1. No category

The Traveling Salesman Problem
in Theory & Practice
Lecture 8: Lin-Kernighan and Beyond
11 March 2014
David S. Johnson
[email protected]
http://davidsjohnson.net
Seeley Mudd 523, Tuesdays and Fridays
Outline
1. k-Opt, k > 3
2. Lin-Kernighan
3. and Beyond….
4. A quick tour of heuristics for the Asymmetric TSP
4-Opt
• Seems to be relatively straightforward to generalize from 3-opt.
– Just need to now consider possibilities for t7 and t8.
– Can further exploit the Partial Sum Theorem.
– Can consider all possibilities for t5, but need to make sure that t7
breaks up the short cycle (or create a new one).
An alternative View: Maintain the path
t1
t3
t4
t2
t4
An alternative View: Maintain the path
t1
t3
t2
t4
An alternative View: Maintain the path, at least eventually…
t1
t4
t3
t5
t6
t2
An alternative View: Maintain the path, at least eventually…
t1
t4
t35
t53
t26
t62
Double Bridge Move
Cannot be produced by any sequential move.
Best one can be found in O(N2) time.
Best Double Bridge Move in Time O(N2)
[Glover, “Finding a best traveling salesman 4-opt move in the same
time as a best 2-opt move,” J. Heuristics 2 (1996), 169-179]
Simpler algorithm due to Johnson & McGeoch, 2002
Normal Form for Double Bridge Move
j
j+1
p+1
p
Smallest index of
the first endpoint
of a deleted edge
q
qp+1
i+1
i
2 1 N N-1
C[h,k] = Cost of move that deletes {h,h+1} and {k,k+1} and adds {h,k+1} and {h+1,k}
The Functions to be Computed
• f(p,j) = min {C[p,q] : j < q ≤ N}, 2 ≤ p < j ≤ N.
• g(i,j) = C[i,j] + min {f[p,j] : i < p < j}, 1 ≤ i < j-1 ≤ N-1.
j
Theorem: The length of the
best 2-bridge move is
min {g(i,j): 1 ≤ i < j-1 ≤ N-2.
j+1
p+1
p
Question: How is this O(N2)?
There are
values of f(p,j)
and g(i,j) to compute, and the
average ones appears to take
time θ(N) to compute.
q
θ(N2)
q+1
i+1
i
2 1 N N-1
The Functions to be Computed
• f(p,j) = min {C[p,q] : j < q ≤ N}, 2 ≤ p < j ≤ N.
• g(i,j) = C[i,j] + min {f[p,j] : i < p < j}, 1 ≤ i < j-1 ≤ N-1.
j
Observation: The only
difference between f(p,j)
and f(p,j-1) is the inclusion
of C[p,j] in the minimization.
p+1
p
The only difference
between g(i,j) and g(i-1,j) is
the inclusion of f(i,j) in the
minimization.
So the values of each can be
computed in constant time
per value.
j+1
q
q+1
i+1
i
2 1 N N-1
4-Opt Conclusions
• Can be implemented to run in O(N2) per iteration.
• This is likely to be significantly slower than our 3-opt
implementation.
• k-opt for k > 4 will be even worse:
– Intermediate solutions can involve a path plus multiple cycles.
– More possibilities for non-sequential moves that must be considered as
special cases (triple-bridge moves, etc.),
• Improvement in tours over what 3-opt can produce may not be
worth the effort (Shen Lin, private communication).
• Alternative approach: a highly-pruned N-opt approach:
The Lin-Kernighan Algorithm
The Lin-Kernighan Algorithm
[Shen Lin & Brian Kernighan, Bell Labs, 1971]
• Built on top of 3-opt, augmented to allow choices of t5 that
create a short cycle, so long as there are legal choices of t6, t7,
and t8 that pull things together again.
• After each choice of t6 (or t8 in in the above special case), we
invoke a deep-dive “LK-search,” assuming the gain criterion from
the Partial Sum Theorem is met.
The source of what follows is [Lin & Kernighan, “An effective heuristic
algorithm for the traveling salesman problem,” Operations Research 21
(1973), 498-516] and the original Lin-Kernighan FORTRAN code.
LK Search
t1
t2i+1 t2i+2
t2i
Candidates for t2i+1 are the cities on the neighbor list for
t2i such that the length of the resulting “one-tree” is less
than the length of the shortest tour seen so far.
[Note that this is the same criterion as provided by the
Partial Sum Theorem.]
We abandon the choice if the edge to be deleted, {t2i+1,t2i+2},
was added to the tour earlier in this search (is not an original
tour edge). This limits the depth of the search to N.
t2i+2
LK Search
t1
t2i+1 t2i
t2i+2
If adding the
Candidates
foredge
t2i+1 {t
are
the} cities
to thison
path
theyields
neighbor
a new
listchampion
for
tour,
1,t2i+2
t2i such
save
that
that
tour.
the length of the resulting “one-tree” is less
than the length of the shortest tour seen so far.
As
thethat
starting
thecriterion
next level
the search,
take the
[Note
this path
is thefor
same
asof
provided
by the
shortest
path
generated by any of the the valid choices for t2i+1.
Partial Sum
Theorem.]
We
the choice
if are
edgeactually
to be deleted,
{t2i+1and
,t2i+2then
}, undone
The abandon
flips illustrated
here
performed,
was
added to
the tourreturning
earlier (isusnot
tour
as needed,
eventually
to an
ouroriginal
starting
point (or the
edge).
This
limits
the depth of the search to N.
improved
tour,
if found).
Differences between Algorithm Tested
and Lin-Kernighan, as described
•
We use pre-computed neighbor lists (k = 20). LK did not similarly restrict t3, t5, t7,
although their code restricted LK search to the 5 nearest neighbors.
•
We use don’t-look-bits and queue order. LK cycled through t1 candidates in input
order.
•
We use randomized Greedy starting tours, whereas LK used random starting tours.
•
Lin-Kernighan not only forbids the deletion of a previously-added tour edge. It also
forbids addition of a previously deleted edge. We allow this latter possibility.
•
Lin-Kernighan also added a search for an improving double-bridge move (one that
does not deleted added edges), used only whenever no further standard improving
moves could be found. They used an exhaustive search, but even our O(N2)
algorithm is unlikely to scale well, so we omit this step.
•
Lin-Kernighan used Array tour representation, while we switch to 2-level trees for
N > 1,000.
•
Lin-Kernighan coded in FORTRAN and ran on memory-constrained machines. The
largest instance they could test had 318 cities and was considered “big” in 1971.
We used C and modern machines, and go considerably bigger.
Show Movie
(Time to find the next improving move or determine there is none.)
•
O(N) choices for t1.
•
Up to 2 choices for t2.
•
O(k) choices for t3, each potentially involving a between + a flip.
•
Up to 2 choices for t4.
•
O(k) choices for t5, each potentially involving a between + a flip.
•
Up to 2 choices for t6.
•
O(k) choices for t7, each potentially involving a between + a flip.
•
Up to 2 choices for t8.
•
O(N) levels of LK-search.
•
O(k) choices for t2i+1, each potentially involving a flip.
In practice much smaller because of don’t-look-bits.
Typically much smaller.
Total = O(N2k4logN) using splay trees.
Further restricted by the locking of edges
In practice much smaller, since tour edges usually go to near neighbors
Worst-Case Running Time per Iteration
How Many Iterations?
•
In practice, typically θ(N) on random Euclidean instances for all of 2-opt, 3opt, and Lin-Kernighan.
•
In theory, what?
•
Theorem [Englert, Röglin, & Vöcking, “Worst case and probabilistic analysis of the 2-opt algorithm
for the TSP,” Algorithmica 68 (2014), 190-264]: For any Lp metric, 1 ≤ p ≤ ∞, and any N ≥
1, there exists a set of 16N points in the unit square for which 2-opt can take
as many as 2N+4 -22 steps.
•
One can get similar exponential lower bounds for k-opt, k > 2, if one considers
instances not obeying the triangle inequality. [Chandra, Karloff, & Tovey, “New results
on the old k-opt algorithm for the TSP,” SIAM J. Comput., 28 (1999), 1998–2029].
•
Note: This is doubly worst-case – not only must one be unfortunate enough to
get the bad instance, one must also be unfortunate enough to pick the bad
sequence of moves. What about average case, at least on the instances?
•
For random Euclidean instances, one of these levels of worst-case can be
removed.
Iterations for Random Euclidean
Instances
Theorem [Kern, 1989]: With high probability, maximum length of a
sequence of improving 2-opt moves is O(N16).
Theorem [Chandra, Karloff, & Tovey, 1999]: The expected length of
a maximum sequence of 2-opt moves is O(N10logN) and O(N6logN)
for the Manhattan (L1) metric.
Theorem [Englert, Röglin, & Vöcking, 2014]: The expected length of
a maximum sequence of 2-opt moves is O(N4+1/3logN) and O(N3.5logN)
for the Manhattan metric.
(This latter result extends parametrically to higher dimensions and
a variety of other point distributions.)
Removing the Move Sequence from the Picture:
PLS-completeness
[Johnson, Papadimitriou, & Yannakakis, “How easy is local search?” J. Comp. Syst. Sci. 37 (1988), 79-100]
Definition: A local search problem L in PLS (polynomial-time local search) consists of
a) A type TL ε {min, max}.
b) A polynomial-time recognizable set DL of instances.
c) For each instance x 𝜖 DL, a set FL(x) of solutions that is recognizable in time
polynomial in |x|.
d) For each solution s 𝜖 FL(x),
1) a non-negative integer cost cL(s,x), and a
2) a subset N(s,x) ⊆ FL(x), called the neighborhood of x.
e) Three polynomial-time algorithms:
1)
AL, which, given x 𝜖 DL, produces a standard (starting) solution AL(x) 𝜖 FL(x).
2)
BL, which, given x 𝜖 DL and s 𝜖 FL(x), computes cL(s,x).
3)
CL, which, given x 𝜖 DL and s 𝜖 FL(x), either returns a solution s’ 𝜖 N(s,x) with a
better cost (e,g., cL(s’,x) < cL(s,x) if TL = min), or reports truthfully that no such
solution exists, and hence s is locally optimal.
Key Computational Question
About a Problem L in PLS:
Is there an algorithm that, given an instance x 𝜖 DL, finds a locally optimal
solution s 𝜖 FL(x) in time polynomial in |x|?
•
True if all sequences of improving moves are polynomially bounded, as they
would be, for instance, if all costs are polynomially bounded in |x|.
Examples include
– Vertex Cover
– Max Clique
– TSP when all edge lengths are less than some constant
(as in the case of our random Euclidean instance generator, where all
coordinates are in [0, 107] and distances are rounded to the nearest integer)
•
Also may be true if there is a better heuristic for choosing the next move
than the one given by CL(s,x).
•
Or if there is some way of finding a locally optimal solution without using a
local search algorithm at all.
– As when L is corresponds to the Simplex neighborhood for Linear Programming.
PLS-Reductions
A reduction from PLS problem L to PLS problem K consists of two polynomialtime computable functions:
1)
f, which maps instances of x 𝜖 DL to instances f(x) 𝜖 DK and
2) g, which maps pairs (x 𝜖 DL, s 𝜖 FK(f(x))) to solutions g(x,s) 𝜖 FL(x),
such that for all x 𝜖 DL, if s is a locally optimal solution for instance f(x) of K,
then g(x,s) is locally optimal for L.
This notion is transitive, and has all the properties needed to define the
concept of PLS-completeness, and yield the following:
Theorem: If we can find locally optimal solutions in polynomial time for any
PLS-complete problem, then we can do so for all problems in PLS.
In addition, the running time of the “standard algorithm” for a PLS-complete
problem is “typically” exponential, and the problem of determining the output
of that algorithm is “typically” NP-hard.
(This is a property of many particular proofs of PLS-completeness, rather than
a known consequence of the definitions).
PLS-Completeness and the TSP
•
[Krentel, “Structure in locally optimal solutions,” FOCS 1989, IEEE
Computer Society, pp. 216-221]: k-opt is PLS-complete for some k
between 1,000 and 10,000.
•
[Papadimtriou, “The complexity of the Lin-Kernighan heuristic for the
traveling salesman problem,” SIAM J. Comput. 7 (1992), 450-465]. LinKernighan is PLS-complete.
This is for the variant in which, for LK-search, instead of not allowing an added
edge to be subsequently deleted, we instead do not allow an edge that has been
deleted to be subsequently added back.
Note that under this variant, the LK-search can take as many as θ(N2) steps, as
compared to the O(N) bound for the other method.
And recall that both criteria are used in the original Lin-Kernighan paper.
Computational Results, Random Euclidean Instances
N=
103
104
105
106
2-Opt [20] % Excess over HK
4.9
5.0
4.9
4.9
0.32
3.8
56.7
928
3.1
3.0
3.0
3.0
0.38
4.6
66.1
1054
2.0
2.0
2.0
2.0
0.77
9.8
151
2650
150 Mhz Secs
3-Opt [20] % Excess
150 Mhz Secs
LK [20] % Excess
150 Mhz Secs
Time on 3.06 Ghz Intel Core i3 processor at N = 106:
25.4 sec (2-opt), 29.5 sec (3-opt), 61.5 (LK)
Of this, 23.5 sec was for Preprocessing
(input reading + neighbor list construction + initial tour generation)
Beating Lin-Kernighan, Part 1:
Simulated Annealing
[Kirkpatrick, Gelatt, & Vecchi, “Optimization by simulated annealing,”
Science 220 (13 May 1983), 671-680]
[Černy, “A thermodynamical approach to the travelling salesman problem:
An efficient simulation algorithm,” J. Optimization Theory and Appl. 45
(1985), 41-51]
[Kirkpatrick, “Optimization by simulated annealing: Quantitative studies,”
J. Stat. Physics 34 (1984), 976-986]
Based on an analogy:
Physical System
Optimization Problem
State
Feasible Solution
Energy
Cost
Ground State
Optimal Solution
Rapid Quenching
Local Optimization
Careful Annealing
Simulated Annealing
Theoretical Results
General Theorem [Proved by many]:
• If you run long enough, and cool slowly enough
• (say letting the temperature be C/log(n), where C is a
constant and n is the number of moves tested so far),
• then, with high probability, you will converge to an
optimal solution.
General Drawback: For this to work, “long enough” will
exceed the time to perform exhaustive search…
Implementations Details
•
Initial temperature so high that most moves are accepted.
•
Exponentials are evaluated by lookup from a pre-computed table.
•
“Frozen” = No more moves being accepted.
•
“At Equilibrium” = Having tried a given fixed number of moves at the current
temperature.
•
“Lower the temperature” = Multiply it by a fixed constant, say 0.95.
A generic simulated annealing implementation reflecting these principles was developed by DSJ,
together with interns Cecilia Aragon, Lyle McGeoch, and Cathy Schevon. We consulted with
Scott Kirkpatrick and so that our implementation would reflect his own. It is described in
[Johnson, Aragon, McGeoch, & Schevon, “Optimization by simulated annealing: an experimental
evaluation; Part I, Graph Partitioning,” Oper. Res 37 (1989), 865-892] and
[Johnson, Aragon, McGeoch, & Schevon, “Optimization by simulated annealing: an experimental
evaluation; Part II, Graph coloring and number partitioning,” Oper. Res 39 (1991), 378-406].
The implementation was adapted to the TSP with Lyle McGeoch. Results are described in
[Johnson & McGeoch, “The traveling salesman problem: A case study in local optimization,” in
Local Search in Combinatorial Optimization, Aarts & Lenstra (editors), John-Wiley and Sons,
Ltd., 1997, pp. 215-310].
TSP-Specific Details
Basic move: 2-opt (as done by Kirkpatrick).
Initial tour: Random tour.
Starting temperature: set so 97% of moves are accepted.
Temperature length = N(N-1), approximately twice the neighborhood size.
Temperature reduction factor = 0.95.
Averages over 10 runs.
Times are on a 150 Mhz processor
“Excess” is percent over HK bound
Algorithm Engineering for Simulated Annealing
Neighborhood Pruning (first suggested by [Bonomi & Lutton, 1984]).
•
In our version, we only consider moves corresponding to an arbitrary
city as t1, one of its tour neighbors as t2, and t3 chosen from the 20
cities on t2’s neighbor list, a total of 20N possibilities..
•
This is further restricted as follows. Let c = t1 and let c’ be the tour
neighbor of c that is farther away. We only consider candidates c’’ for
t3 such that either d(c,c”) ≤ d(c,c’) or the probability of accepting an
uphill move of size d(c,c”) – d(c,c’) is greater than 0.01.
•
At each temperature, we consider 20αN candidate moves (α ≥ 1 a
parameter), so that each potential move has a reasonably probability of
being chosen as a candidate.
Low-temperature starts (suggested by Kirkpatrick).
•
Start with a Nearest Neighbor tour and a temperature yielding an
initial acceptance rate of about 50%. For random Euclidean instances
we follow the suggestion of Bonomi and Lutton to use L/√N, where L is
the length of our “unit” square.
For more structured, instances, however, SA2 can beat LK on a time equalized basis.
In particular, for dsj1000 from TSPLIB (a clustered Euclidean instance which causes
LK by a factor of 5 or more), SA2 has 1.27% excess and time equalized LK gets 1.35%.
dsj1000
More Algorithm Engineering: Using 3-opt Moves
Proposed by Kirkpatrick in 1984, but his implementation did not exploit
what we now know about fast 3-opt, and so his move set only contained
moves where the segment moved was of length 10 or less (a generalization
of Or-opt).
•
In our version, we consider both 2- and 3-opt moves, choosing 2-opts
for the first 10αN candidates at each temperature, and 3-opts for the
last 10αN.
•
A random 3-opt move is chosen as follows. Choose t1 randomly from the
N possibilities, t2 randomly from the two possibilities, t3 randomly from
the (pruned) list of up to 20 possibilities, t4 randomly from the two
possibilities, t5 randomly from the 20 possibilities (with up to four
retries if the initial choices are topologically infeasible), and then, if t6
is not topologically determined, choose it randomly from the two
possibilities. This yields a total of at most 3200N possibilities.
Other Metaheuristics
• Neural Nets [Hopfield & Tank, 1985]
– Hopeless
• DNA computing [Adleman, 1994]
– Parallel exhaustive search in a bathtub – does not scale
• Tabu Search [Glover, 1986]
– Based on an idea implicit in LK-search - not competitive for TSP
• Genetic Algorithms [Holland, 1975]
– Initial ideas were not competitive, but…
Genetic Algorithm Schema
1.
Generate an initial population P consisting of a random set of k
solutions.
2. While not yet converged, create a new generation of P as follows.
1) Select k’ distinct one- or two-item subsets of P (the mating strategy)
2) For each one-element subset, perform a random mutation to obtain a new
solution.
3) For each two-element subset, perform a randomized crossover operation to
obtain a new solution that reflects both parents.
4) Let P’ be the set of solutions generated by (2) and (3).
5) Using a selection strategy, choose k survivors from P∪P’ and replace P by
these survivors.
3. Return the best solution in P.
Standard “convergence” strategy: Stop if you have run for j generations without
improving the best solution.
Standard “selection” strategy: Choose the k best solutions.
Sample Crossover Operation
a b u h i q w o p v n m s z x l k e r y d f g j c
q w e r t y u o p a d
s f
d g
f h
g j
h c
j v
k b
l n
z m
x s
c z
v x
b l
n k
m
•
Pick a segment S from tour A.
•
Delete all the cities in S from tour B (keeping the remaining cities in the
same order).
•
Insert segment S into what is left of tour B.
If you are skeptical that operations like this can lead to a competitive
algorithm, you are right.
A new idea is needed: The “Hybrid Genetic Algorithm”
[Brady, “Optimization strategies gleaned biological evolution,” Nature 317 (1985), 804-806].
Hybrid Genetic Algorithm Schema
1.
Generate an initial population P consisting of a random set of k
solutions.
2.
Apply a given local optimization algorithm A to each solution s in P,
letting the resulting local optimal solution s’ replace s in P.
3.
While not yet converged, create a new generation of P as follows.
1)
Select k’ distinct one- or two-item subsets of P (the mating strategy)
2) For each one-element subset, perform a random mutation to obtain a new
solution.
3) For each two-element subset, perform a randomized crossover operation to
obtain a new solution that reflects both parents.
4) Let P’ be the set of solutions generated by (2) and (3),
after first applying algorithm A to each.
5) Using a selection strategy, choose k survivors from P∪P’ and replace P by these
survivors.
4. Return the best solution in P.
New
Advantages and Disadvantages
+ We can use Lin-Kernighan as the local optimization
algorithm. Presumably we will get better tours than for
basic Lin-Kernighan.
-
For a reasonably population size, we will need to perform
lots of LK’s. Is this the most cost-effective way of
investing all that extra time?
-
We still have the problem of the ad-hoc crossover to
contend with.
In 1989, Martin, Otto, & Felten suggested a way to remove both
drawbacks. (Their paper was published as [“Large-step Markov chains for
the TSP incorporating local search heuristics,” Complex Systems 5 (1991), 299-326].)
Martin, Otto, & Felten’s Approach
• Set population size to 1.
• Only do mutations (no matings).
• For the mutation, perform a random doublebridge 4-opt move.
Note that, although this was originally viewed in the context of
genetic algorithms, there is little left that is genetic about it. The
common name for this approach (using Lin-Kernighan) is now
– “Iterated Lin-Kernighan” or
– “Chained Lin-Kernighan.”
Advantages of Iterated Lin-Kernighan
• Minimal extra programming needed to turn a local search
heuristic into an iterated local search heuristic.
• Preprocessing is amortized (although this is true of multipleindependent-run LK as well).
• After the first run of LK, subsequent ones are all much faster,
since they start with all but 8 don’t-look-bits still on.
• The damage done to the tour by just changing 4 edges still turns
out to open up significant room for LK to find new improvements.
• Surprisingly good performance.
Random Euclidean Instances
.96
1704
3.06 Ghz
Intel Core i3
processor
Random Euclidean Instances
Selected TSPLIB Instances
Except for instance fl3795, ILK(N) is always within 0.05% – 0.25% of optimal.
TSPLIB Instance fl3795
ILK(N), 20 nearest neighbors:
4.33% excess (1080 seconds, 3.06 Ghz processor)
ILK(N), 20 quad neighbors:
1.09% excess ( 205 seconds, 3.06 Ghz processor)*
*Average of three runs, one of which found the optimal and a second found optimal + 1.
Random Distance Matrices
Beating Iterated Lin-Kernighan
Chained Lin-Kernighan [Applegate, Cook, & Rohe, “Chained Lin-Kernighan for
large traveling salesman problems,” INFORMS J. Comput. 15 (2003), 82-92]
•
Broader and deeper search before LK-searches, More selective in choices of
double-bridge moves (mixture of greedier choices, random walks, etc.)
Hybrid Genetic Algorithm [Nguyen, Yoshihara, Yamamori, and Yasunaga,
“Implementation of an effective hybrid GA for large-scale traveling
salesman problems,” IEEE Trans. Syst., Man, Cybernetics B37 (2007), 9299]
•
Genetic algorithm with crossovers + mutations + Lin-Kernighan, and large running
times for bigger instances.
Helsgaun’s algorithm [Helsgaun, “An effective implementation of the LinKernighan traveling salesman heuristic,” European J. Oper. Res. 126 (2000),
106-130. Source code available at http://www.dat.ruk.dk/~keld/]
•
Different restart mechanism, broader initial search, …
Tour merging [Cook & Seymour, “Tour merging via branch-decomposition,”
INFORMS J. Comput. 15 (2003), 233-248]
Random Distance Matrices
N = 1000
3162 10,000
% Excess of HK bound
N = 1000
3162 10,000
Seconds on a 500 Mhz DEC Alpha
Note: Helsgaun’s algorithm is the only heuristic we’ve seen that is not
fazed by these instances.
Heuristics for the Asymmetric TSP
(a quick tour)
Tour Construction:
•
Asymmetric Nearest Neighbor
•
Asymmetric Greedy
•
Match & Patch (Compute minimum-cost cycle cover, repeatedly patch together the
two largest cyles)
•
Contract or Patch
•
Repeated Assignment
•
Zhang’s Algorithm (Truncated Branch & Bound)
Local Optimization:
•
Asymmetric 3-Opt
•
Iterated 3-Opt
•
Kanellakis-Papadimitriou (Mimics Lin-Kernighan, and also uses “best double-bridge
move” as a component)
•
Helsgaun (Convert to symmetric instance and apply Helsgaun’s algorithm.
Matching-and-Patch
Contract-or-Patch
Once all 2-cycles are removed, proceed as in Match & Patch.
Repeated Assignment
Delete all but one city from each cycle, find new matching, and repeat.
The union of all the matching edges is a connected graph in which every vertex has
equal in- and out-degrees. So there is an Euler tour. If we assume the Δ-Inequality,
the Euler tour can be traversed using shortcuts at no extra cost. Each matching is
no longer than an optimal tour, again by the triangle inequality. Since each matching
involves no more than half the previous number of vertices, we have:
Theorem [Frieze, Galbiati, & Maffioli (1982]: Assuming Δ-Inequality,
RA(I) ≤ log(N)OPT(I).
Heuristics for the Asymmetric TSP
(a quick tour)
Tour Construction:
•
Asymmetric Nearest Neighbor
•
Asymmetric Greedy
•
Match & Patch (Compute minimum-cost cycle cover, repeatedly patch together the
two largest cyles)
•
Contract or Patch
•
Repeated Assignment
•
Zhang’s Algorithm (Truncated Branch & Bound)
Local Optimization:
•
Asymmetric 3-Opt (Only 3-opt moves that cause no reversals)
•
Iterated Asymmetric 3-Opt
•
Kanellakis-Papadimitriou (Mimics Lin-Kernighan, and also uses “best double-bridge
move” as a component)
•
Helsgaun (Convert to symmetric instance and apply Helsgaun’s algorithm.
Results for Instance Generators of
[Cirasella, Johnson, McGeoch, & Zhang, “The asymmetric traveling salesman problem:
Algorithms, instance generators, and tests,” ALENEX 2001, Lecture Notes in
Computer Science 2153, Springer, pp. 32-59]
The Coin Collection Problem
The Stacker-Crane Problem
The No-Wait Shop Scheduling Problem
The Disk Scheduling Problem
The Shortest Common Superstring Problem
The Drilling on a Tilted Table Problem
Next Time
Optimization Algorithms