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

2. Roadmap Simultaneous Matchings  Problem definition  Motivation NP-Completeness APX-Completeness A 2/(k+1)-factor Approximation A Comment on the Polytope Conclusion/Open Problems

3. X-Perfect Bipartite Matchings Input: A bipartite graph D X

4. X-Perfect Bipartite Matchings Input: A bipartite graph D X Output: A matching saturating all nodes of X

5. Simultaneous Matchings Input: A bipartite graph D X X1X1 X2X2 A collection of k subsets of X

6. Simultaneous Matchings Output: A set M of edges such that … D X X1X1 X2X2 for each subset X i, the set is an X i -perfect matching.

7. Theoretical Motivation Berge, Edmonds [1950s, 1960s]: Classic results on matching.

8. Theoretical Motivation Berge, Edmonds [1950s, 1960s]: Classic results on matching. Since then: Half a century of research on nuances and variants of matchings.

9. 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 …

10. Practical Motivation Constraint programming: Variables X, values D. E represents ”possible assignments”. Values (D) Variables (X)

11. Practical Motivation An AllDifferent(V={v 1, v 2,…, v n }) constraint is a V-perfect matching problem.  An important and well-studied constraint. V Values (D) Variables (X)

12. Practical Motivation A constraint program with several AllDifferent constraints is a simultaneous matchings problem. V U Values (D) Variables (X)

13. NP-Hardness for k=2 By reduction from SET-PACKING: Input: sets S 1,…,S p and an integer c. Output: Are there c pairwise-disjoint sets? Example: Solution with c=2: No solution with c=3

14. The Reduction - Overview A value for each set A value for each element Gadgets c choice variables Gadgets ensure that only disjoint sets can be chosen

15. The Reduction - Overview A value for each set A value for each element Gadgets The two variable sets are ”red” and ”green”. Choice variables are in both sets.

16. The Gadgets Set value uv

17. The Gadgets Set value Choice variable u v If the set is not chosen, u and v are free.

18. The Gadgets Set value Choice variable u v If the set is chosen, u and v are assigned to variables which are both red and green.

19. Concatenated Gadgets Set value Choice variable u v If the set is chosen, u,u’ and v’ are assigned to variables which are both red and green. u’ v’

20. A full example Choice variables b U={a,b,c,d}. S 1 ={a,b} S 2 ={b,c} S 3 ={c,d} c=2 S1S1 S2S2 S3S3 Gadget for S 1 Gadget for S 2 Gadget for S 3 ad c

21. Complete Bipartite Graphs K=2: R RGRGG D There is a solution if and only if RG+max{R,G} D And larger k?

22. Complete Bipartite Graphs Node 3-coloring: Can the nodes of a graph be colored with three colors such that neighbors have different colors?

23. Complete Bipartite Graphs D= three colors Edge {u,v} is an AllDifferent(u,v) NP-hard even if |D|=3 and |X i |=2!

24. 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 X i, is a matching (not necessarily X i -perfect).

25. 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 X i, 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)

26. A Simple Approximation Algorithm s i = maximum weight of a matching in the subgraph induced by Also: So: I.e., max{s i } is a 1/k-factor approximation.

27. A Simple Approximation Algorithm s i = maximum weight of a matching in the subgraph induced by Also: So: I.e., max{s i } is a 1/k-factor approximation. Ok, not very impressive, but it does imply APX- completeness for any constant k.

28. A Better Approximation A ABB 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.

29. A Better Approximation A ABB 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.

30. With k constraint sets Let So Or: The maximum of them is a 2/(k+1)-approximation. X2X2 X3X3 X1X1 Y2Y2 Y1Y1 Y3Y3

31. 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.

32. 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.

33. Conclusion Better approximation factor? Huge gap: For k=2, upper bound = 3299/3300 and lower bound = 2/3. Interesting special cases?