Approximating Graphic TSP by Matchings

Report
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
1
1
2
2
1
1
4
2
1
3
Value = 1+2+1+1 = 5
Classic Problem
1800’s
1930’s
50’s
60’s
70’s
80’s
90’s
00’s
2392 cities13509 cities
•G.AnDantzig,
optimal
ofform
120Hamilton
cities
(West)
Germany
William
General
R.tour
Fulkerson,
Rowan
of
and
TSPS.of
gets
Johnson
and
popular
Thomas
publish
and
Penyngton
is
a method
promoted
Kirkman
forbysolving the
S. Arora and J. S. studied
B. Mitchell
publish
related
TSPmathematical
andasolve a problems
PTAS for Euclidian49-city
TSP instance to optimality
Major
open problem
to understand
approximabilityalgorithm
of metric TSP:
• Christofides
publishes
the famous the
1.5-approximation
C. H. Papadimitriou and S. Vempala:
• NP-hard
toproposes
approximate
better
than 220/219
Held-Karp
a very
successful
heuristic for calculating
NP-hard
to approximate
220/219
Proctor
and
Gamble ran
awithin
contest
for solving a TSP instance on 33 cities
a lower boundalgorithm
on a tour still best
• Christofides’ 1.5-approximation
Applegate,
Cook,
•• The
lowerBixby,
boundChvatal,
coincides
withand
theto
value
a linear program
known as
Held-Karp
relaxation
is
conjectured
haveofintegrality
gap of 4/3
Karl
Menger
Helsgaun find the shortest tour
of Whitney
Hassler
Merrill Flood
Hamilton
Kirkman
24978 cities Held-Karp
in Sweden or Subtour Elimination relaxation
http://www.tsp.gatech.edu/
Graphic TSP (graph-TSP)
• Given an unweighted graph G(V,E),
• find spanning
the shortest
Eulerian
tour with
multigraph
respectwith
to distances
minimum #edges
Length = 4n/3 -1
1
#edges = 4n/3 -1
1
4
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
2000
2005
2010
Boyd, Gharan,
Sitters, van
der &Star
& Stougie
Oveis
Saberi
Singh
give a give a
(1.5-ε)-approximation algorithm for graph-TSP
Major
open problem to
understand
approximability
of metric TSP:
4/3-approximation
algorithm
forthe
cubic
graphs
Gamarnik, Lewenstein & Sviridenko give a
-• First
improvement
overbetter
Christofides
NP-hard
to approximate
than 220/219
-7/5-approximation
Similar to Christofides,
but instead
of starting
with a minimum
algorithm
for subcubic
graphs
1.487-approximation
for cubic
graphs
• Christofides’
1.5-approximation
still
best connected
MST
they sample
onealgorithm
from thealgorithm
solution
of3-edge
Held-Karp
relaxation
-Conjecture:
Analysis requires several novel ideas, like structure of almost
• Held-Karp relaxation is conjectured to have integrality gap of 4/3
minimum cuts
subcubic 2-edge connected graphs has a tour of length 4n/3 -2/3
Our Results
A 1.461-approximation algorithm for graph-TSP
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
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)
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
Subcubic 2-edge-connected graph has a tour of length at most 4n/3 – 2/3
• Comments on general graphs
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
An edge
gadgets
is intoMmake
with graph
probability
cubic 1/3
2. Sample
Expected
perfect
size ofmatching
M U E is 4/3|E|
M so that each edge is taken with probability 1/3
application
Edmond’s
matching polytope)
3. A(Simple
2-VC graph
has aoftour
of sizeperfect
4/3|E|
3. Return graph with edge set
Using Matchings to Remove Edges
First Idea
1. Observation:
Expected sizeremoving
of returned
graph:
an Eulerian
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
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
We have
algorithm
that |R|as=before
2b -1 and
but |E|
return
= n-1 + b and thus
Result for Graphs of Max Degree 3
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)
• Matchings can be used to also remove edges
• Used structure to increase number of removable edges
“removable pairing”
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
1.0
1.02
1.04
1.06
1.08
1.1
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

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