Approximating Graphic TSP by Matchings

Tobias Mömke and Ola SvenssonKTH 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

1 2

34

1

2

1

1

12

Value = 1+2+1+1 = 5

http://www.tsp.gatech.edu/

Classic Problem1800’s

William Rowan Hamilton and Thomas Penyngton Kirkman studied related mathematical problems

Hamilton Kirkman

1930’s

General form of TSP gets popular and is promoted by

Karl Menger Hassler Whitney Merrill 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

13509 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, andHelsgaun find the shortest tour of 24978 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),

1 1

4

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-TSP2000 2005 2010

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 1.487-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 1.461-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 graphFrederickson & Ja’Ja’82 and Monma, Munson & Pulleyplank’90

1. Use gadgets to make graph cubic2. 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 1/32. 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 EdgesFirst 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 graph3. Same algorithm as before but return

1. Expected size of returned Eulerian graph:

Using Matchings to Remove EdgesSecond Idea

1. Use structure of perfect matchings to increase the set R of removable edges2. 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 thensize of R is 2b-1

Using Matchings to Remove EdgesSecond 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

1.0 1.02 1.04 1.06 1.08 1.1

Christofides

Our

A 1.461-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

• 1.461-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 xe

such that all edges are removable => 4/3–approximation algorithm