lasserre hierarchy, higher eigenvalues and approximation schemes for graph partitioning and psd qip...

Lasserre Hierarchy,Higher Eigenvalues and

Approximation Schemes for Graph Partitioning and PSD QIP

Ali Kemal Sinop(joint work with Venkatesan Guruswami)

Carnegie Mellon University

Outline

• Introduction– Sample Problem: Minimum Bisection– Approximation Algorithms – Our Motivation and Results Overview

• Results– Graph Spectrum– Related Work and Our Results

• Case Study: Minimum Bisection– Lasserre Hierarchy Formulation– Rounding Algorithm– Analysis 210:13 PM

Minimum Bisection• Given graph G=(V,E,W), find subset of size n/2

which cuts as few edges as possible.

• Canonical problem for graph partitioning by allowing arbitrary size:– Small Set Expansion (weight each node by its degree)– Uniform Sparsest Cut (try out all partition sizes in

small increments)– Etc…

• NP-hard. 3

1 2 3 4 1 2 3 4Cost=2

µ

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Approximation Algorithms

• Find an α-factor approximation. – If minimum cost = OPT, • Algorithm always finds a solution with value ≤ α OPT.

• (This work) Round a convex relaxation.

OPT Algorithm α OPT0 Relaxation

4

1

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Motivation

• For many graph partitioning problems (including minimum bisection), huge gap between hardness and approximation results.

• Best known algorithms have factor • Whereas no 1.1 factor hardness is known.• We want to close the gap.

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Our Results: Overview

• For graph partitioning problems including:– Minimum bisection,– Small set expansion,– Uniform sparsest cut,– Minimum uncut,– Their k-way generalizations, etc…

• We give approximation schemes whose running time is dependent on graph spectrum.

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Outline

• Introduction– Sample Problem: Minimum Bisection– Approximation Algorithms – Our Motivation and Results Overview

• Results– Graph Spectrum– Related Work and Our Results

• Case Study: Minimum Bisection– Lasserre Hierarchy Formulation– Rounding Algorithm– Analysis 710:13 PM

Graph Spectrum and Eigenvalues

1 2 3 4

rows and cols indexed by V

8

λ2: Measures expansion of the graph through Cheeger’s inequality .λr: Related to small set expansion [Arora, Barak, Steurer’10], [Gharan, Trevisan’11].

0 = λ1 ≤ λ2 ≤ … ≤ λn ≤2 and λ1 + λ2 + … + λn = n,

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Related Previous Work

• (Minimization form of) Unique Games (k-labeling with permutation constraints):– [AKKSTV’08], [Makarychev, Makarychev’10]

Constant factor approximation for Unique Games on expanders in polynomial time.

– [Kolla’10] Constant factor when λr is large.

• [Arora, Barak, Steurer’10] – For Unique Games and Small Set Expansion,

factor in time – For Sparsest Cut, factor assuming 910:13 PM

Our Results (1)• In time we obtain

• Why approximation scheme?– 0 = λ1 ≤ λ2 ≤ … ≤ λn ≤2 and λ1 + λ2 + … + λn = n,

10

Minimum Bisection*Small Set Expansion*Uniform Sparsest CutTheir k-way generalizations*

Independent Set

* Satisfies constraints within factor of

For r=n, λr >1, λn-r <1

Minimum Uncut

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Our Results for Unique Games

• For Unique Games, a direct bound will involve spectrum of lifted graph, whereas we want to bound using spectrum of original graph.– We give a simple embedding and work directly on

the original graph.• We obtain factor in time .• Concurrent to our work, [Barak, Steurer,

Raghavendra’11] obtained factor in time using a similar rounding.

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Outline

• Introduction– Sample Problem: Minimum Bisection– Approximation Algorithms – Our Motivation and Results Overview

• Results– Graph Spectrum– Related Work and Our Results

• Case Study: Minimum Bisection– Lasserre Hierarchy– Rounding Algorithm– Analysis 1210:13 PM

Case Study: Minimum Bisection

• We will present an approximation algorithm for minimum bisection problem on d-regular unweighted graphs.

• We will show that it achieves factor .• Obtaining factor requires some additional

ideas.

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Lasserre Hierarchy

• Basic idea: Rounding a convex relaxation of minimum bisection.

• [Lasserre’01] Strongest known SDP-relaxation.– (Relaxation of) For each subset S of size ≤ r and

each possible labeling of S, – An indicator vector which is 1 if S is labeled with f – 0 else.

• And all implied consistency constraints.

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Previous Work on Lasserre Hierarchy

• Few algorithmic results known before, including:– [Chlamtac’07], [Chlamtac, Singh’08] nΩ(1) approximation for 3-coloring and

independent set on 3-uniform hypergraphs, – [Karlin, Mathieu, Nguyen’10] (1+1/r) approximation of knapsack for r-rounds.

• Known integrality gaps are:– [Schoenebeck’08], [Tulsiani’09] Most NP-hardness results

carry over to Ω(n) rounds of Lasserre.– [Guruswami, S, Zhou’11] Factor (1+α) integrality gap for

Ω(n) rounds of min-bisection and max-cut.• Not ruled out yet:

“5-rounds of Lasserre relaxation disproves Unique Games Conjecture.”

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Why So Few Positive Results?

• For regular SDP [Goemans, Williamson’95] showed that with hyperplane rounding:

• Prior to our work, no analogue for Lasserre solution.

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Consistency

Lasserre Relaxation for Minimum Bisection

17

• Relaxation for consistent labeling of all subsets of size < r:

Marginalization

Distribution

Partition SIze

Cut cost

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Rounding Algorithm

• Choose S with probability – [Deshpande, Rademacher, Vempala, Wang’06] Volume sampling.

• Label S by choosing f with probability .• Propagate to other nodes:– For each node v,• With probability include v in U.

– Inspired by [AKKSTTV’08] which used propagation from a single node chosen uniformly at random.

• Return U.

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Analysis

• Partition Size– Each node is chosen into U independently– By Chernoff, with high probability

• Number of Edges Cut– After arithmetization, we have the following

bound:

19

Normalized Vector for xS(f)

≤ OPT19

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Matrix ΠS

• Remember xS(f)f are orthogonal. is a projection matrix onto

spanxS(f)f .

• For any

20

Let PS be the corresponding projection

matrix.

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Low Rank Matrix Reconstruction

• The final bound is:

• For any S of size r this is lower bounded by:

• [Guruswami, S’11] Volume sampling columns yield

– And this bound is tight.

best rank-r approximation of X

best rank-r approximation of X

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Relating Reconstruction Error to Graph Spectrum

• Best rank-r approximation is obtained by top r-eigenvectors.

• Using Courant-Fischer theorem,

• Therefore

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Summary

• Gave a randomized rounding algorithm based on propagation from a seed set S so that:

• Related choosing S to low rank matrix reconstruction error.

• Bounded low rank matrix reconstruction error in terms of λr.

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Questions?

• Thanks.

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