139 research outputs found
Prize-collecting Network Design on Planar Graphs
In this paper, we reduce Prize-Collecting Steiner TSP (PCTSP),
Prize-Collecting Stroll (PCS), Prize-Collecting Steiner Tree (PCST),
Prize-Collecting Steiner Forest (PCSF) and more generally Submodular
Prize-Collecting Steiner Forest (SPCSF) on planar graphs (and more generally
bounded-genus graphs) to the same problems on graphs of bounded treewidth. More
precisely, we show any -approximation algorithm for these problems on
graphs of bounded treewidth gives an -approximation
algorithm for these problems on planar graphs (and more generally bounded-genus
graphs), for any constant . Since PCS, PCTSP, and PCST can be
solved exactly on graphs of bounded treewidth using dynamic programming, we
obtain PTASs for these problems on planar graphs and bounded-genus graphs. In
contrast, we show PCSF is APX-hard to approximate on series-parallel graphs,
which are planar graphs of treewidth at most 2. This result is interesting on
its own because it gives the first provable hardness separation between
prize-collecting and non-prize-collecting (regular) versions of the problems:
regular Steiner Forest is known to be polynomially solvable on series-parallel
graphs and admits a PTAS on graphs of bounded treewidth. An analogous hardness
result can be shown for Euclidian PCSF. This ends the common belief that
prize-collecting variants should not add any new hardness to the problems
Spanning trees short or small
We study the problem of finding small trees. Classical network design
problems are considered with the additional constraint that only a specified
number of nodes are required to be connected in the solution. A
prototypical example is the MST problem in which we require a tree of
minimum weight spanning at least nodes in an edge-weighted graph. We show
that the MST problem is NP-hard even for points in the Euclidean plane. We
provide approximation algorithms with performance ratio for the
general edge-weighted case and for the case of points in the
plane. Polynomial-time exact solutions are also presented for the class of
decomposable graphs which includes trees, series-parallel graphs, and bounded
bandwidth graphs, and for points on the boundary of a convex region in the
Euclidean plane. We also investigate the problem of finding short trees, and
more generally, that of finding networks with minimum diameter. A simple
technique is used to provide a polynomial-time solution for finding -trees
of minimum diameter. We identify easy and hard problems arising in finding
short networks using a framework due to T. C. Hu.Comment: 27 page
Optimization in Telecommunication Networks
Network design and network synthesis have been the classical optimization problems intelecommunication for a long time. In the recent past, there have been many technologicaldevelopments such as digitization of information, optical networks, internet, and wirelessnetworks. These developments have led to a series of new optimization problems. Thismanuscript gives an overview of the developments in solving both classical and moderntelecom optimization problems.We start with a short historical overview of the technological developments. Then,the classical (still actual) network design and synthesis problems are described with anemphasis on the latest developments on modelling and solving them. Classical results suchas Menger’s disjoint paths theorem, and Ford-Fulkerson’s max-flow-min-cut theorem, butalso Gomory-Hu trees and the Okamura-Seymour cut-condition, will be related to themodels described. Finally, we describe recent optimization problems such as routing andwavelength assignment, and grooming in optical networks.operations research and management science;
An efficient algorithm for nucleolus and prekernel computation in some classes of TU-games
We consider classes of TU-games. We show that we can efficiently compute an allocation in the intersection of the prekernel and the least core of the game if we can efficiently compute the minimum excess for any given allocation. In the case where the prekernel of the game contains exactly one core vector, our algorithm computes the nucleolus of the game. This generalizes both a recent result by Kuipers on the computation of the nucleolus for convex games and a classical result by Megiddo on the nucleolus of standard tree games to classes of more general minimum cost spanning tree games. Our algorithm is based on the ellipsoid method and Maschler's scheme for approximating the prekernel. \u
Minimizing Movement: Fixed-Parameter Tractability
We study an extensive class of movement minimization problems which arise
from many practical scenarios but so far have little theoretical study. In
general, these problems involve planning the coordinated motion of a collection
of agents (representing robots, people, map labels, network messages, etc.) to
achieve a global property in the network while minimizing the maximum or
average movement (expended energy). The only previous theoretical results about
this class of problems are about approximation, and mainly negative: many
movement problems of interest have polynomial inapproximability. Given that the
number of mobile agents is typically much smaller than the complexity of the
environment, we turn to fixed-parameter tractability. We characterize the
boundary between tractable and intractable movement problems in a very general
set up: it turns out the complexity of the problem fundamentally depends on the
treewidth of the minimal configurations. Thus the complexity of a particular
problem can be determined by answering a purely combinatorial question. Using
our general tools, we determine the complexity of several concrete problems and
fortunately show that many movement problems of interest can be solved
efficiently.Comment: A preliminary version of the paper appeared in ESA 200
The generalized vertex cover problem and some variations
In this paper we study the generalized vertex cover problem (GVC), which is a
generalization of various well studied combinatorial optimization problems. GVC
is shown to be equivalent to the unconstrained binary quadratic programming
problem and also equivalent to some other variations of the general GVC. Some
solvable cases are identified and approximation algorithms are suggested for
special cases. We also study GVC on bipartite graphs and identify some
polynomially solvable cases. We show that GVC on bipartite graphs is equivalent
to the bipartite unconstrained 0-1 quadratic programming problem. Integer
programming formulations of GVC and related problems are presented and
establish half-integrality property on some variables for the corresponding
linear programming relaxations. We also discuss special cases of GVC where all
feasible solutions are independent sets or vertex covers. These problems are
observed to be equivalent to the maximum weight independent set problem or
minimum weight vertex cover problem along with some algorithmic results.Comment: 24 page
A comparison of two approaches for polynomial time algorithms computing basic graph parameters
In this paper we compare and illustrate the algorithmic use of graphs of
bounded tree-width and graphs of bounded clique-width. For this purpose we give
polynomial time algorithms for computing the four basic graph parameters
independence number, clique number, chromatic number, and clique covering
number on a given tree structure of graphs of bounded tree-width and graphs of
bounded clique-width in polynomial time. We also present linear time algorithms
for computing the latter four basic graph parameters on trees, i.e. graphs of
tree-width 1, and on co-graphs, i.e. graphs of clique-width at most 2.Comment: 25 pages, 3 figure
Well-solvable special cases of the TSP : a survey
The Traveling Salesman Problem belongs to the most important and most investigated problems in combinatorial optimization. Although it is an NP-hard problem, many of its special cases can be solved efficiently. We survey these special cases with emphasis on results obtained during the decade 1985-1995. This survey complements an earlier survey from 1985 compiled by Gilmore, Lawler and Shmoys. Keywords: Traveling Salesman Problem, Combinatorial optimization, Polynomial time algorithm, Computational complexity
Overlaid oriented Voronoi diagrams and the 1-Steiner tree problem
Overlaid oriented Voronoi diagrams (OOVDs) are known to provide useful data
for the construction of optimal Euclidean -Steiner trees. The theoretical
time complexity of construction methods exploiting the OOVD is , but a
computational study has never been performed, and robust constructions for
OOVDs have not previously been implemented.
In this paper, we outline a numerically stable implementation for
constructing OOVDs using tools from the Computational Geometry Algorithms
Library (CGAL), and test its performance on random point sets. We then study
the effect that the OOVD data has in reducing the complexity of -Steiner
tree construction when compared to a naive approach. The number of iterations
of the main loop of the 1-Steiner algorithm is directly determined by the
number of faces in the OOVD, and this appears to be linear for the random
inputs we tested. We also discuss methods for processing the OOVD data that
lead to a reduction in construction time by roughly a factor of 12.Comment: 16 pages, 9 figure
Distance-preserving graph contractions
Compression and sparsification algorithms are frequently applied in a
preprocessing step before analyzing or optimizing large networks/graphs. In
this paper we propose and study a new framework contracting edges of a graph
(merging vertices into super-vertices) with the goal of preserving pairwise
distances as accurately as possible. Formally, given an edge-weighted graph,
the contraction should guarantee that for any two vertices at distance , the
corresponding super-vertices remain at distance at least in the
contracted graph, where is a tolerance function bounding the
permitted distance distortion. We present a comprehensive picture of the
algorithmic complexity of the contraction problem for affine tolerance
functions , where and
are arbitrary real-valued parameters. Specifically, we present polynomial-time
algorithms for trees as well as hardness and inapproximability results for
different graph classes, precisely separating easy and hard cases. Further we
analyze the asymptotic behavior of contractions, and find efficient algorithms
to compute (non-optimal) contractions despite our hardness results.Comment: An extended abstract of this work has appeared in the Proceedings of
the 9th Innovations in Theoretical Computer Science Conference (ITCS) 201
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