approximating graphic tsp with matchings tobias momke and ola svensson royal institute of tech.,...

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.

Graphic (unweighted) TSP

Min. the number of edges in the tour.

Find an Eulerian multi-graph with min. number of edges.

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]

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]

This Paper

1.46-approx for Graphic TSP

4/3-approx for cubic (and sub-cubic) graphs.

New techniques …

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

Christofides’ algorithm

Start with a MST (cost at most OPT)

Construct a matching over the odd-degree vertices in the shortest path metric.

Christofides’ algorithm

Cost of matching · OPT/2 Total cost · 1.5 OPT

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

2-connected graphs

Can assume that the graph is 2-connected.

Cubic 2-connected graphs

Any cubic 2-connected graph has a perfect matching.

Adding a perfect matching makes it Eulerian.

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 ?

Edmonds’ Matching Polytope

x(±(v))=1 for all vertices vx(±(S)) ¸ 1 for all odd sets Sxe ¸ 0 for all edges e

Theorem[Edmonds] Any vertex corresponds to a perfect matching.

Edmonds’ Matching Polytope

Set x(e)=1/3 for all edges e.

S : odd set

|±(S)| ¸ 2. |±(S)| must also be odd.

Edmonds’ Matching Polytope

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

2-connected cubic graphs

Take E U M, where M is a random matching drawn from the collection M1, …, Mk

Total number of edges = 2nWhich edges can we remove ?

2-connected cubic graphs

Construct a DFS Tree

The matching M contains exactly one edge incident to v : three cases arise

v

2-connected cubic graphs

v v v

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

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

Removable Pairs

• 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.

G : 2 connectedR : subset of edgesP µ R X R

Removable Pairs

G : 2 connectedR : subset of edgesP µ R X R

R could have edges which are not in any pair.

Removable Pairs

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

Proof idea

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

Proof idea

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

Proof idea

In the cubic graph, pick a random matchingand with prob. 2/3 we can remove 2 edgesfor each pair in P.

Finding Good Removable Pairs

Can start with any DFS Tree.

Finding Good Removable Pairs

Finding Good Removable Pairs

If k (¸ 1) back-edges from Tw to v, can add one pair to P and k+1 edges to R

v

w

Tw

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|

Some Notation

v

w

v

w

Sub-divide tree edges.

|R|=i 2 I 0 or B(i) +1

in-vertices

i

Circulation Problem

v

w

in-vertices

i (1,1)(0,1)

Edges with non-zero (integral) flow form a 2-connected graph.

Min-cost Circulation Problem

v

w

in-vertices

i (1,1)(0,1)

Cost of flow=i 2 I min(0, f(B(i))-1)

Removable Pairs from Circulation

v

w

in-vertices

i (1,1)(0,1)

C=|R|-2|P|

E=n+|R|-|P|

4/3E-2/3R=4/3n+2/3C

Main Theorem

v

w

in-vertices

i (1,1)(0,1)

Given a circulation of cost C, there is a TSP tour of cost at most 4/3n + 2/3C

2-connected sub-cubic graphs

Send 1 unit of flow on all back-edges.

C=0

v

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

Held Karp LP

Min e xe

x(±(S)) ¸ 2 for all S x ¸ 0

Integrality Gap Example

LP Value = 3L, Opt = 4L

L

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

w0.9

0.5

0.3

0.2

Bounding the cost of circulation

For each back-edge e, send xe amount of flow on the unique cycle formed by adding e to the tree.

v

w

0.5Exhibit a circulation of low cost.

Bounding the cost of circulation

If flow fe on a tree edge < 1, then send the remaining (1-fe) unit on any cycle containing e and one back-edge

v

w0.95

First circulationv

w

0.5

At most n back-edges.i

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

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

v

w0.95

Final Theorem

Cost of circulation is at most

e

ex)32(4)n26(1

Open Problems

4/3 approx for general graphs.

Better than 3/2 for weighted graphs.