Simultaneous Matchings Irit Katriel - BRICS, U of Aarhus, Denmark Joint work with Khaled Elabssioni...
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Transcript of Simultaneous Matchings Irit Katriel - BRICS, U of Aarhus, Denmark Joint work with Khaled Elabssioni...
Simultaneous Matchings
Irit Katriel - BRICS, U of Aarhus, Denmark
Joint work with
Khaled Elabssioni and Martin Kutz - MPI, Germany
Meena Mahajan - IMSC, India
Roadmap
Simultaneous Matchings Problem definition Motivation
NP-Completeness APX-Completeness A 2/(k+1)-factor Approximation A Comment on the Polytope Conclusion/Open Problems
X-Perfect Bipartite MatchingsInput: A bipartite graph E)D,(XG
D
X
X-Perfect Bipartite MatchingsInput: A bipartite graph E)D,(XG
D
X
Output: A matching saturating all nodes of X
Simultaneous Matchings
Input: A bipartite graph E)D,(XG
D
X
X1
X2
A collection of k subsets of X
Simultaneous Matchings
Output: A set M of edges such that …
D
X
X1
X2
for each subset Xi, the set
is an Xi-perfect matching.
D)(XM i
Theoretical Motivation
Berge, Edmonds [1950s, 1960s]: Classic results on matching.
Theoretical Motivation
Berge, Edmonds [1950s, 1960s]: Classic results on matching.
Since then:Half a century of research on nuances and variants of matchings.
Theoretical Motivation
Berge, Edmonds [1950s, 1960s]: Classic results on matching.
Since then:Half a century of research on nuances and variants of matching.
Problem variants: Maximum Weight Matching, Minimum Weight Perfect Matching, Stable Matchings, Rank-Maximal Matchings, Popular Matchings …
Special cases: Planar, Bipartite, Convex Bipartite … Models of computation: Sequential, Parallel …
Practical Motivation Constraint programming:
Variables X, values D.E represents ”possible assignments”.
Values (D)
Variables (X)
Practical Motivation
An AllDifferent(V={v1, v2,…, vn}) constraint is a V-perfect matching problem. An important and well-studied constraint.
V
Values (D)
Variables (X)
Practical Motivation A constraint program with several AllDifferent
constraints is a simultaneous matchings problem.
VU
Values (D)
Variables (X)
NP-Hardness for k=2
By reduction from SET-PACKING:Input: sets S1,…,Sp and an integer c. Output: Are there c pairwise-disjoint sets?
Example:
Solution with c=2:
No solution with c=3
The Reduction - Overview
A value foreach set
A value foreach element
Gadgets
c choice variables Gadgets ensure that onlydisjoint sets can be chosen
The Reduction - Overview
A value foreach set
A value foreach element
Gadgets
The two variable sets are ”red” and ”green”. Choice variables are in both sets.
The Gadgets
Set value u v
The Gadgets
Set value
Choicevariable
u v
If the set is not chosen, u and v are free.
The Gadgets
Set value
Choicevariable
u v
If the set is chosen, u and v are assigned to variables which are both red and green.
Concatenated Gadgets
Set value
Choicevariable
u v
If the set is chosen, u,u’ and v’ are assigned to variables which are both red and green.
u’ v’
A full example
Choicevariables
b
U={a,b,c,d}. S1={a,b} S2={b,c} S3={c,d} c=2
S1 S2 S3
Gadget for S1 Gadget for S2 Gadget for S3
a dc
Complete Bipartite Graphs
K=2:
R RG G
D
There is a solution if and only if RG+max{R,G} D
And larger k?
Complete Bipartite Graphs
Node 3-coloring: Can the nodes of a graph be colored with three colors such that neighbors have different colors?
Complete Bipartite Graphs
D= three colorsEdge {u,v} is an AllDifferent(u,v)
NP-hard even if|D|=3 and |Xi|=2!
Optimisation Version
Input: as before, with weight on the edges. Output: (the size of) a maximum weight subset M of
the edges such that For each constraint set Xi,
is a matching (not necessarily X i-perfect). D)(XM i
Optimisation Version: APX-hardness Input: as before, with weight on the edges. Output: (the size of) a maximum weight subset M of
the edges such that For each constraint set Xi,
is a matching (not necessarily X i-perfect).
A simple modification of the reduction we used is an approximation-preserving reduction.
For k=2, inapproximable within better than 1-1/3300 unless P=NP. Using 99/100 hardness factor of 3-SET-PACKING(2)
D)(XM i
A Simple Approximation Algorithm si= maximum weight of a matching in the subgraph
induced by Also:
So:
I.e., max{si} is a 1/k-factor approximation.
.DXi
isOpt i Opt si
}max{sk sOpt }max{s iii
A Simple Approximation Algorithm si= maximum weight of a matching in the subgraph
induced by Also:
So:
I.e., max{si} is a 1/k-factor approximation. Ok, not very impressive, but it does imply APX-
completeness for any constant k.
.DXi
isOpt i Opt si
}max{sk sOpt }max{s iii
A Better Approximation
A AB B
• We computed optimum for A+AB and for AB+B.
• We can also compute optimum for A+B (ignore intersection).
• OPT(A+AB)+OPT(AB+B)+OPT(A+B) is at least 2 OPT.
• Maximum between them is a 2/3-factor approximation.
A Better Approximation
A AB B
• We computed optimum for A+AB and for AB+B.
• We can also compute optimum for A+B (ignore intersection).
• OPT(A+AB)+OPT(AB+B)+OPT(A+B) is at least 2 OPT.
• Maximum between them is a 2/3-factor approximation.
With k constraint sets
2OPT)YOPT()OPT(X ii
• Let
• SoOr:The maximum of them is a 2/(k+1)-approximation.
jijii X\XY
X2 X3
X1
Y2
Y1
Y3
Can We Go Further? We generalize our approach and show that we
cannot. Sketch: There is a linear program such that the value of its optimal
solution is the approximation ratio achieved. There is a feasible solution to the dual with value 2/(k+1).
Note: Most of the details are not in the proceedings version. See full version on our websites.
A Comment on the Polytope
Bipartite matching polytope: Integral vertices. General matching polytope: Half-integral
vertices.
We show (by example) that neither property carries over to the simultaneous matchings polytope.
Conclusion
Better approximation factor?Huge gap: For k=2, upper bound = 3299/3300 and lower bound = 2/3.
Interesting special cases?