Approximating Graphic TSP by Matchings Tobias Mömke and Ola Svensson KTH Royal Institute of Technology Sweden

Travelling Salesman Problem Given – n cities – distance d(u,v) between cities u and v Find shortest tour that visits each city once Value = = 5

Classic Problem 1800’s William Rowan Hamilton and Thomas Penyngton Kirkman studied related mathematical problems HamiltonKirkman 1930’s General form of TSP gets popular and is promoted by Karl Menger Hassler WhitneyMerrill Flood 50’s G. Dantzig, R. Fulkerson, and S. Johnson publish a method for solving the TSP and solve a 49-city instance to optimality 60’s Proctor and Gamble ran a contest for solving a TSP instance on 33 cities An optimal tour of 120 cities of (West) Germany Christofides publishes the famous 1.5-approximation algorithm Held-Karp proposes a very successful heuristic for calculating a lower bound on a tour The lower bound coincides with the value of a linear program known as Held-Karp or Subtour Elimination relaxation 70’s 80’s 2392 cities 90’s cities 00’s S. Arora and J. S. B. Mitchell publish a PTAS for Euclidian TSP C. H. Papadimitriou and S. Vempala: NP-hard to approximate within 220/219 Applegate, Bixby, Chvatal, Cook, and Helsgaun find the shortest tour of cities in Sweden Major open problem to understand the approximability of metric TSP: NP-hard to approximate better than 220/219 Christofides’ 1.5-approximation algorithm still best Held-Karp relaxation is conjectured to have integrality gap of 4/3

Graphic TSP (graph-TSP) Given an unweighted graph G(V,E), Length = 4n/3 -1#edges = 4n/3 -1 find spanning Eulerian multigraph with minimum #edges find the shortest tour with respect to distances

Important Special Case Natural problem to find smallest Eulerian subgraph – Studied for more than 3 decades Easier to study than general metrics but hopefully shed light on them – Still APX-hard – Worst instances for Held-Karp lower bound are graphic – Any difficult instance to Held-Karp lower bound is determined by a weighted graph with at most 2n-3 edges – Until recently, Christofides best approximation algorithm

Recent Advancements on graph-TSP Major open problem to understand the approximability of metric TSP: NP-hard to approximate better than 220/219 Christofides’ 1.5-approximation algorithm still best Held-Karp relaxation is conjectured to have integrality gap of 4/3 Gamarnik, Lewenstein & Sviridenko give a approximation algorithm for cubic 3-edge connected graphs Boyd, Sitters, van der Star & Stougie give a 4/3-approximation algorithm for cubic graphs 7/5-approximation algorithm for subcubic graphs Conjecture: subcubic 2-edge connected graphs has a tour of length 4n/3 -2/3 Oveis Gharan, Saberi & Singh give a (1.5-ε)-approximation algorithm for graph-TSP - First improvement over Christofides - Similar to Christofides, but instead of starting with a minimum MST they sample one from the solution of Held-Karp relaxation - Analysis requires several novel ideas, like structure of almost minimum cuts

Our Results A approximation algorithm for graph-TSP Subcubic 2-edge-connected graph has a tour of length at most 4n/3 – 2/3 A 4/3-approximation algorithm for subcubic/claw-free graphs (matching the integrality gap) Based on techniques used by Frederickson & Ja’Ja’82 and Monma, Munson & Pulleyblank’90 + novel use of matchings: instead of only adding edges to make a graph Eulerian we allow for removal of certain edges

Outline Held-Karp Relaxation Given a 2-vertex connected graph G(V,E) find a spanning Eulerian graph with at most 4/3|E| edges Introduce removable edges and prove Comments on general graphs Subcubic 2-edge-connected graph has a tour of length at most 4n/3 – 2/3

Held-Karp Relaxation A variable x {u,v} for each pair u,v of cities Very well studied: Any extremepoint has support consisting of at most 2n-3 edges Restriction to 2-vertex-connected graphs is w.l.o.g.

Eulerian subgraph of 2-VC graph Frederickson & Ja’Ja’82 and Monma, Munson & Pulleyplank’90 1.Use gadgets to make graph cubic 2.Sample perfect matching M so that each edge is taken with probability 1/3 ( Simple application of Edmond’s perfect matching polytope) 3.Return graph with edge set 1.An edge is in M with probability 1/3 2.Expected size of M U E is 4/3|E| 3.A 2-VC graph has a tour of size 4/3|E|

Using Matchings to Remove Edges First Idea 1.Observation: removing an edge from the matching will still result in even degree vertices 2.If it stays connected we will again have an Eulerian graph 3.Same algorithm as before but return 1.Expected size of returned Eulerian graph:

Using Matchings to Remove Edges Second Idea 1.Use structure of perfect matchings to increase the set R of removable edges 2.If it stays connected we will again have an Eulerian graph 3.Define a “removable pairing” Pair of edges: only one edge in each pair can be removed by a matching Graph obtained by removing removable edges such that at most one edge in each pair is removed is connected R contains all back-edges and tree-edges paired with a back-edge If G has degree at most 3 then size of R is 2b-1

Using Matchings to Remove Edges Second Idea 1.Same algorithm as before but return 1.We have that |R| = 2b -1 and |E| = n-1 + b and thus

Result for Graphs of Max Degree 3 Matchings can be used to also remove edges Used structure to increase number of removable edges “removable pairing” Subcubic 2-edge-connected graph has a tour of length at most 4n/3 – 2/3 A 4/3-approximation algorithm for subcubic/claw-free graphs (matching the integrality gap)

General Case In degree 3 instances each back-edge is paired with a tree edge In general instance this might not be possible LP prevents this situation: Min Cost circulation flow where the cost makes you pay for this situation Analyze by using LP extreme point structure

Final Result Christofides Our A approximation algorithm for graph-TSP

Summary Novel use of matchings – allow us to remove edges leading to decreased cost Bridgeless subcubic graphs have tour of size 4n/3 - 2/3 – Tight analysis of Held-Karp for these graphs approximation algorithm for graph-TSP

Open Problems Find better removable pairing and analysis – If LP=n is there always a 2-vertex connected subgraph of degree 3? Removable paring straight forward to generalize to any metric – However, finding one remains open One idea is to sample extremepoint, for example: – Sample two spanning trees with marginals x e such that all edges are removable => 4/3–approximation algorithm