46 research outputs found
Algorithms, Reductions and Equivalences for Small Weight Variants of All-Pairs Shortest Paths
APSP with small integer weights in undirected graphs [Seidel'95, Galil and
Margalit'97] has an time algorithm, where
is the matrix multiplication exponent. APSP in directed graphs with small
weights however, has a much slower running time that would be
even if [Zwick'02]. To understand this bottleneck, we
build a web of reductions around directed unweighted APSP. We show that it is
fine-grained equivalent to computing a rectangular Min-Plus product for
matrices with integer entries; the dimensions and entry size of the matrices
depend on the value of . As a consequence, we establish an equivalence
between APSP in directed unweighted graphs, APSP in directed graphs with small
integer weights, All-Pairs Longest Paths in DAGs with small
weights, approximate APSP with additive error in directed graphs with small
weights, for and several other graph problems. We also
provide fine-grained reductions from directed unweighted APSP to All-Pairs
Shortest Lightest Paths (APSLP) in undirected graphs with weights and
APSP in directed unweighted graphs (computing counts mod
).
We complement our hardness results with new algorithms. We improve the known
algorithms for APSLP in directed graphs with small integer weights and for
approximate APSP with sublinear additive error in directed unweighted graphs.
Our algorithm for approximate APSP with sublinear additive error is optimal,
when viewed as a reduction to Min-Plus product. We also give new algorithms for
variants of #APSP in unweighted graphs, as well as a near-optimal
-time algorithm for the original #APSP problem in unweighted
graphs. Our techniques also lead to a simpler alternative for the original APSP
problem in undirected graphs with small integer weights.Comment: abstract shortened to fit arXiv requirement
New Parameterized Algorithms for APSP in Directed Graphs
All Pairs Shortest Path (APSP) is a classic problem in graph theory. While for general weighted graphs there is no algorithm that computes APSP in O(n^{3-epsilon}) time (epsilon > 0), by using fast matrix multiplication algorithms, we can compute APSP in O(n^{omega}*log(n)) time (omega < 2.373) for undirected unweighted graphs, and in O(n^{2.5302}) time for directed unweighted graphs. In the current state of matters, there is a substantial gap between the upper bounds of the problem for undirected and directed graphs, and for a long time, it is remained an important open question whether it is possible to close this gap.
In this paper we introduce a new parameter that measures the symmetry of directed graphs (i.e. their closeness to undirected graphs), and obtain a new parameterized APSP algorithm for directed unweighted graphs, that generalizes Seidel\u27s O(n^{omega}*log(n)) time algorithm for undirected unweighted graphs. Given a directed unweighted graph G, unless it is highly asymmetric, our algorithms can compute APSP in o(n^{2.5}) time for G, providing for such graphs a faster APSP algorithm than the state-of-the-art algorithms for the problem
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