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

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

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] 5

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] 6

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

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 8

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

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

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 11

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

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

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 ? 14

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. 15

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

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

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 ? 18

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

2-connected cubic graphs v v v 20

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 21

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 22

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. 23

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

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

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

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

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. 28

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

Finding Good Removable Pairs 30

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

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| 32

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

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

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) 35

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

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 37

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

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 39

Held Karp LP Min e xe x(±(S)) ¸ 2 for all S x ¸ 0

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

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

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

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.

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

First circulation v i At most n back-edges. w 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 e xe

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

Final Theorem Cost of circulation is at most

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