4,114 research outputs found
Finding Simple Shortest Paths and Cycles
The problem of finding multiple simple shortest paths in a weighted directed
graph has many applications, and is considerably more difficult than
the corresponding problem when cycles are allowed in the paths. Even for a
single source-sink pair, it is known that two simple shortest paths cannot be
found in time polynomially smaller than (where ) unless the
All-Pairs Shortest Paths problem can be solved in a similar time bound. The
latter is a well-known open problem in algorithm design. We consider the
all-pairs version of the problem, and we give a new algorithm to find
simple shortest paths for all pairs of vertices. For , our algorithm runs
in time (where ), which is almost the same bound as
for the single pair case, and for we improve earlier bounds. Our approach
is based on forming suitable path extensions to find simple shortest paths;
this method is different from the `detour finding' technique used in most of
the prior work on simple shortest paths, replacement paths, and distance
sensitivity oracles.
Enumerating simple cycles is a well-studied classical problem. We present new
algorithms for generating simple cycles and simple paths in in
non-decreasing order of their weights; the algorithm for generating simple
paths is much faster, and uses another variant of path extensions. We also give
hardness results for sparse graphs, relative to the complexity of computing a
minimum weight cycle in a graph, for several variants of problems related to
finding simple paths and cycles.Comment: The current version includes new results for undirected graphs. In
Section 4, the notion of an (m,n) reduction is generalized to an f(m,n)
reductio
Faster Parametric Shortest Path and Minimum Balance Algorithms
The parametric shortest path problem is to find the shortest paths in graph
where the edge costs are of the form w_ij+lambda where each w_ij is constant
and lambda is a parameter that varies. The problem is to find shortest path
trees for every possible value of lambda.
The minimum-balance problem is to find a ``weighting'' of the vertices so
that adjusting the edge costs by the vertex weights yields a graph in which,
for every cut, the minimum weight of any edge crossing the cut in one direction
equals the minimum weight of any edge crossing the cut in the other direction.
The paper presents fast algorithms for both problems. The algorithms run in
O(nm+n^2 log n) time. The paper also describes empirical studies of the
algorithms on random graphs, suggesting that the expected time for finding a
minimum-mean cycle (an important special case of both problems) is O(n log(n) +
m)
Maximum Skew-Symmetric Flows and Matchings
The maximum integer skew-symmetric flow problem (MSFP) generalizes both the
maximum flow and maximum matching problems. It was introduced by Tutte in terms
of self-conjugate flows in antisymmetrical digraphs. He showed that for these
objects there are natural analogs of classical theoretical results on usual
network flows, such as the flow decomposition, augmenting path, and max-flow
min-cut theorems. We give unified and shorter proofs for those theoretical
results.
We then extend to MSFP the shortest augmenting path method of Edmonds and
Karp and the blocking flow method of Dinits, obtaining algorithms with similar
time bounds in general case. Moreover, in the cases of unit arc capacities and
unit ``node capacities'' the blocking skew-symmetric flow algorithm has time
bounds similar to those established in Even and Tarjan (1975) and Karzanov
(1973) for Dinits' algorithm. In particular, this implies an algorithm for
finding a maximum matching in a nonbipartite graph in time,
which matches the time bound for the algorithm of Micali and Vazirani. Finally,
extending a clique compression technique of Feder and Motwani to particular
skew-symmetric graphs, we speed up the implied maximum matching algorithm to
run in time, improving the best known bound
for dense nonbipartite graphs.
Also other theoretical and algorithmic results on skew-symmetric flows and
their applications are presented.Comment: 35 pages, 3 figures, to appear in Mathematical Programming, minor
stylistic corrections and shortenings to the original versio
Parametric shortest-path algorithms via tropical geometry
We study parameterized versions of classical algorithms for computing
shortest-path trees. This is most easily expressed in terms of tropical
geometry. Applications include shortest paths in traffic networks with variable
link travel times.Comment: 24 pages and 8 figure
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
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