1,121 research outputs found
Exploiting chordal structure in polynomial ideals: a Gr\"obner bases approach
Chordal structure and bounded treewidth allow for efficient computation in
numerical linear algebra, graphical models, constraint satisfaction and many
other areas. In this paper, we begin the study of how to exploit chordal
structure in computational algebraic geometry, and in particular, for solving
polynomial systems. The structure of a system of polynomial equations can be
described in terms of a graph. By carefully exploiting the properties of this
graph (in particular, its chordal completions), more efficient algorithms can
be developed. To this end, we develop a new technique, which we refer to as
chordal elimination, that relies on elimination theory and Gr\"obner bases. By
maintaining graph structure throughout the process, chordal elimination can
outperform standard Gr\"obner basis algorithms in many cases. The reason is
that all computations are done on "smaller" rings, of size equal to the
treewidth of the graph. In particular, for a restricted class of ideals, the
computational complexity is linear in the number of variables. Chordal
structure arises in many relevant applications. We demonstrate the suitability
of our methods in examples from graph colorings, cryptography, sensor
localization and differential equations.Comment: 40 pages, 5 figure
Highly parallel sparse Cholesky factorization
Several fine grained parallel algorithms were developed and compared to compute the Cholesky factorization of a sparse matrix. The experimental implementations are on the Connection Machine, a distributed memory SIMD machine whose programming model conceptually supplies one processor per data element. In contrast to special purpose algorithms in which the matrix structure conforms to the connection structure of the machine, the focus is on matrices with arbitrary sparsity structure. The most promising algorithm is one whose inner loop performs several dense factorizations simultaneously on a 2-D grid of processors. Virtually any massively parallel dense factorization algorithm can be used as the key subroutine. The sparse code attains execution rates comparable to those of the dense subroutine. Although at present architectural limitations prevent the dense factorization from realizing its potential efficiency, it is concluded that a regular data parallel architecture can be used efficiently to solve arbitrarily structured sparse problems. A performance model is also presented and it is used to analyze the algorithms
A Tutorial on Clique Problems in Communications and Signal Processing
Since its first use by Euler on the problem of the seven bridges of
K\"onigsberg, graph theory has shown excellent abilities in solving and
unveiling the properties of multiple discrete optimization problems. The study
of the structure of some integer programs reveals equivalence with graph theory
problems making a large body of the literature readily available for solving
and characterizing the complexity of these problems. This tutorial presents a
framework for utilizing a particular graph theory problem, known as the clique
problem, for solving communications and signal processing problems. In
particular, the paper aims to illustrate the structural properties of integer
programs that can be formulated as clique problems through multiple examples in
communications and signal processing. To that end, the first part of the
tutorial provides various optimal and heuristic solutions for the maximum
clique, maximum weight clique, and -clique problems. The tutorial, further,
illustrates the use of the clique formulation through numerous contemporary
examples in communications and signal processing, mainly in maximum access for
non-orthogonal multiple access networks, throughput maximization using index
and instantly decodable network coding, collision-free radio frequency
identification networks, and resource allocation in cloud-radio access
networks. Finally, the tutorial sheds light on the recent advances of such
applications, and provides technical insights on ways of dealing with mixed
discrete-continuous optimization problems
OV Graphs Are (Probably) Hard Instances
© Josh Alman and Virginia Vassilevska Williams. A graph G on n nodes is an Orthogonal Vectors (OV) graph of dimension d if there are vectors v1, . . ., vn ∈ {0, 1}d such that nodes i and j are adjacent in G if and only if hvi, vji = 0 over Z. In this paper, we study a number of basic graph algorithm problems, except where one is given as input the vectors defining an OV graph instead of a general graph. We show that for each of the following problems, an algorithm solving it faster on such OV graphs G of dimension only d = O(log n) than in the general case would refute a plausible conjecture about the time required to solve sparse MAX-k-SAT instances: Determining whether G contains a triangle. More generally, determining whether G contains a directed k-cycle for any k ≥ 3. Computing the square of the adjacency matrix of G over Z or F2. Maintaining the shortest distance between two fixed nodes of G, or whether G has a perfect matching, when G is a dynamically updating OV graph. We also prove some complementary results about OV graphs. We show that any problem which is NP-hard on constant-degree graphs is also NP-hard on OV graphs of dimension O(log n), and we give two problems which can be solved faster on OV graphs than in general: Maximum Clique, and Online Matrix-Vector Multiplication
Reconfiguration in bounded bandwidth and treedepth
We show that several reconfiguration problems known to be PSPACE-complete
remain so even when limited to graphs of bounded bandwidth. The essential step
is noticing the similarity to very limited string rewriting systems, whose
ability to directly simulate Turing Machines is classically known. This
resolves a question posed open in [Bonsma P., 2012]. On the other hand, we show
that a large class of reconfiguration problems becomes tractable on graphs of
bounded treedepth, and that this result is in some sense tight.Comment: 14 page
Algebraic Methods in the Congested Clique
In this work, we use algebraic methods for studying distance computation and
subgraph detection tasks in the congested clique model. Specifically, we adapt
parallel matrix multiplication implementations to the congested clique,
obtaining an round matrix multiplication algorithm, where
is the exponent of matrix multiplication. In conjunction
with known techniques from centralised algorithmics, this gives significant
improvements over previous best upper bounds in the congested clique model. The
highlight results include:
-- triangle and 4-cycle counting in rounds, improving upon the
triangle detection algorithm of Dolev et al. [DISC 2012],
-- a -approximation of all-pairs shortest paths in
rounds, improving upon the -round -approximation algorithm of Nanongkai [STOC 2014], and
-- computing the girth in rounds, which is the first
non-trivial solution in this model.
In addition, we present a novel constant-round combinatorial algorithm for
detecting 4-cycles.Comment: This is work is a merger of arxiv:1412.2109 and arxiv:1412.266
- …