Integrality Gaps for Sparsest
Cut and Minimum Linear
Arrangement Problems
Nikhil R. Devanur
Subhash A. Khot
Rishi Saket
Nisheeth K. Vishnoi
Sparsest Cut Problem (SCP)
and b-Balanced Cuts (BSP)
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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
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An f(n)-approximation algorithm for SCP can
be applied iteratively to obtain O(f(n))
approximation algorithm for BSP
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[Leighton-Rao, JACM 1999] a linearprogramming relaxation produces O(log n)
approximation to SCP.
Linear Programming (LP) Review
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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
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Simplex method, Ellipsoid method
Approximation results by rounding fractional x
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Lower bound of the approximation factor is
sometimes called “integrality gap”
Semidefinite Programming (SDP)
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Find X that maximizes ∑cij∙xij
Subject to
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Equivalent to vector programming (VP)
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∑aijk∙xij = bk
X is a symmetric and positive semidefinite matrix
Find set of vectors V
X=VTV  xij=vi∙vj
Often SDP approximates better than LP
SDP references
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M. Goemans and D. Williamson
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D. Williamson
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Great lecture notes on SDP
Comprehensive website on SDP
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MAXCUT algorithm [1995]
Extensions to MAX3SAT and MAXDICUT
http://www-user.tu-chemnitz.de/~helmberg/semidef.html
List of papers maintained by Farid Alizadeh
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http://rutcor.rutgers.edu/~alizadeh/Sdppage/papers.html
Difference between LP and SDP
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LP
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SDP
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Useful dual problems
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Same
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Linear functions
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Non-linear functions
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Fractional solution
which has to be rounded
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Usually a vector solution
which has to be matched
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Simplex and ellipsoid
methods are poly-time
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Interior point or general
convex optimization
algorithms, also poly-time
but with large constants
SDP results for graph partitioning
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Arora, Rao, and Vazirani. Expander flows,
geometric embeddings and graph
partitioning. STOC 2004.
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An SDP relaxation of the problem gives
O(sqrt(log n)) approximation
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ARV-conjecture
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Standard SDP relaxation can give constant factor
approximation
Devanur, et al. results
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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
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Implies the bound for SCP
Similar bound for Minimum Linear
Arrangement Problem
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Find a bijection π : V -> {1, …, n} that minimizes
∑e=(u,v) |π(u)-π(v)|
SDP relaxation for SCP
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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)
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The following objective function and
constraints are equal to the sparsity value.
Algorithm for SCP
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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
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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.
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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)
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Value of the b-Balanced sparsest cut is given
by the following objective function
Questions and Comments