Integrality Gaps for Sparsest Cut and Minimum Linear Arrangement ProblemsNikhil R. Devanur

Subhash A. Khot Rishi SaketNisheeth K. Vishnoi

Sparsest Cut Problem (SCP) and b-Balanced Cuts (BSP) Given undirected graph G=(V,E), find subset

of nodes S, |S|<|V|/2 that minimizes

|E(S, V\S)| / |S|·|V\S| b-Balanced cuts ensure that S and V\S are at

least bn in size, where 0≤b≤1/2. b-Balanced Separator Problem (BSP)

satisfies both conditions

Previously known results

An f(n)-approximation algorithm for SCP can be applied iteratively to obtain O(f(n)) approximation algorithm for BSP

[Leighton-Rao, JACM 1999] a linear-programming relaxation produces O(log n) approximation to SCP.

Linear Programming (LP) Review Given matrix A, and vectors b and c, find x Maximize cT·x Subject to A·x≤b, x≥0 NP-hard to find optimal integral solution Relatively easy to find a fractional solution

Simplex method, Ellipsoid method Approximation results by rounding fractional x

Lower bound of the approximation factor is sometimes called “integrality gap”

Semidefinite Programming (SDP) Find X that maximizes ∑cij∙xij

Subject to ∑aijk∙xij = bk X is a symmetric and positive semidefinite matrix

Equivalent to vector programming (VP) Find set of vectors V X=VTV xij=vi∙vj

Often SDP approximates better than LP

SDP references

M. Goemans and D. Williamson MAXCUT algorithm [1995] Extensions to MAX3SAT and MAXDICUT

D. Williamson Great lecture notes on SDP

Comprehensive website on SDP http://www-user.tu-chemnitz.de/~helmberg/semidef.html

List of papers maintained by Farid Alizadeh http://rutcor.rutgers.edu/~alizadeh/Sdppage/papers.html

Difference between LP and SDP LP

Useful dual problems

Linear functions

Fractional solution which has to be rounded

Simplex and ellipsoid methods are poly-time

SDP Same

Non-linear functions

Usually a vector solution which has to be matched

Interior point or general convex optimization algorithms, also poly-time but with large constants

SDP results for graph partitioning Arora, Rao, and Vazirani. Expander flows,

geometric embeddings and graph partitioning. STOC 2004. An SDP relaxation of the problem gives

O(sqrt(log n)) approximation

ARV-conjecture Standard SDP relaxation can give constant factor

approximation

Devanur, et al. results

The standard SDP relaxations of BSP with the triangle inequality constraint have an integrality gap at least Ω(log log n)

Ω(log log n) lower bound for BSP Implies the bound for SCP

Similar bound for Minimum Linear Arrangement Problem Find a bijection π : V -> 1, …, n that minimizes

∑e=(u,v) |π(u)-π(v)|

SDP relaxation for SCP

How to encode any cut of the graph. If node i is left of the cut, set it equal to some

vector w. Otherwise, set it to –w.

SDP relaxation for SCP (con’t) The following objective function and

constraints are equal to the sparsity value.

Algorithm for SCP

Solve the SDP Choose w Obtain a plain orthogonal to w For all nodes i whose vi is on w side of the

plane, place them in S For all other nodes, place them in V\S

SDP relaxation for BSP - Main Theorem There are absolute constants c1, c2 > 0 such

that, for every large enough n there exists a multi-graph G(V;E) on n vertices, and a vector assignment i->vi for every i in V s.t. Every (1/3, 2/3) balanced cut must contain at least

c1∙|E|∙(log log n / log n) The vector assignment gives a low SDP objective

value < c2∙|E|∙(1/log n) Vectors are well-separated Δ-inequality on the vectors holds

SDP relaxation for BSP (con’t) Value of the b-Balanced sparsest cut is given

by the following objective function

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