworkalgorithmsare
apt to be faster than the simplex algorithmfor problemsto which they too can be
applied(Bradley(1975, p. 229)).)
3. Whatis the state of the art? Problems(1)-(5) are admittedlyspecialin nature,
but combinatorialoptimizationis concernedwith special problems.Thus it may be
reasonable to judge the state of the art by what is known about (1)-(5). These
problemsappearin variousways in the next four sections,in whichthe state of the art
is representedby three each of primarylandmarks,secondarylandmarks,exciting
recent advances,and directionsfor futureresearch.If permittedto make a summary
judgment in architecturalterms, we might say that the subject of combinatorial
optimizationhas graduallybeen transformedfrom the rococo to the merelybaroque
(thus reversingthe evolution that occurredin architecture),but is still far from the
simple elegance of the classic ordersor the sleek functionalismof modern architecture.This refersto a definitionof rococo as "a meaninglessassemblageof scrollsand
crimpedconventionalshellwork,wroughtinto all sorts of irregularand indescribable
forms," and of baroqueas "odd, grotesque,bizarre,having unusualformation."The
scrollsand crimpedconventionalshellworkare the many papersin the subjectwhich
repeat,in only slightlydifferentform, algorithmsthat appearin earlierpapers.As the
subjecthas developed,these "scrolls"have fallen away (been forgotten)and interrelationshipshave been discoveredamong much of what remains.The subjecthas even
achieved a considerabledegree of architecturalunity and may now properly be
regardedas a branch of mathematics,or of operations research,or of computer
science. However,the branchis certainlyof "unusualformation,"becausein the long
searchfor "good"algorithmsfor (1)-(5), none has been proved"best"relativeto the
RAM model of computation.
COMBINATORIALOPTIMIZATION: STATE OF THE ART
5
4. Three primarylandmarks. Three lines of investigation stand out for their
intrinsicdepth and their influence on the rest of the field. They are associatedwith
problem (3), with problem (4), and less directly with (5). We consider only the
directedcase of (3) becauseits formulationis slightlysimpler.
A.
The maximum flow problem and its ramifications. For each real function 4)on
A and each node k E N, let
=W- 2
?(k,j)-
(k,j)EA
2
(i,k).
(i,k)EA
The function4 is an (s, t)-flowif it satisfiesthe conservationconditions
(a) lk( )=0 for all k E N - {s, t}.
(at any intermediatenode, the outflow equals the inflow), from which it follows that
,U(J)= - ^(q) (the net outflow at the source s equals the net inflow at the sink t).
The number s,(4))is called the s-valueof 4. The flow 4 is a-feasibleif it satisfies the
capacityconstraints
(b) 0 < 4(i, j) < a(i, j) for all (i, j) E A.
Problem (3) asks for an a-feasible flow of maximum s-value. This is a linear
programmingproblemin the variables0(i, j) with objectivefunction ,s. The feasible
regionis the convex polytope P,, definedby the equalities(a) and the inequalities(b).
Thus (3) can be solved by general linear programmingalgorithms;but it is also
amenableto the special methodsof combinatorialoptimization.
An (s, t)-cut is a set of arcswhose removalfrom G leaves no directedpath froms to
t. The initial question about rail capacitiesled to the max-flowmin-cut theoremof
Ford and Fulkerson(1956) (discoveredalso by Elias et al. (1956)) and to their book
on Flowsin Networks(Ford and Fulkerson(1962)), which stimulatedmuch research
on combinatorialoptimization.The theoremassertsthat the maximumof s-values of
the a-feasible(s, t)-flowsis equal to the minimumof the weightsof the (s, t)-cuts.The
flow-augmentingpaths used to prove this theoremare used also in an algorithmfor
finding a maximumflow. As describedby Ford and Fulkerson(1957), (1962), the
algorithmis not polynomiallyboundedbut is has been appliedsuccessfullyto a large
numberof practicalproblems.Edmondsand Karp (1972)use the methodto produce
an O(na2) algorithm(see pp. 116-120 of Lawler (1976) for an exposition of their
work). However,a differentline of attack, startingfrom the O(n2a) maximumflow
algorithmof Dinic (1970), has led to still faster algorithmsfor finding maximum
flows. See ?6 for references.
Space permitsus to mention only a few of the many consequencesand ramifications of networkflow theory.An earlyobservationwas that if the capacityfunctiona
is integer-valuedthen all verticesof P. have integercoordinatesand hence any linear
integer program over P. can be solved as an ordinary linear program,without
worryingabout the integralityconstraints.That led to the search for other integer
programswhich could be solved as linearprogramsand thus to the study of matrices
that are totally unimodular(each square submatrixhas determinant-1, 0 or + 1).
See Garfinkeland Nemhauser(1972), Hu (1969), Lawler(1976) and their references.
And see especiallythe work of Seymour(1979),which has been used by J. Edmonds
in designinga good algorithmthat tests for total unimodularity.In a differentbut
related direction, the theory of blocking and antiblocking polyhedra (Fulkerson
(1971),Chv/atal(1976,pp. 316-318)) may be regardedas a far-reachinggeneralization
of the max-flow min-cut theorem.There have been several other generalizationsof
this famous result,the deepestbeing that of Seymour(1977).
Suppose,in (3), that each arc (i, j) A has not only a capacity a(i, j) but also a
cost -y(i,j) of sending a unit of flow along the arc, and suppose there exists an
6
VICTOR KLEE
a-feasible flow of given s-value v. The minimum-costflow problemasks for such a
flow q, whose cost (i, jeAy(i, jWi, j) is minimum.The primal-dual approach to
finding minimum cost flows led to the general primal-duallinear programming
algorithmof Dantzig, Ford and Fulkerson(1956). A significantunification of the
maximumflow problem,the minimum-costflow problem,and other partsof network
flow theorywas achievedin the study of minimum-costcirculationsand the "out-ofkilter"method for finding them (Minty (1960), (1961), Fulkerson(1962)). For recent
improvementsof this method,see Zadeh (1980?)and Rockafellar(1981?).
Many other problemsof combinatorialoptimizationcan be reducedto the maximum flow problemor the minimum-costflow problem.See the next paragraphfor
one example and Lawler (1976) for others. Plainly if a single publicationis to be
named as the most importantlandmarkof combinatorialoptimization,it must be the
book of Ford and Fulkerson(1962). It is now outdated in many respects,but that
merely indicates its success in stimulatingthe development of a new branch of
mathematics.
B. Maximummatching. Because of its close relationshipto other combinatorial
topics, specialattentionhas been paid to the case of the maximummatchingproblem
(4) in which each arc is of weight I (cardinalityweighting)and G is bipartite.Let P
and Q denote the two partsof the node set, and form a directedgraphD by adding
new nodes s and t, new arcs(s, p) and (q, t) for all p E P and q E Q, and directingall
old arcs from P to Q. Let each arc have capacity 1. Then the maximummatching
problemfor G is easily seen to be equivalentto the maximumflow problemfor D.
Hence the maximumcardinalitymatchingproblemfor bipartitegraphscan be solved
as a maximumflow problem,but it can be solved in otherways as well.
A theoremof Berge(1957) assertsthat for cardinalityweighting,a matchingM is
maximumif and only if it admits no augmenting
path. This is a simplepath S whose
end nodes are not coveredby M and whose arcs belong alternatelyto A - M and to
M. If S c A is such a path and M' is the symmetric difference M ? S = (M-
S) U
(S-- M) then M' is a matchingwith IM' = IMI+ 1. Suppose,conversely,that there
exists a matchingM' with IM' > IM. FollowingNormanand Rabin(1959),consider
the graphG' = (N,A') whereA' = M e M'. Each node of G' is of valence < 2, so G'
is a node-disjointunion of isolated nodes, circuitswhose arcs alternatebetween M
and M', and alternatingpaths.Since IM'l > M, at least one of the alternatingpaths is
an augmentingpath for M.
For the cardinalityweightedbipartitecase, augmentingpaths are relativelyeasy to
find; an O(na) maximummatchingalgorithmthat is unusuallyeasy to understand,
analyze, and programcan be obtained by specializingthe method of Desler and
Hakimi (1969). By finding several augmentingpaths simultaneously,Hopcroft and
Karp (1973) produce an elegant O(nl/2a) algorithm.When G is bipartitebut the
arc-weightinga is unrestricted,(4) is called the assignment
problem.The best available
algorithmsare of complexityO(pq2), wherep and q are the cardinalitiesof the two
partsof the node-set(Kuhn (1955),Tomizawa(1972), Lawler(1976, pp. 201-207)).
The workof J. Edmondsand otherson nonbipartitematchingconstitutesa second
primarylandmarkin the developmentof combinatorialoptimization.Augmenting
paths play a role here too, but it is muchmoredifficultto find themefficientlythan in
the bipartitecase. For severalyears there was doubt about the existenceof a good
nonbipartitemaximummatchingalgorithm,even for cardinalityweighting.However,
Edmonds(1965a)found an O(n4) algorithmand then (1965b)showedhow to extend
it to arbitraryweightings.His implementationis improvedin the O(n3) algorithmsof
Gabow(1973),(1976)and Lawler(1976,pp. 217-263). See also Feicht,Heck and Pape
(1977). For cardinality weighting only, a fairly simple O(na) implementationis
COMBINATORIALOPTIMIZATION:STATE OF THE ART
7
providedby Kamedaand Munro(1974),and the ideas of Hopcroftand Karp (1973)
are extended by Even and Kariv (1975) to yield an O(n5/2) maximum cardinality
matchingalgorithmfor generalgraphs.It seems safe to say the latter has not been
widely checked,for the joint paperis quite condensedand the detailedexposition,in
the dissertationof Kariv(1976),requiresmore than 200 pages!
The nonbipartitemaximummatchingproblemis certainlyone of the deepest for
which good algorithmsare known,and the rationalebehind Edmonds'sapproachhas
been influentialin other ways that space does not permit us to describe(Edmonds
(1970), Chvatal (1973)). A book on matching is being written by Edmonds and
Pulleyblank(1980?).
C. %916-Completeness.A third primary landmark of combinatorial optimization
was the discoveryby Cook (1970) and Karp(1972),(1975a)of a large class 9LPC of
combinatorialdecisionproblemssuch that
(i) 9L9C includesmany importantand difficultproblemsof combinatorialoptimization(for example,a close relativeof (5))-problems for whichgood algorithmshave
vainly been soughtfor years by many researchers;
(ii) many problemsin 9OYC are at least superficiallysimilarto problemsfor which
good algorithmsare known;
(iii) if even one memberof 9L6C admitsa good algorithmthen they all do.
The precise definition of %LYfC
depends on the notions of a deterministicTuring
machine (DTM), a nondeterministicTuring machine (NTM), the complexity of a TM,
and other notions from the theoryof computation.The details are complicated,but
some may be omittedbecauseit is knownthat a problemadmitsa good algorithmfor
the RAM model of computationif and only if it admits one for the DTM model.
Thus for present purposes a DTM may be regarded as an ordinary computer
program, and an NTM may be regarded as an animate DTM so fecund and
imaginativethat it can, at the start of a computation,make a finite number k of
guessesand immediatelysplit into k replicasof itself (to be run simultaneously),one
to investigatethe consequencesof each guess. The complexityof a TM is a function
that tells how long it takes to processsets of input data of varioussizes.
Let ' <resp.A9@>denote the class of all decision problemssolvable by DTM's
<resp.NTM's>of polynomiallyboundedcomplexity.Then S6 consistsof all decision
problemsthat admit good algorithms.And 91L consists of all decisionproblemsfor
which an affirmativeanswercan be obtained(when correct)by applyinga polynomially bounded "checking algorithm"to the appropriateguess output by a finite
"guessingalgorithm."
For each optimizationproblem there is a closely related decision problem that
includesa number/3 as partof the input.For example,the problemrelatedto (5) asks
whetherG admitsa tour of length < P/.To see this belongs to 9L%P,
let the guessing
the
n!
of
It
is
all
G's
nodes.
easy to check, in
permutations
algorithmproduce
whether
a
a
tour
of
given guess represents
length < P/.
polynomialtime,
The preceding paragraphsgloss over several matters, but should nevertheless
convey the correctideas. (See Aho et al. (1974) and Garey and Johnson(1979) for a
firmerfoundation.)Continuingin this vein, a problemX is said to be (polynomially)
reducibleto a problemY if each good algorithmfor solving Y can be used to produce
a good algorithmfor solvingX. This does not requirethat X or Y actuallyadmits a
good algorithm,but only that if Y does then so does X. In effect, it requiresthe
existenceof a good algorithmfor transformingX into a specialcase of Y, such as the
procedureused earlierto transformthe bipartitecardinalitymatchingprobleminto a
specialsort of maximumflow problem.Note that reducibilityis transitive.A problem
Y is 6?P-hard if every problem in 9LP is reducible to Y; if in addition, Y itself
8
VICTOR KLEE
belongs to 9L?, then Y is 9Lj9-complete. 6X6C is the class of all 9L?i-complete
problems.
A Boolean expression (formed from conjunctions of disjunctions of logical variables or negations of those variables) is satisfiable if values of 0 and 1 can be assigned
to the variables so as to give the entire expression the value 1. (For example,
(x V y V z) A x Ay is satisfiable but x A (y V z) A (x V z) A (x V y V z) is not.) It is
unknown whether there is a good algorithm for deciding the satisfiability of an
arbitrary expression of length n. Cook (1970) showed the satisfiability problem to be
9LP?-complete and then Karp (1972), (1975a) established the ~9 -completeness of
many other problems Y by showing that Y E L?9 (usually easy) and that the
satisfiability problem, or some other problem known to be 9LX6-complete,is reducible
to Y. Garey and Johnson (1979) have not only the most extensive list of 9L?i-complete
problems, but also an excellent discussion of various strategies for proving 69Pcompleteness.
In addition to the decision problem most closely related to (5), the following is also
o99 -complete.
(5') Determine whether the graph G = (N,A) admits a tour.
It is easy to reduce (5') to the special case of (5) in which all arc-lengths are 0 or 1.
For let G' be the complete graph with node-set N and let each arc in A <resp. not in
A> be of length 0 <resp. 1)>.Then G admits a tour if and only if G' admits a tour of
length 0.
Note that if even one 91P-hardproblem admits a good algorithm, then 9P= ~9L.
In particular, 6? = _9P if and only if (5') admits a good algorithm. These facts, (i)
above, and the fact that NTM's seem intuitively to be much more powerful than
DTM's, are regarded by many researchers as almost conclusive evidence that
9 = A9C. However, it is also conceivable that (5) or (5') admits a good algorithm but
does not admit one of complexity less than O(nP) where p is very large and the
algorithm therefore so complicated that it is unlikely ever to be found. Good
algorithms have been useful in practice because for problems of practical interest they
have seldom been worse than 0(n6) and the multipliers c have not been excessively
large. Does this say more about the complexity of the interesting problems in 9 or
about the limitations of human ingenuity in devising algorithms?
We mention only a few more 91?P-complete combinatorial problems, mainly to
illustrate the aspect (ii) stated above and to prepare for later comments. In precise
treatments, the term "P9 -complete" is usually reserved for decision problems such as
(5') and not applied to optimization problems such as those stated below. That
distinction is ignored here, for our aim is only to provide a rough understanding of
L9P-completeness. The problems listed below all appear (in slightly different forms)
in the list of Garey and Johnson (1979), and all but (2') are treated by Karp (1972),
(1975a).
The following problem should be compared with (1). It is 9P9-complete even in the
cardinality weighted version.
(1') For two given nodes s and t of G, find a longest simple path from s to t.
The following should be compared with (2). For each fixed d > 4, it is 99Pcomplete even in the version in which all arc weights are 0 or 1. The problem arose in
the design of telephone networks.
(2') Given that G admits a spanning tree in which no simple path has more than d
arcs, find such a spanning tree of minimum weight.
For the cardinality weighting, (4) is equivalent to asking for a smallest set of arcs
that covers all nonisolated nodes of G. Though (4) belongs to VP,the following
problem is 9X?-complete.
(4') Find a smallest set of nodes that intersects all arcs of G.
COMBINATORIALOPTIMIZATION:STATE OF THE ART
9
A set X C N is independentif no two of its members are adjacent (joined by an edge).
Since X is independent if and only if N - X intersects all arcs, the following problem
is equivalent to (4').
(4") Find a largest independent set of nodes in G.
The assignment problem, mentioned earlier, can be stated as follows in terms of a
nonnegative n x n matrix (aij): Among all sets S of n pairs (i, j) that include at most
one pair from each row and column, find an S for which (i/,] esag is maximum.
This problem admits an O(n3) algorithm, but the 3-dimensional assignmentproblem is
9L9 -complete. It asks, for a nonnegative n X n x n array (aijk):
(4"') Among all sets S of n triples (i, j, k) such that no two members of S agree in
any coordinate, find one for which (i j, k)e Saijk is maximum.
Suppose, in fact, that each aijk is 0 or 1 and the problem is merely to determine
whether the maximum in (4"') is equal to n. Even that restricted form of the problem
is 9Lt6-complete.
5. Three secondary landmarks. Among the 9L9 -hard problems, the traveling
salesman problem (5) and its relatives stand out for the amount of attention they have
received and the variety of practical situations in which they have arisen. The same is
true of the shortest-path problem (1) among those for which good algorithms are
known. The collections of algorithms dealing with these problems form two secondary
landmarks of combinatorial optimization. A third is the study of optimization in
independence systems, which orginated from the minimum spanning tree problem (2).
A fourth, not discussed here, is the part of polyhedral combinatorics that studies
interrelationship among combinatorial optimization, linear programming duality, and
the facial structure of convex polyhedra (Edmonds (1965), (1970), Fulkerson (1971),
(1973), Lovaisz (1972), (1977), Chvatal (1973), (1975), Balinski and Hoffman (1978)).
A. The traveling salesman. Some important papers from the early history of the
traveling salesman problem are those of Eastman (1958), Dantzig et al. (1959), Little
et al. (1963), Lin (1965), Gomory (1966) and Shapiro (1966). A comprehensive survey
of work up to 1968 is provided by Bellmore and Nemhauser (1968). The present
situation is summarized below.
Though Gilmore and Gomory (1964), Systo (1973) and Garfinkel (1977) have found
good algorithms for some useful special cases of (5), developments of this sort are
limited by the fact that (5) is %LAP-hard
even for the special case in whichi G is
undirected and (a) each arc-length is 0 or 1, or (b) G is complete and its node-set lies
in the Euclidean plane, the length of each arc being the distance between its nodes.
The second result is due to Garey, Graham and Johnson (1976) and Papadimitriou
(1977). The first follows from 9LP-completeness of the undirected version of (5'),
which holds even for the special case in which G is a 3-valent 3-connected planar
graph (Garey, Johnson and Tarjan (1976)).
When a problem calls for minimization of a positive function f, an algorithm for the
problem is E-approximateif each "solution" output by the algorithm has f-value at
most 1 + E times the optimum value. Now suppose that VP=A%A and A. is (for some
e) an E-approximate algorithm for the undirected case of (5). Then Sahni and
Gonzalez (1976) show A. is not polynomially bounded, while Papadimitriou and
Steiglitz (1977) show that if A. is a local search algorithm of the sort described below,
even the search phase is not polynomially bounded.
When G is undirected the traveling salesman problem may be approached by the
local search methods of Lin (1965), Reiter and Sherman (1965), Karg and Thompson
(1964) as improved by Raymond (1969), Lin and Kernighan (1973) (generally regarded as the most effective), and many others. See Bellmore and Nemhauser (1968),
Garfinkel and Nemhauser (1972), Savage et al. (1976), and Papadimitriou and
10
VICTOR KLEE
Steiglitz (1977) for additional references. In these algorithms,one starts with an
arbitrarytour and then, by searchinga set of tourswhichare consideredits neighbors,
either finds a shortertour and uses it as a new startingpoint or, if no shortertour is
found in this way, terminates.(For example,the neighborsof a tour T might be all
tours T' obtainable from T by replacing three successive arcs {w,x)}, {x, y) and
{y,z} of T by the arcs {w,y), {y,x} and {x,z).) The resultingtour is locally
optimal relativeto the neighborhoodsystem underlyingthe searchprocedure.Since
the locally optimalsolutionsoften turnout to be globallyoptimalor close to that, and
since the computationtime is often modest, these methods are frequentlyused in
attackinglarge instancesof (5). Intrinisiclimitationsare indicatedby the theoremof
Papadimitriouand Steiglitz(1977) and by the fact that even if 9 = 9L6y any exact
(0-approximate)local searchmethod must involve very large neighborhoods(Savage
et al. (1976)). See Golden (1977) for statisticalanalysisof anotherheuristicapproach
to (5).
Christofides(1976) has an O(n3) -approximatealgorithmfor the special case of
(5) in which G is a completeundirectedgraphwhose arc-lengthssatisfy the triangle
inequality(a(i, j) < a(i,k) + a(k, j)). It firstfinds a minimumspanningtree and then
solves a matchingproblem.His analysisis sharpenedby Cornuejolsand Nemhauser
(1978).
When G is undirectedand one seeks a tour that is definitelyoptimal,the approach
of Held and Karp (1970), (1971) is called for. It combines a branch-and-bound
procedure(for surveysof such procedures,see Lawlerand Wood (1966),Agin (1966),
Mitten (1970), Geoffrion and Marsten(1972), Knuth (1975), and Garfinkel(1978))
with an ascent method that involves 1-trees, where these consist of a tree with
node-set {2, ..., n) togetherwith two arcs incidentto node 1. Since toursare simply
1-treesin which each node is of valence 2, a minimum1-treethat is a tour is in fact a
minimumtour.The ascent methodis based on the fact that minimum1-treesare easy
to find, and that certain transformationsof arc-lengthpreserveminimumtours but
may producenew minimum1-trees.The approachof Held and Karphas been refined
by Held et al. (1974),HelbigHansenand Krarup(1974),Smithand Thompson(1977),
and Yamamoto(1978).
For the traveling salesman problem (5), unlike the shortest path problem (1),
transition from the "symmetric"(undirected) case to the general "asymmetric"
(directed)case results in a significantreductionin the size of instancesthat can be
solved in a reasonableamountof time. For the generalcase, the best exact methodis
apparentlya modification,due to Smith et al. (1977), of earliermethodsof Eastman
(1958),Shapiro(1977),and Bellmoreand Malone(1968).They all rely on the fact that
there is a good algorithm for minimizingthe sum a(7) = =la(i, (r(i)) over all
permutationsS: N -- N (this is a form of the assignmentproblem),while (5) requires
minimizationof o(7r)over the cyclic permutations.
Some approximatealgorithmsfor (5) that have good average-timebehavior are
mentionedin ?6.
B. Shortestpaths. Shortestpath algorithmswere surveyedby Dreyfus(1969) and
Gilsinn and Witzgall(1973),and discussedin detail by Domschke(1972) and Lawler
(1976).In view of these studies,the more recentsurveysby Bradley(1975)and Lawler
(1978), and the bibliographiescollected by Pierce (1976) and Pape (1977), we make
only a few remarks here rather than attempting to summarizea large body of
material.
Problem(1) is usuallysolved as part of
(1*) For a given node s of G, find shortestpaths froms to all othernodes accessible
from s.
COMBINATORIALOPTIMIZATION: STATE OF THE ART
11
One might think (1) could be solved significantlyfaster than (1*). That is true for
some interestingspecial classes of graphs(Hadlock (1977), (1979)) but probablynot
for generalgraphs.
Several good algorithmsare available for (1*), the best-knownbeing the 0(n2)
procedureof Dijkstra (1959). By a simple use of priority queues, it can also be
implementedin time 0(alogn) (Lawler(1978)), an advantagefor sufficientlysparse
graphs. By a more complicateduse of priorityqueues, Johnson (1977) shows that
for each fixed positive integer k there is an implementationthat runs in time
O(min(n1+ 1/k + a, alogn)}). When all arc lengths are positive integers the algorithm
of Dial (1969) and Wagner(1976) solves (1*) in time 0(max {n,a,d)), whered is the
maximum of the lengths of shortest paths from s to other nodes. For exploiting
limitationson the arc lengths and on the combinatorialstructureof G, the sharpest
methodsappearto be those of Denardoand Fox (1979).
ApplyingDijkstra'sproceduren times yields an 0(n3) solutionof
(1**) For each pair (s,t) of nodes of G, find a shortestpath from s to t.
The clever but complicated approach of Fredman (1976) shows that with
O(n3loglogn/logn) preprocessingan 0(n5/2) solution of (1**) is available. The
preprocessingconsists of compiling a table that is then used for all instances of a
given size n.
The most elegant solution of (1**) is the 0(n3) algorithmof Floyd (1962). Extend
the definition of arc-length by setting a(i, j) = oo when (i, j) is not an arc, and then
proceedas follows:
begin
for k *- 1 until n do
for i - 1 until n do
- 1 until n do
for j
a(i,j) <-min (a(i,j),a(i,k)
+ a(k, j)
end.
When this computationterminates,a(i,i) is the length of a shortestcircuit through
node i, and when i =j a(i, j) is the length of a shortest path from i to j. From this
informationit is easy to find the paths themselves.
In contrast to the shortest-pathalgorithmsmentioned earlier, Floyd's algorithm
does not requirenonnegativearc lengths.It appliesto an arbitrarydirectednetworkin
which no circuitis of negativelength. A relatedalgorithmof Yuval (1976),using the
fast multiplicationof matricesdue to Strassen(1969)and based on an extensionof the
RAM model that permits infinite-precisionreal arithmetic, solves (1**) in time
0(n2'81).Klee and Larman (1979) extend Floyd's method to find shortest paths
among those satisfyingvarioussorts of restrictions.For networksin which no circuit
is of negative length, Johnson (1977), Lawler(1978), and Denardo and Fox (1979)
show how to take advantageof sparsenessin the directedcase, and Lawler(1976)
shows how to use matchingalgorithmsto find shortestpaths in the undirectedcase.
See ?6 for reference to a solution of (1**) whose average-casecomplexity is
O(n21og2n).
C. Spanning trees and optimization in independencesystems. Let us turn now to
the minimum spanning tree problem (2). For a connected undirectedgraph G =
(N,A) and nonnegativearc-lengthfunction a, we seek a spanningtree of minimum
if it contains no circuit.And for x E X C A,
weight. Call a subset of A independent
say that x is a first memberof X if a(x) < a(x') for all x' E X. Kruskal(1956)showed
12
VICTOR KLEE
a minimumspanning tree T can be constructedas follows, where U is the set of
unexploredarcs.
begin
T-Q0; Uk-A;
while U 4 0 do begin
u <- a first member of U;
U<-U-
{u};
if T U {u} is independent then T<- T U f u
end;
output T
end.
Kruskal'salgorithmcan be implementedin time 0(alogn). A later algorithmof
Prim (1957) and Dijkstra (1959) solves (2) in time 0(n2) or, as implementedby
Kerschenbaumand Van Slyke (1972), in time 0(alogn). For sparsegraphsthe best
algorithmsare those of Yao (1975) and Cheriton and Tarjan (1976), which are
0(aloglogn). On the other hand, for each positive integerk there are algorithmsof
Johnson(1975)and Cheritonand Tarjan(1976)which,when appliedto graphshaving
a > cn' + /k for some fixed c > 0, solve (2) in time 0(a).
Actually, Kruskal'sprocedurefinds minimumspanningtrees without any restriction on the sign of a, so the same procedurecan be used to find maximumspanning
trees;simplyredefinefirst memberto mean of maximuma-value. The name of greedy
algorithm,proposed by Edmonds (1971), is then especially appropriate,for one
attemptsat each stage to swallow(add to T) the largestmemberu of U, refraining
only if u is immediatelyunpalatable(destroysindependence).No attentionis paid to
the possibilitythat this greedy choice, howevertemptingat the moment,may in the
long run cause indigestion(suboptimality)!And, in fact, the greedy algorithmnot
only solves (2) but worksin the more generalmatroidsettingdescribedbelow.
The theory of matroids,which can be axiomatizedin many equivalentways, was
inventedby Whitney(1933) as a generalizationof the theoryof linearindependence.
Let us define an independencesystem as a finite family 9 of finite sets, called
sets, such that every subset of an independentset is independent.(In the
independent
exampleabove, a set of arcs is independentif it containsno circuit.)A matroidis an
independencesystem 9 such that for each subset S of Ug, all independentsubsetsof
S are of the same cardinality.An equivalentcondition is that wheneverI and J are
independentsets with {I} < IJI,there existsj E J - I such that I U {j) is independent. A base of a matroid9 is a maximalmemberof l. (In the example,the bases are
the spanningtrees of G.) Lawler(1976) lists several books and other referenceson
matroid theory, the latest book being that of Welsh (1976). Bruno and Weinberg
(1976a,b) presentmatroidtheoryas the properfoundationfor the study of electrical
networks,an importantadvantagebeing that the well-developedduality theory of
matroidscan replacethe purelygraph-theoreticdualitynotions whichare satisfactory
only for planargraphs.
If 9 is a matroid,< is a linearorderingof U5, and "first"is interpretedin termsof
<, then with A replacedby U. the above algorithmproducesa base T of 9 that is
optimalin the followingstrongsense: for any other base T' of l there is a bijection
p: T-> T' with t < ?(t) for all t E T. This was establishedby Rado (1957), rediscovered independentlyby Gale (1968),Welsh (1968) and Edmonds(1971),and extended
to infinitematroidsby Klee (1971).In additionto the spanningtree problem,Lawler
COMBINATORIALOPTIMIZATION: STATE OF THE ART
13
(1976) describes a "semimatching" problem, a job-sequencing problem, and an
experimental design problem that have underlying matroid structures and hence can
be solved efficiently by the greedy algorithm. Of course, the complexity of an
implementation depends on the difficulty of recognizing members of 1.
Each independence system g can be expressed in various ways as the intersection of
a finite number of matroids. Many important problems of combinatorial optimization
can be cast in the following form: For an independence system g that is the
intersection of matroids l, . . ., k of specified sorts, and for a nonnegative function
a on U4, find a member of 4 that is of maximum a-weight-that is, maximize
ZiEla(i) over all I E 1. Lawler (1976, chapter 8) observes that the traveling salesman
problem (5) and the 3-dimensional assignment problem (4"') can be formulated in this
way for k = 3, so good exact algorithms should not be expected for general threematroid optimization problems. However, for k = 2 Lawler (1975), (1976) has both a
"primal" algorithm analogous to a procedure for finding minimum cost network flows
and a "primal-dual" algorithm analogous to a procedure for finding maximum
matchings. The directed analogue of the minimum spanning tree problem (2) can be
handled this way, but is solved more efficiently by the special methods of Chu and
Liu (1965) and Edmonds (1967). See also Lawler (1976) and his references, and Tarjan
(1977).
When 4, k and a are as in the preceding paragraph, and "first" means of maximum
a-value, the greedy procedure described earlier can be used as a heuristic for finding
independent sets of large weight. Results of Jenkyns (1976) and of Korte and
Hausmann (1978) show that if I0 is a member of 5 of maximum a-weight and Ig is the
output of the greedy heuristic as applied to 9 and a, then
lr(S)
> min
scu
ur(S)
iEioaO(i)
iE.e/i (i)
1
> -,
k
where lr(S) <resp. ur(S)> is the minimum <resp. maximum> of the cardinalities of the
independent subsets of S. Both inequalities are sharp in certain senses. See Korte
(1979) for a fuller description of "greedy" heuristics, their uses and limitations, and
for additional references.
6. Three recent advances. As is clear from the recent dates of many of the cited
references, the field of combinatorial optimization is in a very active stage of
development. Three of the most exciting recent advances have been improvements in
Dinic's maximum flow algorithm, Minty's good algorithm for finding maximum
independent sets in a large class of graphs, and a collection of algorithms whose
average-case behavior is remarkably good.
A. Maximum flow algorithms. Like other maximum flow algorithms, the algorithm of Dinic (1970) starts with a feasible flow 4, uses 4 and the underlying network
to produce an auxiliary network, and by a computation in the latter attempts to find
an improvement 4' of 4. If an improvement q' is found, the computation is repeated
with q' in the role of 4, and when no improvement is found then 4> is optimal.
However, in Dinic's approach at most n improvements are necessary because each
improvement is made in a "global" manner rather than being based on a single
flow-augmenting path. Karsanov (1974) shows how to carry out each improvement in
time 0(n2), and the same is accomplished much more simply by Malhotra, Kumar
and Maheshwari (1978). The overall computation time of 0(n3) is the best known for
finding maximum flows in dense networks. For sparse networks, Cherkasky (1977) has
an 0(n2a 1/2) implementation of Dinic's method, improved to O(n5/3a2/3) by Galil
(1978) and to 0(nalog2a) by Galil and Naamad (1979) and Shiloach (1979?).
14
VICTOR KLEE
B. Minty's algorithmfor maximum independentsets. The maximum independent
set problem (4") is ?%V-complete, even when restricted to 3-valent planar graphs
(Garey, Johnson and Stockmeyer (1976), Garey and Johnson (1977), Maier and Storer
(1977)) but one may hope for good algorithms that solve the problem for special
classes of graphs. For references to such algorithms, and to other restrictions under
which the problem remains %96?-complete,see Garey and Johnson (1979). In the
bipartite case, (4") is well-solved by matching (Lawler (1976, pp. 189-196)). The most
successful attack on the nonbipartite case is due to Minty (1979?), who shows that
matching techniques are also applicable to all graphs that are claw-free. A claw is a
quadruple of nodes (v, w, x, y) such that each node in the set S = w, x, y) is adjacent
to v but no two members of S are adjacent.
To appreciate how close Minty's result comes to the boundary of %9L6-completeness, consider the restricted form of the 3-dimensional assignment problem
mentioned at the end of ?4. Let G be the graph whose nodes are the ordered triples
(i, j,k) of integers between 1 and n for which ajk = 1, two nodes being adjacent if
they agree in at least one coordinate. Then G may have claws but it is free of
quintuples (v, w, x, y, z) such that each node in the set S = { w, x, y, z) is adjacent to v
but no two members of S are adjacent. The problem of deciding, for each such G,
whether G contains an independent set of cardinality n, is %L?-complete.
Minty also considers the weighted version of (4"), in which a node-weighting a is
given and one seeks an independent set of maximum weight. He shows that for
claw-free graphs this problem is reducible to the weighted matching problem and
hence solvable in polynomial time.
C. Some algorithms with good average-case behavior. From the worst-case viewpoint, some well-solved problems of combinatorial optimization have been so intensively studied that further significant improvement in usable solution methods seems
unlikely; also, most researchers doubt the existence of good algorithms for the
96 -complete problems. However, neither of these comments applies to the averagecase viewpoint. This fact, along with the belief that average-case behavior is more
important than worst-case behavior in many applications, has motivated the search
for algorithms of good average-case behavior.
Spira (1973) and Karp (1978) consider a complete directed network ((N,A),a)
whose arc lengths a(i,j) are all drawn independently from the same probability
distribution in ]0, oo[. Spira proposes an algorithm for the all-pairs shortest path
problem (1**) and shows that for any distribution the expected running time is
O(n21log2n). Fredman (1976) shows how to reduce the expected number of comparisons (but not the overall complexity) to O(n2logn). Carson and Law (1977) report on
computations comparing Spira's method for various distributions with an improvement of the method of Dijkstra (1959) due to Yen (1972). See also Noshita, Masuda
and Machida (1978).
Karp (1978) proposes an O(n3) approximate algorithm for the traveling salesman
problem (5). His algorithm first solves the n x n assignment problem for the matrix
(a(i, j)) and then patches together the cycles of the resulting permutation to form a
tour. He shows that if the a(i, j) are drawn independently from the uniform distribution on ]O,1[ then, with probability tending to 1 as n - oo, the ratio of the length of
the tour produced by the algorithm to that of an optimum tour is < 1 + E(n), where
E(n)-0 as n - oo. A tool in his proof is a result of Walkup (1979) on the expected
value of a random assignment problem. Similar patching schemes appear in algorithms proposed by Karp (1977) for the traveling salesman problem in the Euclidean
plane.
In the results of Spira and Karp stated above, the underlying graph is always
COMBINATORIALOPTIMIZATION:STATE OF THE ART
15
complete;only the arc lengthsvary at random.For other resultson the average-case
behavior of algorithms,a notion of a random undirectedgraph that has node-set
N = (1,... n) and has a specifieddensityp (or has a specifiednumberE of arcs) is
required.Let U, denote the set of all n(n - 1)/2 unorderedpairs of distinctintegers
between 1 and n, each pair representinga possiblearc of a graphwith node-setN. In
the "p" model, a sample graph is formed by including each member of Un with
probabilityp, independentof what other arcs are chosen. In the "E" model, the
samplespaceconsistsof all A C Unfor which IAI = E, all such arc-setsA being drawn
with equal probabilities.When E = pn(n - 1)/2, the two models are closely related
(Erdosand Renyi, (1961));Angluinand Valiant(1977) show that if in a certainsense
an algorithmworksfor one model then it worksfor the other one also.
A matchingis perfectif it covers all nodes. Erdos and Renyi (1966) show that if
l
+ o(n)n and if PM(n) is the probability that a random graph with n
E(n) = nlogn
nodes and E(n) arcs admits a perfect matching, then as (even) n - oo, PM(n) - 1 if
w(n)- oo. Using a model of computation similar to but more flexible than the RAM
model, Angluin and Valiant (1977) show there exists an algorithm that has the
followingproperties:
(a) acceptingas input an undirectedgraph G with n nodes, it outputs a perfect
matchingor concludescorrectlythat G does not admit one;
(b) there exist positive constants c and k such that for each n and for each
E > cnlogn, the expected run-timeis less than knlogn for random graphs with n
nodes and E arcs.
P6sa (1976) shows that if E(n) = fn logn with ,B > 1, and if T(n) is the probability
that a random graph with n nodes and E(n) arcs admits a tour, then T(n)-> 1 as
n -> oo. Korsunov(1976) has a still sharperresult.Karp (1976) uses Posa's argument
as the basis of a polynomialtime algorithmfor finding toursalmost surely.
Consideringspanningsimple paths as well as tours, Angluin and Valiant (1977)
describean algorithmthat has the followingproperties:
(a) acceptingas input an undirectedgraphG with n nodes and a pair of nodes s, t
with 1 < s < t < n, it attemptsto find a simplepath froms to t that uses all nodes (a
tour when s = t);
(b) for each y > 0 there exist positive constantsc and k such that for each n, for
each E > cn log n, and for each pair s,t, the probability is at least 1 - O(n-~) (for
randomgraphswith n nodes and E arcs) that the desiredpath is found in fewer than
knlogn steps.
Essentiallythe same results are obtained by Angluin and Valiant (1977) for the
problem of finding a tour or a spanning simple path in a directed graph. Their
complexity estimates are based on a model of computationin which randomized
decisionsare possible,but are worsenedonly by an additionalfactor of logn when
that possibilityis removedfrom the model.
See Karp (1976) for the results of probabilisticanalyses of other combinatorial
searchalgorithms,includinguse of the greedyalgorithmto find largeindependentsets
of nodes. And see Evans (1976) for a treatmentof maximum flow problems in
probabilisticgraphs.Also, see the last two paragraphsof ?7.
7. Three directionsfor research. Since the field of combinatorialoptimizationis
developing so rapidly, it provides many directionsfor researchand many specific
problemsthat beg for solution.Ratherthan discussingindividualproblems,we focus
on the three directionsof researchthat seem respectivelyto be most promisingfor:
(a) elucidationof the fundamentaltheoreticalissues;
(b) unificationof the existingtheory;
(c) productionof useful or interestingalgorithms.
16
VICTOR KLEE
A. Finding lower bounds. The goal here is simply stated: Find sharp lower
bounds for the computationalcomplexityof the importantproblemsof combinatorial
optimization,preferablywith respect to the RAM or a closely related model of
randomaccess computation.When a problemdeals with networks((N,A), a), we say
the problemis of complexityQ2((n,a)) if there is a constant c > 0 such that every
algorithmfor solving the problemrequiresat least cr(n,a) steps for worst-caseinput
with parametersn and a. This lower bound is sharpif the problemcan be solved by
an algorithmof complexityO(r(n,a)), so that the worst-casebehaviorof the problem
is determinedto within a multiplicativeconstant.
At least with respect to the RAM model, this directionof researchappearsto be
very difficult.In particular,it involves decidingwhether6y =%9P. Nevertheless,it is
describedas "promising"becauseit deals with such importantquestions.Until sharp
lowerboundsare found for a varietyof problemsof combinatorialoptimization,there
will be no clearunderstandingof why some problemsare easy and otherssuperficially
resemblingthem are very hard. An upper bound on the intrinsiccomplexityof a
problemcan be establishedby designingand analyzingan algorithmfor its solution.
If the design involvessome ingenuityand the analysisis done with care, the resultis
likely to be interestingand perhapseven useful. At the very least, it is likely to be
"nontrivial"in an intuitivesense. Nontriviallower bounds are much harderto come
by, becausethey requireconsiderationof all conceivablealgorithmsand thus demand
a muchclearerunderstandingof the underlyinglogicalissues.In fact, as Weide(1977)
remarks,triviallowerboundsare often the only ones available.Consider,for example,
the problemof finding a longest simplepath between two given nodes of a complete
undirectednetworkwith n nodes. The problemis 6LP-hardand hence probablydoes
O
not admit a good algorithm-not even one of complexity
(nlI??? ). However,the
best known lowerbound (with respectto the RAM model) is the trivialone of
S2(n2),
obtained by noting that none of the n(n - 1)/2 arc lengths can be ignoredby any
algorithmthat solves the problemfor all sets of input data.
Weide (1977) discusses a few methods that have been used to establish lower
bounds,with respectto variousmodels of computation,on the complexityof combinatorialproblems.The interestedreadershould consult Kerr (1970), Spira and Pan
(1975), and Graham,Yao and Yao (1979?)concerningshortestpaths, Spiraand Pan
(1975)and Shamosand Hoey (1975)concerningminimumspanningtrees,Harperand
Savage(1972) concerningmaximummatching,Harperand Savage(1972)concerning
shortest tours, Holt and Reingold (1972) concerning the detection of circuits or
connectedness,and Rivest and Vuillemin (1975) concerning the detection of an
arbitrarynontrivialmonotonegraphproperty(a propertypossessedby some but not
all graphsand preservedby additionof arcs).
DeMillo and Lipton (1979) provide a partial explanation of the difficulty of
showingthat certainproblemsare not solvablein polynomialtime. For establishing
nontriviallowerboundson the complexityof problemsof combinatorialoptimization,
the most promisinggeneralapproachesinvolvethe geometryof convexpolytopes.See
especiallyRabin (1972),Dobkin and Lipton(1978),and Yao and Rivest(1979?);also
Spira(1972), Spiraand Pan (1974), Klee (1975),and Yao (1976).
B.
Optimization in oriented matroids. The theory of matroids is a combinatorial
generalizationof the theoryof lineardependencein a vectorspace over a field, and of
the theory of linear equalities. Similarly,the theory of oriented matroids may be
viewed as a combinatorialgeneralizationof the theory of positive linear dependence
in a vector space over an orderedfield, and of the theoryof linear inequalities.The
notion of an oriented matroid developed from the digraphoidsof Minty (1966)
through the generalizationsof Camion (1968), Fulkerson (1968) and Rockafellar
COMBINATORIALOPTIMIZATION: STATE OF THE ART
17
(1969), to the present axiomaticversions of Bland (1974), Las Vergnas(1975), and
Folkman and Lawrence(1978). A good basic referenceis Bland and Las Vergnas
(1978).
Realizinga possibilityfirst suggestedby Rockafellar(1969), Bland (1974), (1976)
and Lawrence(1975) show the basic parts of linear programmingtheory can be
extended to oriented matroids.Both have oriented matroid formulationsof linear
programmingduality, and Bland (1976), (1977) also has a finite pivoting method
which specializes,in the context of linearprogramming,to Dantzig'ssimplexmethod
with a new pivot selection rule. The study of optimization in the frameworkof
orientedmatroidsseems to offer the best chance for the unificationof large parts of
linear and combinatorialoptimization.This is related to the approach of Lawler
(1976, chapter 9) through matroids with parity conditions, and it is important to
clarify the relationshipsbetween the two approaches.A deep contributionto the
matroidparityproblemis that of Lovasz(1979?).
C. Developmentof algorithms. A strikingaspect of combinatorialmathematics,
and of combinatorialoptimizationin particular,is its endless supply of interesting
problems. To become convinced that the design and analysis of algorithms for
combinatorialoptimizationwill continue for a long time, one need only glance at a
few issues of Annals of Discrete Mathematics, Information Processing Letters, Journal
of the Associationfor ComputingMachinery, Mathematical Programming,Mathematical
Programming Studies, Networks, Operations Research, Proceedings of A CM Symposia
on the Theory of Computing, Proceedings of IEEE Symposia on the Foundations of
Computer Science, or the SIAM Journal on Computing. Below are described some
directionsof algorithmdevelopmentthat seem at presentto be especiallypromising.
Exploring the boundary of 69L9-completeness. By following the advice in the
excellentchapterof Gareyand Johnson(1979)on using 9L6-completenessto analyze
problems,designersof algorithmscan help to "explorethe boundary"of LgeC, the
class of all 9L9-completeproblems.The generalidea is that, having establishedthe
membershipof a problemin one of the classes 9 and 9L9C, one should not rest
before finding a closely relatedproblemin the other class. For each of the problems
(1)-(4), Gareyand Johnson(1979)describeseveralrelativesin L9C, and for (5) there
are relativesin VP.The readerhas met some of these relativesearlier,and a few more
are describedbelow.
The Chinesepostmanis a relativeof the travelingsalesman.In servinghis route he
must traverseeach street at least once, one-way streetsonly in the properdirection,
and returnto his startingpoint. And of course he wants to minimize the distance
traveled.The postman'sproblemis in ? if all streetsare directedor all are undirected
(Edmondsand Johnson(1973)),but the mixed case is 9L9-complete(Papadimitriou
(1976)).
When G = (N,A) and H = (V, E) are undirected graphs, a perfect H-packing of G is
a partitionof N into pairwisedisjointsubsetseach of which is of cardinalityIVI and
induces in G a subgraphisomorphicto H. For fixed H, the problem of deciding
whetherG admitsa perfectH-packingbelongs to 6? if IVI < 2 (use matching)but is
9L6-completewheneverIVI > 3 (Kirkpatrickand Hell (1978)).
Let '9DL = 9L9 -((9 U 9L9 C), the class of all problems in 9L6 that are of
"intermediate"difficulty in the sense that, even though they can't be solved by
polymomiallybounded algorithms,they're at least not so difficult as to be L9?complete. Under the assumption that 9 gL96, Ladner (1975) proves 6X9 is
nonemptybut no one has produceda memberof 919 4 that is of practicalimportance
18
VICTORKLEE
IS. Linearprogramming(?6@)
or intrinsicinterest aside from its membershipin L9
was a prime candidate,but has recentlybeen shown by Khachiyan(1979) to be in 6.
With respect to "exploringthe boundary,"multicommodityflow problemsare of
special interest.For 2-commodityflows in undirectednetworks,Itai (1978) improves
the procedureof Hu (1963) to obtain a maximumflow in time O(n3). But in shifting
to the directed case of maximum multicommodityflows, or asking for maximum
integerflows, or imposinglower and upperboundson the arc flows and askingabout
feasibility,other degreesof complexityare encountered.In the table below, adapted
from Itai (1978), ?6ymeansthe problemis polynomiallyequivalentto linearprogramming.
Problem
1-commodity
2-commodity
Maximumflow
Undirected
Directed
Maximumintegerflow
Undirectedor directed
Feasibleflow withlower
and upperbounds
Undirected
Directed
O(n3)
O(n3)
O(n3)
te
O(n3)
66e
e
%6P
O(n3)
e
?(
For m-commodityflows with m > 3, the situation is the same as for 2-commodity
flows, except that the complexity of the m-commodityundirectedmaximum flow
problemis unsettled.
Approximatealgorithms. We saw in the result of Sahni and Gonzalez(1976) that
unless 9?= %LP,the generalundirectedtravelingsalemanproblem(5) does not admit
a polynomiallybounded algorithmthat is E-approximate,
no matter how large E is
chosen. On the other hand, when G's arc lengths satisfy the triangleinequality,the
0(n3) method of Christofides(1976) is -approximateand it may even turn out that
for each c > 0 this special case of (5) admits a polynomiallyboundedc-approximate
algorithm.In short, though the %69-completeproblemsare computationallyequivalent so far as solvabilityby polynomiallyboundedexact algorithmsis concerned,this
does not apply to approximatealgorithms.Much is known about approximate
algorithmsbut much moreremainsto be discovered.Good sourcesof informationare
the annotatedbibliographyof Garey and Johnson(1976),the surveyarticlesof Sahni
and Horowitz(1978)and of Korte(1979),and the book of Gareyand Johnson(1979).
Average-case and probabilistic analysis of algorithms. In Part C of ?6, three sorts of
algorithmsare lumped together as "algorithmswith good average-casebehavior."
Some are precisely that, while others are with high probabilitye(n)-approximate,
where E(n)-> 0 as n -> oo. A period of considerable activity in the average-case and
probabilisticanalysisof algorithmsfor combinatorialoptimizationis now underway.
Four basic papersare those of Karp(1976),DeWitt (1977),Weide(1978)and Lueker
(1979).
The papers on random graphs by Erdos and Renyi do not treat algorithms
explicitly,but they containmany facts that are essentialfor the designerof algorithms
in this area; severalof these papersare collectedin Erdos(1973).Otherrecentpapers
that are not algorithmicin nature,but may be especiallyuseful in analyzingalgorithms,are those of Grimmettand McDiarimid(1975),Matula(1976),Walkup(1977),
Cohen et al. (1979),and Klee and Larman(1979).
STATEOF THEART
OPTIMIZATION:
COMBINATORIAL
19
8. What are the best sources of further information? As is clear from our frequent
reference to them, the two indispensablebooks for the study of combinatorial
optimizationare those of Lawler(1976)for problemsin P and of Gareyand Johnson
(1979)for problemsin A C6.But they cannot cover everythingin detail,and many of
the referencescited by them or in our list of referencesare also necessary for a
thoroughunderstandingof the field.
We close by mentioningtwo very active areasof combinatorialoptimizationwhose
omission here causes regret. Though both are very importantfor the solution of
practical problems, they have been omitted because of space considerationsand
because of our decision to base the entireexpositionon problems(I)-(5). For access
to the tremendousvolume of materialon Scheduling(several books, hundredsof
researchpapers), a good startingpoint is the special issue of OperationsResearch
devoted to the subject.It is listed here underFlorian(1978).
For discrete location problems, see Krarup and Pruzan (1979) and their references.
Acknowledgements.Preparationof this reportwas supportedin part by the Office
of Naval Research.Helpful commentswere suppliedby M. Garey, A. Mandel, G.
Minty, and two referees.An earlierversion appearsin InformationLinkagebetween
AppliedMathematicsand Industry,AcademicPress,New York, 1979.
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