Hypergraph Matching by Linear and Semidefinite Programming. Yves Brise, ETH Zürich, Based on 2010 paper by Chan and Lau

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1 Hypergraph Matching by Linear and Semidefinite Programming Yves Brise, ETH Zürich, Based on 2010 paper by Chan and Lau

2 Introduction Vertex set V : V = n Set of hyperedges E Hypergraph matching: find maximum subset of disjoint hyperedges. k-set packing: hypergraph matching on k-uniform hypergraphs. Theorem (Halldórsson, Kratochvil, Telle, 1998): Hypergraph matching can be approximated within a factor of Θ( n). Theorem (Hazan, Safra, Schwartz, 2003): k-set packing is hard to approximate within a factor of O(k/ log k).

3 Variants of k-set Packing k-dimensional Matching aka k-partite Matching e 3 e 4 e 1 e 1 e 3 e 2 e 4 e 2 Bounded degree independend set

4 Local Search Algorithms Unweighted Ratio k =3 Hurkens, Schrijver, 1989 k Weighted Arkin, Hassin, 1997 k Candra, Halldórsson, (k+1) Berman, 2000 k Berman, Krysta, k

5 Standard Linear Program (LP) max s.t. e E w ex e ev x e 1 v V x e 0 e E Theorem (Füredi, 1981): The integrality gap of LP is k 1+1/k for unweighted hypergraphs. Theorem (Füredi, Kahn, Seymour, 1993): The integrality gap of LP is k 1+1/k for weighted hypergraphs. But: Not algorithmic, does not imply approximation algorithm

6 Standard Linear Program (LP) max s.t. e E w ex e ev x e 1 x e 0 Best known! Gets rid of the. v V e E Theorem (Chan, Lau, 2010): (i) There is a k 1+1/k approximation algorithm for k-uniform hypergraph matching. (ii) There is a k 1 approximation algorithm for k-partite hypergraph matching. Corollary There is a 2-approximation algorithm for (weighted) 3-partite matching.

7 3-Partite Matching Corollary There is a 2-approximation algorithm for (weighted) 3-partite matching. 1. Compute basic solution. 2. Find a good ordering of the edges iteratively. The same proof works for all variants (weighted/unweighted, k-partite and k-uniform) 3. Use local ratio to compute an approximation.

8 1. Basic Solution (LP) max s.t. e E w ex e ev x e 1 x e 0 v V e E We can assume x e > 0 for all edges (otherwise delete edge). Only vertex constraints are tight. Fact from LP theory: for any basic LP solution, #non-zero variables #tight contraints Lemma In a basic solution, there is a vertex of degree 2.

9 1. Basic Solution Lemma In a basic solution, there is a vertex of degree 2. Proof: Let T be the set of tight vertices, i.e., ev x e = 1. Recall that x e > 0 for all edges e E. Suppose not, then v T deg(v) 3 T Since the graph is 3-uniform 3 E = v V deg(v) v T deg(v) =3 3 T In any basic solution E T (LP fact), so E = T

10 1. Basic Solution v V deg(v) =3 T Every edge consists of vertices in T only Graph is 3-uniform, 3-regular, and 3-partite. Constraints are not linearly independent, i.e., solution cannot be basic.

11 2. Small Fractional Neighborhood v x b Let v be a vertex of degree at most 2. x a And let b be the edge of largest x-value, i.e., x b x a. (x b )+( x b )+( 1 x b )+( 1 x b ) 2 Pick edge b. This gives 2-approximation in the unweighted case.

12 Standard Linear Program (LP) max s.t. e E w ex e ev x e 1 x e 0 Best known! Gets rid of the. v V e E Theorem (Chan, Lau, 2010): (i) There is a k 1+1/k approximation algorithm for k-uniform hypergraph matching. (ii) There is a k 1 approximation algorithm for k-partite hypergraph matching. Corollary There is a 2-approximation algorithm for (weighted) 3-partite matching.

13 The Bound is Tight Projective plane of order k 1 k 2 k + 1 hyperedges Degree k on each vertex Pairwise intersecting Exists when k 1is prime power k = 3: Fano plane (order 2) LP solution: 1/k on every edge gives k 1+1/k Integral solution: 1 (intersecting) Integrality gap: k 1+1/k

14 Fano Linear Program (Fano-LP) max s.t. e E w ex e ev x e 1 e F x e 1 x e 0 v V F V 7, F Fano e E 1 Proof idea: Theorem (Chan, Lau, 2010): The Fano-LP for unweighted 3-uniform hypergraphs has integrality gap exactly 2. Show that any extreme point solution of Fano-LP contains no Fano plane. Apply result by Füredi.

15 Adams-Sherali Hirarchy Idea: add more local constraints... We add local constraints on edges after rounds ev x e 1 i I x i (1 x j ) 0 j J linearize and project where I and J are disjoint edge subsets, I J No integrality gap for any set of edges e.g. Fano constraint will be added in round 7 1

16 Bad Example for Sherali-Adams A modified projective plane Still an intersecting family opt = 1 Fractional solution k 2 Theorem (Chan, Lau, 2010): The Sherali-Adams gap is at least k 2afterΩ(n/k 3 ) rounds. Sherali-Adams cannot yield a better polynomial time approximation algorithm.

17 Global Constraints (better LPs) Theorem (Chan, Lau, 2010): There is an LP (of exponential size) with integrality gap at most k+1 2. Add constraint x(k) 1for all intersecting families. Proof: relate to 2-optimal solution. Theorem (Chan, Lau, 2010): For k constant, there exists a polynomial size LP with integrality gap at most k+1 2. Proof: Replaceintersectingfamily constraints by kernel constraints. Theorem (Calczynska-Karlowicz, 1964): For every k there exists an f (k) s.t.every k-uniform intersecting family K has a kernel S V of size at most f (k). Neither approach is algorithmic, no rounding algorithm provided.

18 Semidefinite Relaxation (k + 1)/2 2-local OPT OPT SDP Clique LP Lovász ϑ-function is an SDP formulation of the independent set problem. max i,j V w i w j s.t. w i w j =0 (i, j) E n i=1 w2 i =1 e E Known facts: ϑ-function is a stronger relaxation than the clique LP w i R n i V Theorem (Chan, Lau, 2010): Lovász ϑ-function has integrality gap (k + 1)/2

19 Conclusion What we have seen (at least partly): Algorithmic proof of integrality gap k 1+1/k for k-uniform matching, and k 1fork-partite matching for the standard LP. Fano plane achieves worst case integrality gap for the standard LP. Strengthening by local constraints cannot do the trick. Modified projective plane is bad for Sherali-Adams. For constant k there exists LP with better integrality gap. There exists a SDP with better integrality gap. What would be interesting: Rounding algorithm for SDP relaxtion Examples for SDP with integrality gap Ω(k/ log k) asimpliedby hardness result.

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21 Local Search Algorithms Idea: improve locally by adding t edges, remove fewer edges t =2 t =3 Local optimum (t-opt solution) Greedy solution is 1-opt and k-approximate Running time and performance depend on t

22 3. Local Ratio Method Lemma There is an ordering of the edges e 1,...,e m s.t. x(n[e i ] {e i+1,...,e m }) 2 According to this ordering, split up the weight vector w = w 1 + w 2 on small fractional neigborhoods. Theorem (Bar-Yehuda, Bendel, Freund, Rawitz, 2004) If x is r-approximate w.r.t. w 1 and w.r.t. w 2, then it is also r-approximate w.r.t. w. Apply inductively, and wave hands...

23 Weighted Case It s not so easy in the weighted case... w e = 80 x e = 0.2 w e =2 x e = 0.8 w e = 10 x e = 0.2 w e =1 x e = 0.2 Pick green edge: Gain 2, lose (up to) 91

24 Weighted Case Idea: Write LP solution as a linear combination of matchings. x e = x e = 0.7 x e = If sum of coefficients is small, by averaging, there is a matching of large weight.

25 Variants of k-set Packing e 3 e 4 e 1 e 1 e 3 e 2 e 4 e 2 Bounded degree independend set k-dimensional Matching aka k-partite Matching col j row i k row i, col j row i, color k col j, color k Latin Square completion