1,208 research outputs found
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
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
Inapproximability of Maximum Biclique Problems, Minimum -Cut and Densest At-Least--Subgraph from the Small Set Expansion Hypothesis
The Small Set Expansion Hypothesis (SSEH) is a conjecture which roughly
states that it is NP-hard to distinguish between a graph with a small subset of
vertices whose edge expansion is almost zero and one in which all small subsets
of vertices have expansion almost one. In this work, we prove inapproximability
results for the following graph problems based on this hypothesis:
- Maximum Edge Biclique (MEB): given a bipartite graph , find a complete
bipartite subgraph of with maximum number of edges.
- Maximum Balanced Biclique (MBB): given a bipartite graph , find a
balanced complete bipartite subgraph of with maximum number of vertices.
- Minimum -Cut: given a weighted graph , find a set of edges with
minimum total weight whose removal partitions into connected
components.
- Densest At-Least--Subgraph (DALS): given a weighted graph , find a
set of at least vertices such that the induced subgraph on has
maximum density (the ratio between the total weight of edges and the number of
vertices).
We show that, assuming SSEH and NP BPP, no polynomial time
algorithm gives -approximation for MEB or MBB for every
constant . Moreover, assuming SSEH, we show that it is NP-hard
to approximate Minimum -Cut and DALS to within factor
of the optimum for every constant .
The ratios in our results are essentially tight since trivial algorithms give
-approximation to both MEB and MBB and efficient -approximation
algorithms are known for Minimum -Cut [SV95] and DALS [And07, KS09].
Our first result is proved by combining a technique developed by Raghavendra
et al. [RST12] to avoid locality of gadget reductions with a generalization of
Bansal and Khot's long code test [BK09] whereas our second result is shown via
elementary reductions.Comment: A preliminary version of this work will appear at ICALP 2017 under a
different title "Inapproximability of Maximum Edge Biclique, Maximum Balanced
Biclique and Minimum k-Cut from the Small Set Expansion Hypothesis
Beyond pairwise clustering
We consider the problem of clustering in domains where the affinity relations are not dyadic (pairwise), but rather triadic, tetradic or higher. The problem is an instance of the hypergraph partitioning problem. We propose a two-step algorithm for solving this problem. In the first step we use a novel scheme to approximate the hypergraph using a weighted graph. In the second step a spectral partitioning algorithm is used to partition the vertices of this graph. The algorithm is capable of handling hyperedges of all orders including order two, thus incorporating information of all orders simultaneously. We present a theoretical analysis that relates our algorithm to an existing hypergraph partitioning algorithm and explain the reasons for its superior performance. We report the performance of our algorithm on a variety of computer vision problems and compare it to several existing hypergraph partitioning algorithms
The Ising Partition Function: Zeros and Deterministic Approximation
We study the problem of approximating the partition function of the
ferromagnetic Ising model in graphs and hypergraphs. Our first result is a
deterministic approximation scheme (an FPTAS) for the partition function in
bounded degree graphs that is valid over the entire range of parameters
(the interaction) and (the external field), except for the case
(the "zero-field" case). A randomized algorithm (FPRAS)
for all graphs, and all , has long been known. Unlike most other
deterministic approximation algorithms for problems in statistical physics and
counting, our algorithm does not rely on the "decay of correlations" property.
Rather, we exploit and extend machinery developed recently by Barvinok, and
Patel and Regts, based on the location of the complex zeros of the partition
function, which can be seen as an algorithmic realization of the classical
Lee-Yang approach to phase transitions. Our approach extends to the more
general setting of the Ising model on hypergraphs of bounded degree and edge
size, where no previous algorithms (even randomized) were known for a wide
range of parameters. In order to achieve this extension, we establish a tight
version of the Lee-Yang theorem for the Ising model on hypergraphs, improving a
classical result of Suzuki and Fisher.Comment: clarified presentation of combinatorial arguments, added new results
on optimality of univariate Lee-Yang theorem
Approximate Hypergraph Coloring under Low-discrepancy and Related Promises
A hypergraph is said to be -colorable if its vertices can be colored
with colors so that no hyperedge is monochromatic. -colorability is a
fundamental property (called Property B) of hypergraphs and is extensively
studied in combinatorics. Algorithmically, however, given a -colorable
-uniform hypergraph, it is NP-hard to find a -coloring miscoloring fewer
than a fraction of hyperedges (which is achieved by a random
-coloring), and the best algorithms to color the hypergraph properly require
colors, approaching the trivial bound of as
increases.
In this work, we study the complexity of approximate hypergraph coloring, for
both the maximization (finding a -coloring with fewest miscolored edges) and
minimization (finding a proper coloring using fewest number of colors)
versions, when the input hypergraph is promised to have the following stronger
properties than -colorability:
(A) Low-discrepancy: If the hypergraph has discrepancy ,
we give an algorithm to color the it with colors.
However, for the maximization version, we prove NP-hardness of finding a
-coloring miscoloring a smaller than (resp. )
fraction of the hyperedges when (resp. ). Assuming
the UGC, we improve the latter hardness factor to for almost
discrepancy- hypergraphs.
(B) Rainbow colorability: If the hypergraph has a -coloring such
that each hyperedge is polychromatic with all these colors, we give a
-coloring algorithm that miscolors at most of the
hyperedges when , and complement this with a matching UG
hardness result showing that when , it is hard to even beat the
bound achieved by a random coloring.Comment: Approx 201
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