329 research outputs found
Swap-vertex based neighborhood for Steiner tree problems
Steiner tree problems (STPs) are very important in both theory and practice. In this paper, we introduce a powerful swap-vertex move operator which can be used as a basic element of any neighborhood search heuristic to solve many STP variants. Given the incumbent solution tree T, the swap-vertex move operator exchanges a vertex in T with another vertex out of T, and then attempts to construct a minimum spanning tree, leading to a neighboring solution (if feasible). We develop a series of dynamic data structures, which allow us to efficiently evaluate the feasibility of swap-vertex moves. Additionally, in order to discriminate different swap-vertex moves corresponding to the same objective value, we also develop an auxiliary evaluation function. We present a computational assessment based on a number of challenging problem instances (corresponding to three representative STP variants) which clearly shows the effectiveness of the techniques introduced in this paper. Particularly, as a key element of our KTS algorithm which participated in the 11th DIMACS implementation challenge, the swap-vertex operator as well as the auxiliary evaluation function contributed significantly to the excellent performance of our algorithm
Solving Medium to Large Sized Euclidean Generalized Minimum Spanning Tree Problems
The generalized minimum spanning tree problem is a generalization of the minimum spanning tree problem. This network design problems finds several practical applications, especially when one considers the design of a large-capacity backbone network connecting several individual networks. In this paper we study the performance of six neighborhood search heuristics based on tabu search and variable neighborhood search on this problem domain. Our principal finding is that a tabu search heuristic almost always provides the best quality solution for small to medium sized instances within short execution times while variable neighborhood decomposition search provides the best quality solutions for most large instances.
Local Search Approximation Algorithms for Clustering Problems
In this research we study the use of local search in the design of approximation algorithms for NP-hard optimization problems. For our study we have selected several well-known clustering problems: k-facility location problem, minimum mutliway cut problem, and constrained maximum k-cut problem.
We show that by careful use of the local optimality property of the solutions produced by local search algorithms it is possible to bound the ratio between solutions produced by local search approximation algorithms and optimum solutions. This ratio is known as the locality gap.
The locality gap of our algorithm for the k-uncapacitated facility location problem is 2+sqrt(3) +epsilon for any constant epsilon \u3e0. This matches the approximation ratio of the best known algorithm for the problem, proposed by Zhang but our algorithm is simpler. For the minimum multiway cut problem our algorithm has locality gap 2-2/k, which matches the approximation ratio of the isolation heuristic of Dahlhaus et al; however, our experimental results show that in practice our local search algorithm greatly outperforms the isolation heuristic, and furthermore it has comparable performance as that of the three currently best algorithms for the minimum multiway cut problem. For the constrained maximum k-cut problem on hypergraphs we proposed a local search based approximation algorithm with locality gap 1-1/k for a variety of constraints imposed on the k-cuts. The locality gap of our algorithm matches the approximation ratio of the best known algorithm for the max k-cut problem on graphs designed by Vazirani, but our algorithm is more general
Single-Strip Triangulation of Manifolds with Arbitrary Topology
Triangle strips have been widely used for efficient rendering. It is
NP-complete to test whether a given triangulated model can be represented as a
single triangle strip, so many heuristics have been proposed to partition
models into few long strips. In this paper, we present a new algorithm for
creating a single triangle loop or strip from a triangulated model. Our method
applies a dual graph matching algorithm to partition the mesh into cycles, and
then merges pairs of cycles by splitting adjacent triangles when necessary. New
vertices are introduced at midpoints of edges and the new triangles thus formed
are coplanar with their parent triangles, hence the visual fidelity of the
geometry is not changed. We prove that the increase in the number of triangles
due to this splitting is 50% in the worst case, however for all models we
tested the increase was less than 2%. We also prove tight bounds on the number
of triangles needed for a single-strip representation of a model with holes on
its boundary. Our strips can be used not only for efficient rendering, but also
for other applications including the generation of space filling curves on a
manifold of any arbitrary topology.Comment: 12 pages, 10 figures. To appear at Eurographics 200
The Homogeneous Broadcast Problem in Narrow and Wide Strips
Let be a set of nodes in a wireless network, where each node is modeled
as a point in the plane, and let be a given source node. Each node
can transmit information to all other nodes within unit distance, provided
is activated. The (homogeneous) broadcast problem is to activate a minimum
number of nodes such that in the resulting directed communication graph, the
source can reach any other node. We study the complexity of the regular and
the hop-bounded version of the problem (in the latter, must be able to
reach every node within a specified number of hops), with the restriction that
all points lie inside a strip of width . We almost completely characterize
the complexity of both the regular and the hop-bounded versions as a function
of the strip width .Comment: 50 pages, WADS 2017 submissio
Hyperbolic intersection graphs and (quasi)-polynomial time
We study unit ball graphs (and, more generally, so-called noisy uniform ball
graphs) in -dimensional hyperbolic space, which we denote by .
Using a new separator theorem, we show that unit ball graphs in
enjoy similar properties as their Euclidean counterparts, but in one dimension
lower: many standard graph problems, such as Independent Set, Dominating Set,
Steiner Tree, and Hamiltonian Cycle can be solved in
time for any fixed , while the same problems need
time in . We also show that these algorithms in
are optimal up to constant factors in the exponent under ETH.
This drop in dimension has the largest impact in , where we
introduce a new technique to bound the treewidth of noisy uniform disk graphs.
The bounds yield quasi-polynomial () algorithms for all of the
studied problems, while in the case of Hamiltonian Cycle and -Coloring we
even get polynomial time algorithms. Furthermore, if the underlying noisy disks
in have constant maximum degree, then all studied problems can
be solved in polynomial time. This contrasts with the fact that these problems
require time under ETH in constant maximum degree
Euclidean unit disk graphs.
Finally, we complement our quasi-polynomial algorithm for Independent Set in
noisy uniform disk graphs with a matching lower bound
under ETH. This shows that the hyperbolic plane is a potential source of
NP-intermediate problems.Comment: Short version appears in SODA 202
Solution Methods for the \u3cem\u3ep\u3c/em\u3e-Median Problem: An Annotated Bibliography
The p-median problem is a graph theory problem that was originally designed for, and has been extensively applied to, facility location. In this bibliography, we summarize the literature on solution methods for the uncapacitated and capacitated p-median problem on a graph or network
Phylogenetic Trees and Their Analysis
Determining the best possible evolutionary history, the lowest-cost phylogenetic tree, to fit a given set of taxa and character sequences using maximum parsimony is an active area of research due to its underlying importance in understanding biological processes. As several steps in this process are NP-Hard when using popular, biologically-motivated optimality criteria, significant amounts of resources are dedicated to both both heuristics and to making exact methods more computationally tractable. We examine both phylogenetic data and the structure of the search space in order to suggest methods to reduce the number of possible trees that must be examined to find an exact solution for any given set of taxa and associated character data. Our work on four related problems combines theoretical insight with empirical study to improve searching of the tree space. First, we show that there is a Hamiltonian path through tree space for the most common tree metrics, answering Bryant\u27s Challenge for the minimal such path. We next examine the topology of the search space under various metrics, showing that some metrics have local maxima and minima even with perfect data, while some others do not. We further characterize conditions for which sequences simulated under the Jukes-Cantor model of evolution yield well-behaved search spaces. Next, we reduce the search space needed for an exact solution by splitting the set of characters into mutually-incompatible subsets of compatible characters, building trees based on the perfect phylogenies implied by these sets, and then searching in the neighborhoods of these trees. We validate this work empirically. Finally, we compare two approaches to the generalized tree alignment problem, or GTAP: Sequence alignment followed by tree search vs. Direct Optimization, on both biological and simulated data
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