Post on 25-Jul-2020
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