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Held Karp Relaxation

The Traveling Salesman Problem (TSP) is undoubtedly the most important and well-studied problem in Combinatorial Optimization. Today’s post is a quick overview of the Held-Karp Relaxation of TSP.

TSP : Given a complete undirected graph G(V,E) with non-negative costs c_e for each edge e \in E, find a hamiltonian cycle of G with minimum cost. It is well-known that this problem is NP-Complete.

Exercise : There is no \alpha-approximation algorithm for TSP (for any \alpha \geq 1) unless P=NP.

Metric TSP : In Metric-TSP, the edge costs satisfy triangle inequality i.e., for all u,v,w \in V, c(u,w) \leq c(u,v) + c(v,w). Metric-TSP is also NP-complete. Henceforth, we shall focus on metric TSP.

Symmetric TSP (STSP) : In STSP, the edge costs are symmetric i.e., c(u,v) = c(v,u). Approximation algorithms with factor 2 (find a minimum spanning tree (MST) of G and use shortcuts to obtain a tour) and factor 3/2 (find an MST, find a perfect matching on the odd degree nodes of  the MST to get a eulerian graph and obtain a tour) are well-known. The factor 3/2 algorithm, known as Christofides Algorithm [Christofides'76], is the best known approximation factor for STSP. No improvement in the last three decades !!

Following is the Held-Karp Relaxation for STSP with the cut constraints and the degree constraints. The variables are x_e, one for each edge e \in E. For a subset S \subset V, \delta(S) denotes the edges incident to S. Let x(\delta(S)) denote the sum of values of x_e of the edges with exactly one endpoint in S. For more details of Held-Karp relaxation see [HK'70, HK'71]

hk-stsp

Exercise : In the following instance of STSP the cost between vertices u and v is the length of the shortest path between u and v. The three long paths are of length k. Prove that this instance achieves an integrality ratio arbitrarily close to 4/3 (as k is increased).

stsp

Asymmetric TSP (ATSP) : In ATSP, the edge costs are not necessarily symmetric i.e., the underlying graph is directed. The Held-Karp relaxation for ATSP is as follows :

hk-atsp

Charikar, Goemans and Karloff [CGK'04] showed that the integrality of Held-Karp relaxation for ATSP is at least 2-\epsilon. Frieze, Galbiati and Maffioli [FGM'82] gave a simple O({\log}_2{n})-approximation algorithm for ATSP in 1982, where n is the number of vertices. In the last eight years, this was improved to a guarantee of 0.999 {\log}_2{n} by Blaser [Blaser'02], and to \frac{4}{3}{\log}_3{n} Kaplan et al [KLSS'03] and to \frac{2}{3}{\log}_2{n} by Feige and Singh [FS'07]. So we have an approximation factor better than {\ln}n !!

Open Problems :

  • The long-standing open problem is to determine the exact integrality gap of Held-Karp relaxation. Many researchers conjecture that the integrality gap of Held-Karp relaxation for STSP is 4/3 and for ATSP it is bounded by a constant. The best known upper bounds are 3/2 and O(logn) respectively.
  • The size of the integrality gap instance of ATSP (constructed by [CGK'04]) is exponential in 1/\epsilon to achieve an integrality gap of 2-\epsilon. Is there a polynomial-sized (in 1/\epsilon) instance achieving an integrality gap of 2-\epsilon ?

References :

Nicos Christofides, Worst-case analysis of a new heuristic for the travelling salesman problem, Report 388, Graduate School of Industrial Administration, CMU, 1976.

  • [HK'70] Micheal Held and Richard M. Karp, The Traveling Salesman Problem and Minimum Spanning Trees, Operations Research 18, 1970, 1138–1162.
  • [HK'71] Michael Held and Richard Karp, The Traveling-Salesman Problem and Minimum Spanning Trees: Part II, Mathematical Programming 1, 1971, 6–25.
  • [Christofides'76] Nicos Christofides, Worst-case analysis of a new heuristic for the travelling salesman problem, Report 388, Graduate School of Industrial Administration, CMU, 1976.
  • [FGM'82] A. M. Frieze, G. Galbiati and M. Maffioli, On the Worst-Case Performance of Some Algorithms for the Asymmetric Traveling Salesman Problem, Networks 12, 1982, 23–39.
  • [Blaser'02] M. Blaser, A New Approximation Algorithm for the Asymmetric TSP with Triangle Inequality, Proceedings of the 14th Annual ACM-SIAM Symposium on Discrete Algorithms, 2002, 638–645.
  • [KLSS'03] H. Kaplan, M. Lewenstein, N. Shafir and M. Sviridenko, Approximation Algorithms for Asymmetric TSP by Decomposing Directed Regular Multidigraphs, Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science, 2003, 56–67.
  • [CGK'04] Moses Charikar, Michel X. Goemans, Howard J. Karloff: On the Integrality Ratio for Asymmetric TSP. FOCS 2004: 101-107
  • [FS'07] Uriel Feige, Mohit Singh: Improved Approximation Ratios for Traveling Salesperson Tours and Paths in Directed Graphs. APPROX-RANDOM 2007: 104-118