Hypergraph Matching by Linear and Semidefinite

Programming

Yves Brise, ETH Zürich, 20110329Based on 2010 paper by Chan and Lau

IntroductionVertex 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 (Halldorsson, 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).

Variants of k-set Packing

e1

e2e3

e4 e1e2e3e4

Bounded degreeindependend set

k-dimensional Matching akak-partite Matching

Local Search Algorithms

Unweighted Ratio k = 3

Hurkens, Schrijver, 1989k2 + 3

2 +

Weighted

Arkin, Hassin, 1997 k − 1 + 2 +

Candra, Halldorsson, 19992(k+1)

3 + 83 +

Berman, 2000k+12 + 2 +

Berman, Krysta, 2003 ∼ 2k3 + 2 +

Standard Linear Program

max

e∈E wexe

s.t.

ev xe ≤ 1 ∀v ∈ V

xe ≥ 0 ∀e ∈ E

(LP)

Theorem (Furedi, 1981):The integrality gap of LP is k − 1 + 1/k for unweighted hypergraphs.

Theorem (Furedi, Kahn, Seymour, 1993):The integrality gap of LP is k − 1 + 1/k for weighted hypergraphs.

But: Not algorithmic, does not imply approximation algorithm

Standard Linear Programmax

e∈E wexe

s.t.

ev xe ≤ 1 ∀v ∈ V

xe ≥ 0 ∀e ∈ E

(LP)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.

Best known!

Gets rid

of the .

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.

3. Use local ratio to compute

an approximation.

The same proof works for allvariants (weighted/unweighted,k-partite and k-uniform)

1. Basic Solution

max

e∈E wexe

s.t.

ev xe ≤ 1 ∀v ∈ V

xe ≥ 0 ∀e ∈ E

(LP)

LemmaIn a basic solution, there is a vertex of degree ≤ 2.

Fact from LP theory: for any basic LP solution,#non-zero variables ≤ #tight contraints

We can assume

x∗e > 0 for all edges

(otherwise delete edge).

⇒ Only vertex

constraints are tight.

1. Basic SolutionLemmaIn a basic solution, there is a vertex of degree ≤ 2.

Proof:

Recall that xe > 0 for all edges e ∈ E .

Let T be the set of tight vertices, i.e.,

ev xe = 1.

• 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 · |T |

• In any basic solution |E | ≤ |T | (LP fact), so |E | = |T |

deg(v) = 3

1. Basic Solution

⇒ Every edge consists of vertices in T only

v∈V deg(v) = 3 · |T |

Graph is 3-uniform, 3-regular,and 3-partite.

Constraints are not linearly independent, i.e., solution cannot be basic.

2. Small Fractional Neighborhood

xb

xa

Let v be a vertexof degree at most 2.v

And let b be the edgeof largest x-value, i.e.,xb ≥ xa.

(xb) + (≤ xb) + (≤ 1− xb) + (≤ 1− xb) ≤ 2

Pick edge b. This gives 2-approximation in the unweighted case.

Standard Linear Programmax

e∈E wexe

s.t.

ev xe ≤ 1 ∀v ∈ V

xe ≥ 0 ∀e ∈ E

(LP)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.

Best known!

Gets rid

of the .

The Bound is Tight

Projective plane of order k − 1

k = 3: Fano plane (order 2)

• k2 − k + 1 hyperedges

• Degree k on each vertex

• Pairwise intersecting

• Exists when k − 1 is

prime power

Integral solution: 1 (intersecting)

LP solution: 1/k on every edge gives k − 1 + 1/k

⇒ Integrality gap: k − 1 + 1/k

Fano Linear Program(Fano-LP)

max

e∈E wexe

s.t.

ev xe ≤ 1 ∀v ∈ V

e∈F xe ≤ 1 ∀F ∈ V 7,F Fano

xe ≥ 0 ∀e ∈ E

≤ 1

Theorem (Chan, Lau, 2010):The Fano-LP for unweighted 3-uniformhypergraphs has integrality gap exactly 2.

Proof idea:

• Show that any extreme

point solution of Fano-LP

contains no Fano plane.

• Apply result by Furedi.

Adams-Sherali HirarchyIdea: add more local constraints...

ev

xe − 1

i∈I

xi

j∈J

(1− xj) ≤ 0

where I and J are disjoint edge subsets, |I ∪ J| ≤

We add local constraints on edges after rounds

≤ 1

• No integrality gap for any set of ≤ edges

• e.g. Fano constraint will be added in round 7

linearizeand project

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 leastk − 2 after Ω(n/k3) rounds.

Sherali-Adams cannot yielda better polynomial timeapproximation algorithm.

Global Constraints (better LPs)Theorem (Chan, Lau, 2010):

There is an LP (of exponential size) with

integrality gap at most k+12 .

Theorem (Chan, Lau, 2010):

For k constant, there exists a polynomial

size LP with integrality gap at most k+12 .

Neither approach is algorithmic, no rounding algorithm provided.

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).

Add constraint x(K ) ≤ 1 for

all intersecting families.

Proof: relate to 2-optimal solution.

Proof: Replace intersecting family

constraints by kernel constraints.

Semidefinite Relaxation

Clique LPSDPOPT2-local OPT

≤ (k + 1)/2

max

i ,j∈V wi ·wj

s.t. wi ·wj = 0 ∀(i , j) ∈ En

i=1 w2i = 1 ∀e ∈ E

wi ∈ Rn ∀i ∈ V

Lovasz ϑ-function is an SDP formulation of the independent set problem.

Known facts:

• ϑ-function is a stronger

relaxation than the clique LP

Theorem (Chan, Lau, 2010):

Lovasz ϑ-function has integrality gap ≤ (k + 1)/2

Conclusion

• Rounding algorithm for SDP relaxtion

What would be interesting:

What we have seen (at least partly):

• Fano plane achieves worst case integrality gap for the standard LP.

• Algorithmic proof of integrality gap k − 1 + 1/k for k-uniform

matching, and k − 1 for k-partite matching for the standard LP.

• For constant k there exists LP with better integrality gap.

There exists a SDP with better integrality gap.

• Examples for SDP with integrality gap Ω(k/ log k) as implied by

hardness result.

• Strengthening by local constraints cannot do the trick.

Modified projective plane is bad for Sherali-Adams.

Local Search Algorithms

Local optimum (t-opt solution)

Greedy solution is 1-opt and k-approximate

Running time and performance depend on t

Idea: improve locally by adding ≤ t edges, remove fewer edges

t = 2

t = 3

3. Local Ratio Method

Lemma There is an ordering of the edgese1, ... , em s.t. x(N[ei ] ∩ ei+1, ... , em) ≤ 2

According to this ordering, split up the weight vectorw = w1 + w2 on small fractional neigborhoods.

Theorem (Bar-Yehuda, Bendel, Freund, Rawitz, 2004)If x∗ is r -approximate w.r.t. w1 and w.r.t. w2,then it is also r -approximate w.r.t. w .

Apply inductively, and wave hands...

Weighted Case

we = 80xe = 0.2

we = 2xe = 0.8

we = 1xe = 0.2

we = 10xe = 0.2

Pick green edge:Gain 2, lose (up to) 91

It’s not so easy in the weighted case...

Weighted Case

xe = 0.3

xe = 0.7

xe = 0.4

×0.3 ×0.3

×0.3×0.4

Idea: Write LP solution as a linear combination of matchings.

If sum of coefficients is small, by averaging, there is a matching of large weight.

Variants of k-set Packing

1 4

2 k

3 2

3 4

col j

row irow i, col j

row i, color k

col j, color k

e1

e2e3

e4 e1e2e3e4

Bounded degreeindependend set

k-dimensional Matching akak-partite Matching

Latin Squarecompletion