Transcript Maximum Coverage with Group Budget Constraints and

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)
1.5 approx
[Papadimitriou, Vempala 2006]
[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 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 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 = 2n
Which edges can we remove ?
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 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
G : 2 connected
R : subset of edges
PµRXR
•
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 Pairs
G : 2 connected
R : subset of edges
PµRXR
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 matching
and with prob. 2/3 we can remove 2 edges
for each pair in P.
Finding Good Removable Pairs
Can start with any DFS Tree.
Finding Good Removable Pairs
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 Pairs
Given a DFS Tree,
Make it 2-connected by adding as few
back-edges as possible. 4/3|E|-2/3|R|
The back-edges should be “well-distributed”
for many tree-edges, there should be
corresponding back-edges.
Some Notation
v
v
in-vertices
w
Sub-divide tree edges.
|R|=i 2 I 0 or B(i) +1
w
i
Circulation Problem
v
in-vertices
(0,1)
w
i
(1,1)
Edges with non-zero (integral) flow form
a 2-connected graph.
Min-cost Circulation Problem
v
in-vertices
(0,1)
w
i
(1,1)
Cost of flow=i 2 I min(0, f(B(i))-1)
Removable Pairs from Circulation
v
in-vertices
(0,1)
w
i
C=|R|-2|P|
E=n+|R|-|P|
4/3E-2/3R=4/3n+2/3C
(1,1)
Main Theorem
v
in-vertices
(0,1)
w
i
(1,1)
Given a circulation of cost C, there is a
TSP tour of cost at most 4/3n + 2/3C
2-connected sub-cubic graphs
v
Send 1 unit of flow on all back-edges.
C=0
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
L
LP Value = 3L, Opt = 4L
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
0.5
0.3
w
0.9
0.2
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 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
v
0.5
At most n back-edges.
w
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
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
Cost of circulation is at most
6(1  2)n  (4 2  3) xe
e
Open Problems
4/3 approx for general graphs.
Better than 3/2 for weighted graphs.