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Approximating Graphic TSP with Matchings

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Approximating Graphic TSP with Matchings

Tobias Momke and Ola Svensson

Royal Institute of Tech., Stockholm

Presented by Amit Kumar (IIT Delhi)

Given weighted graph G, find a tour

visiting all vertices of min. cost.

Find min. cost Hamiltonian cycle in the

metric completion of G.

Min. the number of edges in the tour.

Find an Eulerian multi-graph with min.

number of edges.

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]

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]

1.46-approx for Graphic TSP

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

New techniques …

- 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

Start with a MST (cost at most OPT)

Construct a matching over the odd-degree

vertices in the shortest path metric.

Cost of matching · OPT/2

Total cost · 1.5 OPT

- 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

Can assume that the graph is 2-connected.

Any cubic 2-connected graph has a

perfect matching.

Adding a perfect matching makes it Eulerian.

3/2n + 1/2n = 2n edges get used.

Can we remove some edges ?

so that only 4/3 n edges remain ?

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.

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

S : odd set

|±(S)| ¸ 2.

|±(S)| must also be odd.

There exist polynomial number of matchings

M1, …, Mk such that any edge appears in

exactly 1/3 of these matchings.

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 ?

v

Construct a DFS Tree

The matching M contains exactly one edge

incident to v : three cases arise

v

v

v

v

v

v

Expected number of edges removed

= n/2 . 2/3 . 2 = 2n/3

Number of remaining edges = 2n-2n/3=4n/3

- 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

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.

G : 2 connected

R : subset of edges

P µ R X R

R could have edges which are not in any pair.

Theorem : There is aTSP tour with at most

4/3 |E| - 2/3 |R| edges.

Transform G to a 2-connected cubic graph G’,

such that (R,P) maps to a removable pair.

Transform G to a 2-connected cubic graph G’,

such that (R,P) maps to a removable pair.

In the cubic graph, pick a random matching

and with prob. 2/3 we can remove 2 edges

for each pair in P.

Can start with any DFS Tree.

v

w

Tw

If k (¸ 1) back-edges from Tw to v,

can add one pair to P and k+1 edges to R

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|

v

in-vertices

v

i

w

w

Sub-divide tree edges.

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

v

in-vertices

i

(1,1)

(0,1)

w

Edges with non-zero (integral) flow form

a 2-connected graph.

v

in-vertices

i

(1,1)

(0,1)

w

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

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

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

v

Send 1 unit of flow on all back-edges.

C=0

- 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

Min exe

x(±(S)) ¸ 2 for all S

x ¸ 0

L

LP Value = 3L, Opt = 4L

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

When at a vertex v, pick the next edge

with the highest xe value.

v

0.5

0.2

0.9

w

0.3

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.

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

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

v

0.95

w

If not enough flow on a tree-edge, the LP

solution must be putting high x value on this

edge.

Cost of circulation is at most

4/3 approx for general graphs.

Better than 3/2 for weighted graphs.