Showing papers by "Madhav V. Marathe published in 1996"

Spanning Trees---Short or Small

Abstract: We study the problem of finding small trees. Classical network design problems are considered with the additional constraint that only a specified number $k$ of nodes are required to be connected in the solution. A prototypical example is the $k$MST problem in which we require a tree of minimum weight spanning at least $k$ nodes in an edge-weighted graph. We show that the $k$MST problem is NP-hard even for points in the Euclidean plane. We provide approximation algorithms with performance ratio $2\sqrt{k}$ for the general edge-weighted case and $O(k^{1/4})$ for the case of points in the plane. Polynomial-time exact solutions are also presented for the class of treewidth-bounded 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 $k$-trees of minimum diameter. We identify easy and hard problems arising in finding short networks using a framework due to T. C. Hu.

Service-Constrained Network Design Problems

Abstract: Several practical instances of network design problems often require the network to satisfy multiple constraints. In this paper, we focus on the following problem (and its variants): find a low-cost network, under one cost function, that services every node in the graph, under another cost function, (i.e., every node of the graph is within a prespecified distance from the network). This study has important applications to the problems of optical network design and the efficient maintenance of distributed databases.

Modifying Networks to Obtain Low Cost Trees

Abstract: We consider the problem of reducing the edge lengths of a given network so that the modified network has a spanning tree of small total length. It is assumed that each edge e of the given network has an associated function c e that specifies the cost of shortening the edge by a given amount and that there is a budget B on the total reduction cost. The goal is to develop a reduction strategy satisfying the budget constraint so that the total length of a minimum spanning tree in the modified network is the smallest possible over all reduction strategies that obey the budget constraint.

Service-constrained network design problems

Abstract: Several practical instances of network design problems often require the network to satisfy multiple constraints. In this paper, we focus on the following problem (and its variants): find a low-cost network, under one cost function, that services every node in the graph, under another cost function, (i.e., every node of the graph is Within a prespecified distance from the network). Such problems find applications in optical network design and the efficient maintenance of distributed databases.We utilize the framework developed in Marathe et al. [1995] co formulate these problems as bicriteria network design problems, and present approximation algorithms for a class of service-constrained network design problems. We also present lower bounds on the approximability of these problems that demonstrate that our performance ratios are close to best possible.

Approximation algorithms for maximum two-dimensional pattern matching

Abstract: We introduce the following optimization version of the classical pattern matching problem (referred to as the maximum pattern matching problem). Given a two-dimensional rectangular text and a two-dimensional rectangular pattern, find the maximum number of non-overlapping occurrences of the pattern in the text. Unlike the classical two-dimensional pattern matching problem, the maximum two-dimensional pattern matching problem is NP-complete. We devise polynomial time approximation algorithms and approximation schemes for this problem. We also briefly discuss how the approximation algorithms can be extended to include a number of other variants of the problem.

On optimal strategies for upgrading networks

Abstract: We study {ital budget constrained optimal network upgrading problems}. Such problems aim at finding optimal strategies for improving a network under some cost measure subject to certain budget constraints. Given an edge weighted graph {ital G(V,E)}, in the {ital edge based upgrading model}, it is assumed that each edge {ital e} of the given network has an associated function {ital c(e)} that specifies for each edge {ital e} the amount by which the length {ital l(e)} is to be reduced. In the {ital node based upgrading model} a node {ital v} can be upgraded at an expense of cost {ital (v)}. Such an upgrade reduces the cost of each edge incident on {ital v} by a fixed factor {rho}, where 0 < {rho} < 1. For a given budget, {ital B}, the goal is to find an improvement strategy such that the total cost of reduction is a most the given budget {ital B} and the cost of a subgraph (e.g. minimum spanning tree) under the modified edge lengths is the best over all possible strategies which obey the budget constraint. Define an ({alpha},{beta})-approximation algorithm as a polynomial-time algorithm that produces a solution within {alpha} times the optimal function value, violating the budget constraint by a factor of at most {Beta}. The results obtained in this paper include the following 1. We show that in general the problem of computing optimal reduction strategy for modifying the network as above is {bold NP}-hard. 2. In the node based model, we show how to devise a near optimal strategy for improving the bottleneck spanning tree. The algorithms have a performance guarantee of (2 ln {ital n}, 1). 3. for the edge based improvement problems we present improved (in terms of performance and time) approximation algorithms. 4. We also present pseudo-polynomial time algorithms (extendible to polynomial time approximation schemes) for a number of edge/node based improvement problems when restricted to the class of treewidth-bounded graphs.

Node weighted network upgrade problems

Abstract: Consider a network where nodes represent processors and edges represent bidirectional communication links. The processor at a node v can be upgraded at an expense of cost(v). Such an upgrade reduces the delay of each link emanating from v by a fixed factor x, where 0 < x < 1. The goal is to find a minimum cost set of nodes to be upgraded so that the resulting network has a spanning tree in which edge is of delay at most a given value {delta}. The authors provide both hardness and approximation results for the problem. They show that the problem is NP-hard and cannot be approximated within any factor {beta} < ln n, unless NP {improper_subset} DTIME(n{sup log log n}), where n is the number of nodes in the network. They then present the first polynomial time approximation algorithms for the problem. For the general case, the approximation algorithm comes within a factor of 2 ln n of the minimum upgrading cost. When the cost of upgrading each node is 1, they present an approximation algorithm with a performance guarantee of 4(2 + ln {Delta}), where {Delta} is the maximum node degree. Finally, they present a polynomial time algorithm for the class of treewidth-bounded graphs.

Approximation Algorithms for Maximum Two-Dimensional Pattern Matching

Abstract: We introduce the following optimization version of the classical pattern matching problem (referred to as the maximum pattern matching problem). Given a two-dimensional rectangular text and a 2-dimensional rectangular pattern find the maximum number of non-overlapping occurrences of the the pattern in the text.