Simultaneous Matchings Irit Katriel - BRICS, U of Aarhus, Denmark Joint work with Khaled Elabssioni and Martin Kutz - MPI, Germany Meena Mahajan - IMSC,

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?