813 research outputs found
Cut-Matching Games on Directed Graphs
We give O(log^2 n)-approximation algorithm based on the cut-matching
framework of [10, 13, 14] for computing the sparsest cut on directed graphs.
Our algorithm uses only O(log^2 n) single commodity max-flow computations and
thus breaks the multicommodity-flow barrier for computing the sparsest cut on
directed graph
Approximation Algorithms for Hypergraph Small Set Expansion and Small Set Vertex Expansion
The expansion of a hypergraph, a natural extension of the notion of expansion
in graphs, is defined as the minimum over all cuts in the hypergraph of the
ratio of the number of the hyperedges cut to the size of the smaller side of
the cut. We study the Hypergraph Small Set Expansion problem, which, for a
parameter , asks to compute the cut having the least
expansion while having at most fraction of the vertices on the smaller
side of the cut. We present two algorithms. Our first algorithm gives an
approximation. The second algorithm finds
a set with expansion in a --uniform hypergraph with maximum degree
(where is the expansion of the optimal solution).
Using these results, we also obtain algorithms for the Small Set Vertex
Expansion problem: we get an
approximation algorithm and an algorithm that finds a set with vertex expansion
(where is the vertex expansion of the optimal
solution).
For , Hypergraph Small Set Expansion is equivalent to the
hypergraph expansion problem. In this case, our approximation factor of
for expansion in hypergraphs matches the corresponding
approximation factor for expansion in graphs due to ARV
Approximation Algorithms for Partially Colorable Graphs
Graph coloring problems are a central topic of study in the theory of algorithms. We study the problem of partially coloring partially colorable graphs. For alpha = alpha |V| such that the graph induced on S is k-colorable. Partial k-colorability is a more robust structural property of a graph than k-colorability. For graphs that arise in practice, partial k-colorability might be a better notion to use than k-colorability, since data arising in practice often contains various forms of noise.
We give a polynomial time algorithm that takes as input a (1 - epsilon)-partially 3-colorable graph G and a constant gamma in [epsilon, 1/10], and colors a (1 - epsilon/gamma) fraction of the vertices using O~(n^{0.25 + O(gamma^{1/2})}) colors. We also study natural semi-random families of instances of partially 3-colorable graphs and partially 2-colorable graphs, and give stronger bi-criteria approximation guarantees for these family of instances
Many Sparse Cuts via Higher Eigenvalues
Cheeger's fundamental inequality states that any edge-weighted graph has a
vertex subset such that its expansion (a.k.a. conductance) is bounded as
follows: \phi(S) \defeq \frac{w(S,\bar{S})}{\min \set{w(S), w(\bar{S})}}
\leq 2\sqrt{\lambda_2} where is the total edge weight of a subset or a
cut and is the second smallest eigenvalue of the normalized
Laplacian of the graph. Here we prove the following natural generalization: for
any integer , there exist disjoint subsets ,
such that where
is the smallest eigenvalue of the normalized Laplacian and
are suitable absolute constants. Our proof is via a polynomial-time
algorithm to find such subsets, consisting of a spectral projection and a
randomized rounding. As a consequence, we get the same upper bound for the
small set expansion problem, namely for any , there is a subset whose
weight is at most a \bigO(1/k) fraction of the total weight and . Both results are the best possible up to constant
factors.
The underlying algorithmic problem, namely finding subsets such that the
maximum expansion is minimized, besides extending sparse cuts to more than one
subset, appears to be a natural clustering problem in its own right
Planted Models for the Densest k-Subgraph Problem
Given an undirected graph G, the Densest k-subgraph problem (DkS) asks to compute a set S ? V of cardinality |S| ? k such that the weight of edges inside S is maximized. This is a fundamental NP-hard problem whose approximability, inspite of many decades of research, is yet to be settled. The current best known approximation algorithm due to Bhaskara et al. (2010) computes a ?(n^{1/4 + ?}) approximation in time n^{?(1/?)}, for any ? > 0.
We ask what are some "easier" instances of this problem? We propose some natural semi-random models of instances with a planted dense subgraph, and study approximation algorithms for computing the densest subgraph in them. These models are inspired by the semi-random models of instances studied for various other graph problems such as the independent set problem, graph partitioning problems etc. For a large range of parameters of these models, we get significantly better approximation factors for the Densest k-subgraph problem. Moreover, our algorithm recovers a large part of the planted solution
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