Exercises. Set 2 of 2. To be handed in latest on Friday, 10-10-2014. Exercises are in random order. When the question is to give an approximation algorithm, then also give a proof for this. For some exercises you may find a (partial) answer in the literature / on the internet. Always, give a complete answer in your own words. It should be clear that you understand your own answer. 1 An instance of the metric TSP-path problem is defined the same as an instance for the metric TSP but in stead of a tour we need to find a path that visits every vertex exactly once, i.e., a TSP-tour minus one edge. We may consider different variants of the problem, depending on the number of endpoints that are given in the instance. (a) Give a 3/2-approximation algorithm if no endpoints are specified. (b) Give a 3/2-approximation algorithm if one endpoint is specified. (c) Give a 5/3-approximation algorithm if both endpoints are specified. Hint: Use the idea of Christofides’ algorithm. 2 A special case of the TSP-problem is the one in which cij ∈ {1, 2} for all pairs i, j. Give a 4/3-approximation for this problem. You may use the fact that a minimum 2-matching can be found in polynomial time. A 2-matching is a subset of the edges such that each vertex is adjacent to exactly two of them and the cost of the 2-matching is defined as the sum of the edge costs. 3 Show that if we do not require that the triangle inequality holds in the kcenter problem, then no α-approximation algorithm exists for any constant α. 4 Give a 2-approximation algorithm for the following variant of the k-center problem. An instance is given by an integer k and a complete graph G = (V, E) with edge cost c(u, v) > 0 for any pair of vertices u, v ∈ V such that the triangle inequality is satisfied. A solution is a partition of V in k sets, V1 , . . . , Vk . The cost of the solution is the maximum distance within a set: max c(u, v). 1≤i≤k u,v∈Vi 5 Give a PTAS for the following problem. Minimum quotient problem: Instance: Positive integers a1 < a2 < · · · < an Solution: S1 , S2 ⊂ {1,P2, . . . , n}, with P S1 , S2 non-empty and disjoint. Assume that a ≥ i i∈S1 i∈S2 ai . P Cost: Pi∈S1 i∈S2 Goal : ai ai Minimize cost of the solution. Hint: first give a pseudo-polynomial time algorithm and then round the instance (as we did in the PTAS for knapsack). 1

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