3,848 research outputs found
Algorithmic Applications of Baur-Strassen's Theorem: Shortest Cycles, Diameter and Matchings
Consider a directed or an undirected graph with integral edge weights from
the set [-W, W], that does not contain negative weight cycles. In this paper,
we introduce a general framework for solving problems on such graphs using
matrix multiplication. The framework is based on the usage of Baur-Strassen's
theorem and of Strojohann's determinant algorithm. It allows us to give new and
simple solutions to the following problems:
* Finding Shortest Cycles -- We give a simple \tilde{O}(Wn^{\omega}) time
algorithm for finding shortest cycles in undirected and directed graphs. For
directed graphs (and undirected graphs with non-negative weights) this matches
the time bounds obtained in 2011 by Roditty and Vassilevska-Williams. On the
other hand, no algorithm working in \tilde{O}(Wn^{\omega}) time was previously
known for undirected graphs with negative weights. Furthermore our algorithm
for a given directed or undirected graph detects whether it contains a negative
weight cycle within the same running time.
* Computing Diameter and Radius -- We give a simple \tilde{O}(Wn^{\omega})
time algorithm for computing a diameter and radius of an undirected or directed
graphs. To the best of our knowledge no algorithm with this running time was
known for undirected graphs with negative weights.
* Finding Minimum Weight Perfect Matchings -- We present an
\tilde{O}(Wn^{\omega}) time algorithm for finding minimum weight perfect
matchings in undirected graphs. This resolves an open problem posted by
Sankowski in 2006, who presented such an algorithm but only in the case of
bipartite graphs.
In order to solve minimum weight perfect matching problem we develop a novel
combinatorial interpretation of the dual solution which sheds new light on this
problem. Such a combinatorial interpretation was not know previously, and is of
independent interest.Comment: To appear in FOCS 201
NC Algorithms for Computing a Perfect Matching and a Maximum Flow in One-Crossing-Minor-Free Graphs
In 1988, Vazirani gave an NC algorithm for computing the number of perfect
matchings in -minor-free graphs by building on Kasteleyn's scheme for
planar graphs, and stated that this "opens up the possibility of obtaining an
NC algorithm for finding a perfect matching in -free graphs." In this
paper, we finally settle this 30-year-old open problem. Building on recent NC
algorithms for planar and bounded-genus perfect matching by Anari and Vazirani
and later by Sankowski, we obtain NC algorithms for perfect matching in any
minor-closed graph family that forbids a one-crossing graph. This family
includes several well-studied graph families including the -minor-free
graphs and -minor-free graphs. Graphs in these families not only have
unbounded genus, but can have genus as high as . Our method applies as
well to several other problems related to perfect matching. In particular, we
obtain NC algorithms for the following problems in any family of graphs (or
networks) with a one-crossing forbidden minor:
Determining whether a given graph has a perfect matching and if so,
finding one.
Finding a minimum weight perfect matching in the graph, assuming
that the edge weights are polynomially bounded.
Finding a maximum -flow in the network, with arbitrary
capacities.
The main new idea enabling our results is the definition and use of
matching-mimicking networks, small replacement networks that behave the same,
with respect to matching problems involving a fixed set of terminals, as the
larger network they replace.Comment: 21 pages, 6 figure
Deterministically Isolating a Perfect Matching in Bipartite Planar Graphs
We present a deterministic way of assigning small (log bit) weights to the
edges of a bipartite planar graph so that the minimum weight perfect matching
becomes unique. The isolation lemma as described in (Mulmuley et al. 1987)
achieves the same for general graphs using a randomized weighting scheme,
whereas we can do it deterministically when restricted to bipartite planar
graphs. As a consequence, we reduce both decision and construction versions of
the matching problem to testing whether a matrix is singular, under the promise
that its determinant is 0 or 1, thus obtaining a highly parallel SPL algorithm
for bipartite planar graphs. This improves the earlier known bounds of
non-uniform SPL by (Allender et al. 1999) and by (Miller and Naor 1995,
Mahajan and Varadarajan 2000). It also rekindles the hope of obtaining a
deterministic parallel algorithm for constructing a perfect matching in
non-bipartite planar graphs, which has been open for a long time. Our
techniques are elementary and simple
Matching Is as Easy as the Decision Problem, in the NC Model
Is matching in NC, i.e., is there a deterministic fast parallel algorithm for
it? This has been an outstanding open question in TCS for over three decades,
ever since the discovery of randomized NC matching algorithms [KUW85, MVV87].
Over the last five years, the theoretical computer science community has
launched a relentless attack on this question, leading to the discovery of
several powerful ideas. We give what appears to be the culmination of this line
of work: An NC algorithm for finding a minimum-weight perfect matching in a
general graph with polynomially bounded edge weights, provided it is given an
oracle for the decision problem. Consequently, for settling the main open
problem, it suffices to obtain an NC algorithm for the decision problem. We
believe this new fact has qualitatively changed the nature of this open
problem.
All known efficient matching algorithms for general graphs follow one of two
approaches: given by Edmonds [Edm65] and Lov\'asz [Lov79]. Our oracle-based
algorithm follows a new approach and uses many of the ideas discovered in the
last five years.
The difficulty of obtaining an NC perfect matching algorithm led researchers
to study matching vis-a-vis clever relaxations of the class NC. In this vein,
recently Goldwasser and Grossman [GG15] gave a pseudo-deterministic RNC
algorithm for finding a perfect matching in a bipartite graph, i.e., an RNC
algorithm with the additional requirement that on the same graph, it should
return the same (i.e., unique) perfect matching for almost all choices of
random bits. A corollary of our reduction is an analogous algorithm for general
graphs.Comment: Appeared in ITCS 202
Belief-Propagation for Weighted b-Matchings on Arbitrary Graphs and its Relation to Linear Programs with Integer Solutions
We consider the general problem of finding the minimum weight \bm-matching
on arbitrary graphs. We prove that, whenever the linear programming (LP)
relaxation of the problem has no fractional solutions, then the belief
propagation (BP) algorithm converges to the correct solution. We also show that
when the LP relaxation has a fractional solution then the BP algorithm can be
used to solve the LP relaxation. Our proof is based on the notion of graph
covers and extends the analysis of (Bayati-Shah-Sharma 2005 and Huang-Jebara
2007}.
These results are notable in the following regards: (1) It is one of a very
small number of proofs showing correctness of BP without any constraint on the
graph structure. (2) Variants of the proof work for both synchronous and
asynchronous BP; it is the first proof of convergence and correctness of an
asynchronous BP algorithm for a combinatorial optimization problem.Comment: 28 pages, 2 figures. Submitted to SIAM journal on Discrete
Mathematics on March 19, 2009; accepted for publication (in revised form)
August 30, 2010; published electronically July 1, 201
The Matching Problem in General Graphs is in Quasi-NC
We show that the perfect matching problem in general graphs is in Quasi-NC.
That is, we give a deterministic parallel algorithm which runs in
time on processors. The result is obtained by a
derandomization of the Isolation Lemma for perfect matchings, which was
introduced in the classic paper by Mulmuley, Vazirani and Vazirani [1987] to
obtain a Randomized NC algorithm.
Our proof extends the framework of Fenner, Gurjar and Thierauf [2016], who
proved the analogous result in the special case of bipartite graphs. Compared
to that setting, several new ingredients are needed due to the significantly
more complex structure of perfect matchings in general graphs. In particular,
our proof heavily relies on the laminar structure of the faces of the perfect
matching polytope.Comment: Accepted to FOCS 2017 (58th Annual IEEE Symposium on Foundations of
Computer Science
A Decomposition Theorem for Maximum Weight Bipartite Matchings
Let G be a bipartite graph with positive integer weights on the edges and
without isolated nodes. Let n, N and W be the node count, the largest edge
weight and the total weight of G. Let k(x,y) be log(x)/log(x^2/y). We present a
new decomposition theorem for maximum weight bipartite matchings and use it to
design an O(sqrt(n)W/k(n,W/N))-time algorithm for computing a maximum weight
matching of G. This algorithm bridges a long-standing gap between the best
known time complexity of computing a maximum weight matching and that of
computing a maximum cardinality matching. Given G and a maximum weight matching
of G, we can further compute the weight of a maximum weight matching of G-{u}
for all nodes u in O(W) time.Comment: The journal version will appear in SIAM Journal on Computing. The
conference version appeared in ESA 199
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