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Approximating Graphic TSP with Matchings

Approximating Graphic TSP with Matchings. Tobias Momke and Ola Svensson Royal Institute of Tech., Stockholm. Presented by Amit Kumar (IIT Delhi). Traveling Salesman Problem (TSP). Given weighted graph G, find a tour visiting all vertices of min. cost. TSP.

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Approximating Graphic TSP with Matchings

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  1. Approximating Graphic TSP with Matchings Tobias Momke and Ola Svensson Royal Institute of Tech., Stockholm Presented by Amit Kumar (IIT Delhi)

  2. Traveling Salesman Problem (TSP) Given weighted graph G, find a tour visiting all vertices of min. cost.

  3. TSP Find min. cost Hamiltonian cycle in the metric completion of G.

  4. Graphic (unweighted) TSP Min. the number of edges in the tour. Find an Eulerian multi-graph with min. number of edges.

  5. Some History Apx-Hard. (1.0046) [Papadimitriou, Vempala 2006] 1.5 approx [Christofides 1976] Held-Karp LP Relaxation (1970). Best lower bound on integrality gap : 4/3 upper bound : 1.5 [Williamson, Shmoys 1990]

  6. Some History (Graphic TSP) 1.487-approx for cubic 3-edge connected [Gamarnik et. al. 2005] 4/3-approx for cubic graphs, and 7/5-approx for sub-cubic graphs [Boyd et. al. 2011], [Garg, Gupta 2011] 1.5-10-12 approx. [Gharan, Saberi, Singh 2011]

  7. This Paper 1.46-approx for Graphic TSP 4/3-approx for cubic (and sub-cubic) graphs. New techniques …

  8. Talk Outline • Christofides’ algorithm • 4/3-approx for cubic graphs • Idea of removable pairs, and how to • find large number of such pairs • 4/3-approx for sub-cubic graphs • Help-Karp LP Relaxation • Extension to general graphs

  9. Christofides’ algorithm Start with a MST (cost at most OPT) Construct a matching over the odd-degree vertices in the shortest path metric.

  10. Christofides’ algorithm Cost of matching · OPT/2 Total cost · 1.5 OPT

  11. Talk Outline • Christofides’ algorithm • 4/3-approx for cubic graphs • Idea of removable pairs, and how to • find large number of such pairs • 4/3-approx for sub-cubic graphs • Help-Karp LP Relaxation • Extension to general graphs

  12. 2-connected graphs Can assume that the graph is 2-connected.

  13. Cubic 2-connected graphs Any cubic 2-connected graph has a perfect matching. Adding a perfect matching makes it Eulerian.

  14. Cubic 2-connected graphs 3/2n + 1/2n = 2n edges get used. Can we remove some edges ? so that only 4/3 n edges remain ?

  15. Edmonds’ Matching Polytope x(±(v))=1 for all vertices v x(±(S)) ¸ 1 for all odd sets S xe¸ 0 for all edges e Theorem[Edmonds] Any vertex corresponds to a perfect matching.

  16. Edmonds’ Matching Polytope Set x(e)=1/3 for all edges e. S : odd set |±(S)| ¸ 2. |±(S)| must also be odd.

  17. Edmonds’ Matching Polytope There exist polynomial number of matchings M1, …, Mk such that any edge appears in exactly 1/3 of these matchings.

  18. 2-connected cubic graphs Take E U M, where M is a random matching drawn from the collection M1, …, Mk Total number of edges = 2n Which edges can we remove ?

  19. 2-connected cubic graphs v Construct a DFS Tree The matching M contains exactly one edge incident to v : three cases arise

  20. 2-connected cubic graphs v v v

  21. 2-connected cubic graphs v v v Expected number of edges removed = n/2 . 2/3 . 2 = 2n/3 Number of remaining edges = 2n-2n/3=4n/3

  22. Talk Outline • Christofides’ algorithm • 4/3-approx for cubic graphs • Idea of removable pairs, and how to • find large number of such pairs • 4/3-approx for sub-cubic graphs • Help-Karp LP Relaxation • Extension to general graphs

  23. Removable Pairs G : 2 connected R : subset of edges P µ R X R • each edge in R is in at most one pair in P • the edges in a pair are incident to a vertex of degree >= 3 • removing a subset of R such that at most one edge from • each pair is removed does not disconnect G.

  24. Removable Pairs G : 2 connected R : subset of edges P µ R X R R could have edges which are not in any pair.

  25. Removable Pairs Theorem : There is aTSP tour with at most 4/3 |E| - 2/3 |R| edges.

  26. Proof idea Transform G to a 2-connected cubic graph G’, such that (R,P) maps to a removable pair.

  27. Proof idea Transform G to a 2-connected cubic graph G’, such that (R,P) maps to a removable pair.

  28. Proof idea In the cubic graph, pick a random matching and with prob. 2/3 we can remove 2 edges for each pair in P.

  29. Finding Good Removable Pairs Can start with any DFS Tree.

  30. Finding Good Removable Pairs

  31. Finding Good Removable Pairs v w Tw If k (¸ 1) back-edges from Tw to v, can add one pair to P and k+1 edges to R

  32. Finding Good Removable Pairs Given a DFS Tree, Make it 2-connected by adding as few back-edges as possible. The back-edges should be “well-distributed” for many tree-edges, there should be corresponding back-edges. 4/3|E|-2/3|R|

  33. Some Notation v in-vertices v i w w Sub-divide tree edges. |R|=i 2 I 0 or B(i) +1

  34. Circulation Problem v in-vertices i (1,1) (0,1) w Edges with non-zero (integral) flow form a 2-connected graph.

  35. Min-cost Circulation Problem v in-vertices i (1,1) (0,1) w Cost of flow=i 2 I min(0, f(B(i))-1)

  36. Removable Pairs from Circulation v in-vertices i (1,1) (0,1) w C=|R|-2|P| E=n+|R|-|P| 4/3E-2/3R=4/3n+2/3C

  37. Main Theorem v in-vertices i (1,1) (0,1) w Given a circulation of cost C, there is a TSP tour of cost at most 4/3n + 2/3C

  38. 2-connected sub-cubic graphs v Send 1 unit of flow on all back-edges. C=0

  39. Talk Outline • Christofides’ algorithm • 4/3-approx for cubic graphs • Idea of removable pairs, and how to • find large number of such pairs • 4/3-approx for sub-cubic graphs • Help-Karp LP Relaxation • Extension to general graphs

  40. Held Karp LP Min exe x(±(S)) ¸ 2 for all S x ¸ 0

  41. Integrality Gap Example L LP Value = 3L, Opt = 4L

  42. Obtaining a circulation Solve the Held-Karp LP A basic solution will have non-zero xe values for at most 2n-1 edges. Using this basic solution, construct a DFS Tree Bound the cost of circulation by LP value

  43. Constructing the DFS Tree When at a vertex v, pick the next edge with the highest xe value. v 0.5 0.2 0.9 w 0.3

  44. Bounding the cost of circulation v Exhibit a circulation of low cost. 0.5 w For each back-edge e, send xe amount of flow on the unique cycle formed by adding e to the tree.

  45. Bounding the cost of circulation v 0.95 w If flow fe on a tree edge < 1, then send the remaining (1-fe) unit on any cycle containing e and one back-edge

  46. First circulation v 0.5 i At most n back-edges. w No. of back-edges into i at least f(B(i))/xvw Allows us to bound i min(f(B(i))-1,0) in terms of exe

  47. Second circulation v 0.95 w If not enough flow on a tree-edge, the LP solution must be putting high x value on this edge.

  48. Final Theorem Cost of circulation is at most

  49. Open Problems 4/3 approx for general graphs. Better than 3/2 for weighted graphs.

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