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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). Traveling Salesman Problem (TSP). Given weighted graph G, find a tour visiting all vertices of min. cost. TSP.

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Approximating graphic tsp with matchings

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
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
Graphic (unweighted) TSP

Min. the number of edges in the tour.

Find an Eulerian multi-graph with min.

number of edges.


Some history
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
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
This Paper

1.46-approx for Graphic TSP

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

New techniques …


Talk outline
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
Christofides’ algorithm

Start with a MST (cost at most OPT)

Construct a matching over the odd-degree

vertices in the shortest path metric.


Christofides algorithm1
Christofides’ algorithm

Cost of matching · OPT/2

Total cost · 1.5 OPT


Talk outline1
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
2-connected graphs

Can assume that the graph is 2-connected.


Cubic 2 connected graphs
Cubic 2-connected graphs

Any cubic 2-connected graph has a

perfect matching.

Adding a perfect matching makes it Eulerian.


Cubic 2 connected graphs1
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
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.


Edmonds matching polytope1
Edmonds’ Matching Polytope

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

S : odd set

|±(S)| ¸ 2.

|±(S)| must also be odd.


Edmonds matching polytope2
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
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 ?


2 connected cubic graphs1
2-connected cubic graphs

v

Construct a DFS Tree

The matching M contains exactly one edge

incident to v : three cases arise



2 connected cubic graphs3
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 outline2
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
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.


Removable pairs1
Removable Pairs

G : 2 connected

R : subset of edges

P µ R X R

R could have edges which are not in any pair.


Removable pairs2
Removable Pairs

Theorem : There is aTSP tour with at most

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


Proof idea
Proof idea

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

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


Proof idea1
Proof idea

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

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


Proof idea2
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.


Finding good removable pairs
Finding Good Removable Pairs

Can start with any DFS Tree.



Finding good removable pairs2
Finding Good Removable Pairs

v

w

Tw

If k (¸ 1) back-edges from Tw to v,

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


Finding good removable pairs3
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
Some Notation

v

in-vertices

v

i

w

w

Sub-divide tree edges.

|R|=i 2 I 0 or B(i) +1


Circulation problem
Circulation Problem

v

in-vertices

i

(1,1)

(0,1)

w

Edges with non-zero (integral) flow form

a 2-connected graph.


Min cost circulation problem
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)


Removable pairs from circulation
Removable Pairs from Circulation

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


Main theorem
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


2 connected sub cubic graphs
2-connected sub-cubic graphs

v

Send 1 unit of flow on all back-edges.

C=0


Talk outline3
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
Held Karp LP

Min exe

x(±(S)) ¸ 2 for all S

x ¸ 0


Integrality gap example
Integrality Gap Example

L

LP Value = 3L, Opt = 4L


Obtaining a circulation
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
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
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 circulation1
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
First circulation

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


Second circulation
Second circulation

v

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
Final Theorem

Cost of circulation is at most


Open problems
Open Problems

4/3 approx for general graphs.

Better than 3/2 for weighted graphs.


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