11,913 research outputs found
The cavity approach for Steiner trees packing problems
The Belief Propagation approximation, or cavity method, has been recently
applied to several combinatorial optimization problems in its zero-temperature
implementation, the max-sum algorithm. In particular, recent developments to
solve the edge-disjoint paths problem and the prize-collecting Steiner tree
problem on graphs have shown remarkable results for several classes of graphs
and for benchmark instances. Here we propose a generalization of these
techniques for two variants of the Steiner trees packing problem where multiple
"interacting" trees have to be sought within a given graph. Depending on the
interaction among trees we distinguish the vertex-disjoint Steiner trees
problem, where trees cannot share nodes, from the edge-disjoint Steiner trees
problem, where edges cannot be shared by trees but nodes can be members of
multiple trees. Several practical problems of huge interest in network design
can be mapped into these two variants, for instance, the physical design of
Very Large Scale Integration (VLSI) chips. The formalism described here relies
on two components edge-variables that allows us to formulate a massage-passing
algorithm for the V-DStP and two algorithms for the E-DStP differing in the
scaling of the computational time with respect to some relevant parameters. We
will show that one of the two formalisms used for the edge-disjoint variant
allow us to map the max-sum update equations into a weighted maximum matching
problem over proper bipartite graphs. We developed a heuristic procedure based
on the max-sum equations that shows excellent performance in synthetic networks
(in particular outperforming standard multi-step greedy procedures by large
margins) and on large benchmark instances of VLSI for which the optimal
solution is known, on which the algorithm found the optimum in two cases and
the gap to optimality was never larger than 4 %
Approximation Algorithms for Union and Intersection Covering Problems
In a classical covering problem, we are given a set of requests that we need
to satisfy (fully or partially), by buying a subset of items at minimum cost.
For example, in the k-MST problem we want to find the cheapest tree spanning at
least k nodes of an edge-weighted graph. Here nodes and edges represent
requests and items, respectively.
In this paper, we initiate the study of a new family of multi-layer covering
problems. Each such problem consists of a collection of h distinct instances of
a standard covering problem (layers), with the constraint that all layers share
the same set of requests. We identify two main subfamilies of these problems: -
in a union multi-layer problem, a request is satisfied if it is satisfied in at
least one layer; - in an intersection multi-layer problem, a request is
satisfied if it is satisfied in all layers. To see some natural applications,
consider both generalizations of k-MST. Union k-MST can model a problem where
we are asked to connect a set of users to at least one of two communication
networks, e.g., a wireless and a wired network. On the other hand, intersection
k-MST can formalize the problem of connecting a subset of users to both
electricity and water.
We present a number of hardness and approximation results for union and
intersection versions of several standard optimization problems: MST, Steiner
tree, set cover, facility location, TSP, and their partial covering variants
The Steiner Multi Cycle Problem with Applications to a Collaborative Truckload Problem
We introduce a new problem called Steiner Multi Cycle Problem that extends the Steiner Cycle problem in the same way the Steiner Forest extends the Steiner Tree problem. In this problem we are given a complete weighted graph G=(V,E), which respects the triangle inequality, a collection of terminal sets {T_1,..., T_k}, where for each a in [k] we have a subset T_a of V and these terminal sets are pairwise disjoint. The problem is to find a set of disjoint cycles of minimum cost such that for each a in [k], all vertices of T_a belong to a same cycle. Our main interest is in a restricted case where |T_a| = 2, for each a in [k], which models a collaborative less-than-truckload problem with pickup and delivery. In this problem, we have a set of agents where each agent is associated with a set T_a containing a pair of pickup and delivery vertices. This problem arises in the scenario where a company has to periodically exchange goods between two different locations, and different companies can collaborate to create a route that visits all its pairs of locations sharing the total cost of the route. We show that even the restricted problem is NP-Hard, and present some heuristics to solve it. In particular, a constructive heuristic called Refinement Search, which uses geometric properties to determine if agents are close to each other. We performed computational experiments to compare this heuristic to a GRASP based heuristic. The Refinement Search obtained the best solutions in little computational time
Generalized Buneman pruning for inferring the most parsimonious multi-state phylogeny
Accurate reconstruction of phylogenies remains a key challenge in
evolutionary biology. Most biologically plausible formulations of the problem
are formally NP-hard, with no known efficient solution. The standard in
practice are fast heuristic methods that are empirically known to work very
well in general, but can yield results arbitrarily far from optimal. Practical
exact methods, which yield exponential worst-case running times but generally
much better times in practice, provide an important alternative. We report
progress in this direction by introducing a provably optimal method for the
weighted multi-state maximum parsimony phylogeny problem. The method is based
on generalizing the notion of the Buneman graph, a construction key to
efficient exact methods for binary sequences, so as to apply to sequences with
arbitrary finite numbers of states with arbitrary state transition weights. We
implement an integer linear programming (ILP) method for the multi-state
problem using this generalized Buneman graph and demonstrate that the resulting
method is able to solve data sets that are intractable by prior exact methods
in run times comparable with popular heuristics. Our work provides the first
method for provably optimal maximum parsimony phylogeny inference that is
practical for multi-state data sets of more than a few characters.Comment: 15 page
Minimum Makespan Multi-vehicle Dial-a-Ride
Dial a ride problems consist of a metric space (denoting travel time between
vertices) and a set of m objects represented as source-destination pairs, where
each object requires to be moved from its source to destination vertex. We
consider the multi-vehicle Dial a ride problem, with each vehicle having
capacity k and its own depot-vertex, where the objective is to minimize the
maximum completion time (makespan) of the vehicles. We study the "preemptive"
version of the problem, where an object may be left at intermediate vertices
and transported by more than one vehicle, while being moved from source to
destination. Our main results are an O(log^3 n)-approximation algorithm for
preemptive multi-vehicle Dial a ride, and an improved O(log t)-approximation
for its special case when there is no capacity constraint. We also show that
the approximation ratios improve by a log-factor when the underlying metric is
induced by a fixed-minor-free graph.Comment: 22 pages, 1 figure. Preliminary version appeared in ESA 200
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