Hypergraph Matching by Linear and Semidefinite Programming Yves Brise, ETH Zürich, 20110329 Based on 2010 paper by Chan and Lau
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