124 research outputs found
Finding the K shortest hyperpaths using reoptimization
The shortest hyperpath problem is an extension of the classical shortest path problem and has applications in many different areas. Recently, algorithms for finding the K shortest hyperpaths in a directed hypergraph have been developed by Andersen, Nielsen and Pretolani. In this paper we improve the worst-case computational complexity of an algorithm for finding the K shortest hyperpaths in an acyclic hypergraph. This result is obtained by applying new reoptimization techniques for shortest hyperpaths. The algorithm turns out to be quite effective in practice and has already been successfully applied in the context of stochastic time-dependent networks, for finding the K best strategies and for solving bicriterion problems.Network programming; Directed hypergraphs; K shortest hyperpaths; K shortest paths
Understanding the complexity of #SAT using knowledge compilation
Two main techniques have been used so far to solve the #P-hard problem #SAT.
The first one, used in practice, is based on an extension of DPLL for model
counting called exhaustive DPLL. The second approach, more theoretical,
exploits the structure of the input to compute the number of satisfying
assignments by usually using a dynamic programming scheme on a decomposition of
the formula. In this paper, we make a first step toward the separation of these
two techniques by exhibiting a family of formulas that can be solved in
polynomial time with the first technique but needs an exponential time with the
second one. We show this by observing that both techniques implicitely
construct a very specific boolean circuit equivalent to the input formula. We
then show that every beta-acyclic formula can be represented by a polynomial
size circuit corresponding to the first method and exhibit a family of
beta-acyclic formulas which cannot be represented by polynomial size circuits
corresponding to the second method. This result shed a new light on the
complexity of #SAT and related problems on beta-acyclic formulas. As a
byproduct, we give new handy tools to design algorithms on beta-acyclic
hypergraphs
Distance-two coloring of sparse graphs
Consider a graph and, for each vertex , a subset
of neighbors of . A -coloring is a coloring of the
elements of so that vertices appearing together in some receive
pairwise distinct colors. An obvious lower bound for the minimum number of
colors in such a coloring is the maximum size of a set , denoted by
. In this paper we study graph classes for which there is a
function , such that for any graph and any , there is a
-coloring using at most colors. It is proved that if
such a function exists for a class , then can be taken to be a linear
function. It is also shown that such classes are precisely the classes having
bounded star chromatic number. We also investigate the list version and the
clique version of this problem, and relate the existence of functions bounding
those parameters to the recently introduced concepts of classes of bounded
expansion and nowhere-dense classes.Comment: 13 pages - revised versio
Block Crossings in Storyline Visualizations
Storyline visualizations help visualize encounters of the characters in a
story over time. Each character is represented by an x-monotone curve that goes
from left to right. A meeting is represented by having the characters that
participate in the meeting run close together for some time. In order to keep
the visual complexity low, rather than just minimizing pairwise crossings of
curves, we propose to count block crossings, that is, pairs of intersecting
bundles of lines.
Our main results are as follows. We show that minimizing the number of block
crossings is NP-hard, and we develop, for meetings of bounded size, a
constant-factor approximation. We also present two fixed-parameter algorithms
and, for meetings of size 2, a greedy heuristic that we evaluate
experimentally.Comment: Appears in the Proceedings of the 24th International Symposium on
Graph Drawing and Network Visualization (GD 2016
Point-width and Max-CSPs
International audienceThe complexity of (unbounded-arity) Max-CSPs under structural restrictions is poorly understood. The two most general hypergraph properties known to ensure tractability of Max-CSPs, β-acyclicity and bounded (incidence) MIM-width, are incomparable and lead to very different algorithms. We introduce the framework of point decompositions for hypergraphs and use it to derive a new sufficient condition for the tractability of (structurally restricted) Max-CSPs, which generalises both bounded MIM-width and β-acyclicity. On the way, we give a new characterisation of bounded MIM-width and discuss other hypergraph properties which are relevant to the complexity of Max-CSPs, such as β-hypertreewidth
Edge deletion to tree-like graph classes
For a fixed property (graph class) , given a graph and an integer
, the -deletion problem consists in deciding if we can turn into a
graph with the property by deleting at most edges of . The
-deletion problem is known to be NP-hard for most of the well-studied
graph classes (such as chordal, interval, bipartite, planar, comparability and
permutation graphs, among others), with the notable exception of trees.
Motivated by this fact, in this work we study the deletion problem for some
classes close to trees. We obtain NP-hardness results for several classes of
sparse graphs, for which we prove that deletion is hard even when the input is
a bipartite graph. In addition, we give sufficient structural conditions for
the graph class for NP-hardness. In the case of deletion to cactus, we
show that the problem becomes tractable when the input is chordal, and we give
polynomial-time algorithms for quasi-threshold graphs.Comment: 12 pages, no figure
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