Integrality Gaps for Sparsest Cut and Minimum Linear Arrangement Problems Nikhil R. Devanur Subhash...
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Transcript of Integrality Gaps for Sparsest Cut and Minimum Linear Arrangement Problems Nikhil R. Devanur Subhash...
![Page 1: Integrality Gaps for Sparsest Cut and Minimum Linear Arrangement Problems Nikhil R. Devanur Subhash A. Khot Rishi Saket Nisheeth K. Vishnoi.](https://reader036.fdocuments.in/reader036/viewer/2022083004/56649dc75503460f94abc120/html5/thumbnails/1.jpg)
Integrality Gaps for Sparsest Cut and Minimum Linear Arrangement ProblemsNikhil R. Devanur
Subhash A. Khot Rishi SaketNisheeth K. Vishnoi
![Page 2: Integrality Gaps for Sparsest Cut and Minimum Linear Arrangement Problems Nikhil R. Devanur Subhash A. Khot Rishi Saket Nisheeth K. Vishnoi.](https://reader036.fdocuments.in/reader036/viewer/2022083004/56649dc75503460f94abc120/html5/thumbnails/2.jpg)
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
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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.
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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”
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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
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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
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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
![Page 8: Integrality Gaps for Sparsest Cut and Minimum Linear Arrangement Problems Nikhil R. Devanur Subhash A. Khot Rishi Saket Nisheeth K. Vishnoi.](https://reader036.fdocuments.in/reader036/viewer/2022083004/56649dc75503460f94abc120/html5/thumbnails/8.jpg)
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
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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)|
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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.
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SDP relaxation for SCP (con’t) The following objective function and
constraints are equal to the sparsity value.
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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
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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
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SDP relaxation for BSP (con’t) Value of the b-Balanced sparsest cut is given
by the following objective function
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