17 research outputs found
On Coloring Resilient Graphs
We introduce a new notion of resilience for constraint satisfaction problems,
with the goal of more precisely determining the boundary between NP-hardness
and the existence of efficient algorithms for resilient instances. In
particular, we study -resiliently -colorable graphs, which are those
-colorable graphs that remain -colorable even after the addition of any
new edges. We prove lower bounds on the NP-hardness of coloring resiliently
colorable graphs, and provide an algorithm that colors sufficiently resilient
graphs. We also analyze the corresponding notion of resilience for -SAT.
This notion of resilience suggests an array of open questions for graph
coloring and other combinatorial problems.Comment: Appearing in MFCS 201
The Densest k-Subhypergraph Problem
The Densest -Subgraph (DS) problem, and its corresponding minimization
problem Smallest -Edge Subgraph (SES), have come to play a central role
in approximation algorithms. This is due both to their practical importance,
and their usefulness as a tool for solving and establishing approximation
bounds for other problems. These two problems are not well understood, and it
is widely believed that they do not an admit a subpolynomial approximation
ratio (although the best known hardness results do not rule this out).
In this paper we generalize both DS and SES from graphs to hypergraphs.
We consider the Densest -Subhypergraph problem (given a hypergraph ,
find a subset of vertices so as to maximize the number of
hyperedges contained in ) and define the Minimum -Union problem (given a
hypergraph, choose of the hyperedges so as to minimize the number of
vertices in their union). We focus in particular on the case where all
hyperedges have size 3, as this is the simplest non-graph setting. For this
case we provide an -approximation (for arbitrary constant )
for Densest -Subhypergraph and an -approximation for
Minimum -Union. We also give an -approximation for Minimum
-Union in general hypergraphs. Finally, we examine the interesting special
case of interval hypergraphs (instances where the vertices are a subset of the
natural numbers and the hyperedges are intervals of the line) and prove that
both problems admit an exact polynomial time solution on these instances.Comment: 21 page
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
Graph Isomorphism and the Lasserre Hierarchy
In this paper we show lower bounds for a certain large class of algorithms
solving the Graph Isomorphism problem, even on expander graph instances.
Spielman [25] shows an algorithm for isomorphism of strongly regular expander
graphs that runs in time exp(O(n^(1/3)) (this bound was recently improved to
expf O(n^(1/5) [5]). It has since been an open question to remove the
requirement that the graph be strongly regular. Recent algorithmic results show
that for many problems the Lasserre hierarchy works surprisingly well when the
underlying graph has expansion properties. Moreover, recent work of Atserias
and Maneva [3] shows that k rounds of the Lasserre hierarchy is a
generalization of the k-dimensional Weisfeiler-Lehman algorithm for Graph
Isomorphism. These two facts combined make the Lasserre hierarchy a good
candidate for solving graph isomorphism on expander graphs. Our main result
rules out this promising direction by showing that even Omega(n) rounds of the
Lasserre semidefinite program hierarchy fail to solve the Graph Isomorphism
problem even on expander graphs.Comment: 22 pages, 3 figures, submitted to CC
Faster SDP hierarchy solvers for local rounding algorithms
Convex relaxations based on different hierarchies of linear/semi-definite
programs have been used recently to devise approximation algorithms for various
optimization problems. The approximation guarantee of these algorithms improves
with the number of {\em rounds} in the hierarchy, though the complexity of
solving (or even writing down the solution for) the 'th level program grows
as where is the input size.
In this work, we observe that many of these algorithms are based on {\em
local} rounding procedures that only use a small part of the SDP solution (of
size instead of ). We give an algorithm to
find the requisite portion in time polynomial in its size. The challenge in
achieving this is that the required portion of the solution is not fixed a
priori but depends on other parts of the solution, sometimes in a complicated
iterative manner.
Our solver leads to time algorithms to obtain the same
guarantees in many cases as the earlier time algorithms based on
rounds of the Lasserre hierarchy. In particular, guarantees based on rounds can be realized in polynomial time.
We develop and describe our algorithm in a fairly general abstract framework.
The main technical tool in our work, which might be of independent interest in
convex optimization, is an efficient ellipsoid algorithm based separation
oracle for convex programs that can output a {\em certificate of infeasibility
with restricted support}. This is used in a recursive manner to find a sequence
of consistent points in nested convex bodies that "fools" local rounding
algorithms.Comment: 30 pages, 8 figure
Finding Pseudorandom Colorings of Pseudorandom Graphs
We consider the problem of recovering a planted pseudorandom 3-coloring in expanding and low threshold-rank graphs. Alon and Kahale [SICOMP 1997] gave a spectral algorithm to recover the coloring for a random graph with a planted random 3-coloring. We show that their analysis can be adapted to work when coloring is pseudorandom i.e., all color classes are of equal size and the size of the intersection of the neighborhood of a random vertex with each color class has small
variance. We also extend our results to partial colorings and low threshold-rank graphs to show the following:
* For graphs on n vertices with threshold-rank r, for which there exists a 3-coloring that is eps-pseudorandom and properly colors the induced subgraph on (1-gamma)n vertices, we show how to recover the coloring for (1 - O(gamma + eps)) n vertices in time (rn)^{O(r)}.
* For expanding graphs on n vertices, which admit a pseudorandom 3-coloring properly coloring all the vertices, we show how to recover such a coloring in polynomial time.
Our results are obtained by combining the method of Alon and Kahale, with eigenspace enumeration methods used for solving constraint satisfaction problems on low threshold-rank graphs