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38
Approximation schemes for covering and packing problems in image processing and VLSI
 J. ACM
, 1985
"... A unified and powerful approach is presented for devising polynomial approximation schemes for many strongly NPcomplete problems. Such schemes consist of families of approximation algorithms for each desired performance bound on the relative error c> 0, with running time that is polynomial whe ..."
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Cited by 249 (0 self)
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A unified and powerful approach is presented for devising polynomial approximation schemes for many strongly NPcomplete problems. Such schemes consist of families of approximation algorithms for each desired performance bound on the relative error c> 0, with running time that is polynomial when c is fixed. Thougb the polynomiality of these algorithms depends on the degree of approximation e being fixed, they cannot be improved, owing to a negative result stating that there are no fully polynomial approximation schemes for strongly NPcomplete problems unless NP = P. The unified technique that is introduced here, referred to as the shifting strategy, is applicable to numerous geometric covering and packing problems. The method of using the technique and how it varies with problem parameters are illustrated. A similar technique, independently devised by B. S. Baker, was shown to be applicable for covering and packing problems on planar graphs.
The complexity of searching a graph
"... T. Parsons originally proposed and studied the following pursuitevasion problem on graphs: Members of a team of searchers traverse the edges of a graph G in pursuit of a fugitive, who moves along the edges of the graph with complete knowledge of the locations of the pursuers. What is the smallest ..."
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Cited by 153 (0 self)
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T. Parsons originally proposed and studied the following pursuitevasion problem on graphs: Members of a team of searchers traverse the edges of a graph G in pursuit of a fugitive, who moves along the edges of the graph with complete knowledge of the locations of the pursuers. What is the smallest number s(G) of searchers that will suffice for guaranteeing capture of the fugitive? It is shown that determining whether s(G) 5 K, for a given integer K, is NPcomplete for general graphs but can be solved in linear time for trees. We also provide a structural characterization of those graphs G with s(G) I KforK = 1, 2, 3.
On Finding Multiconstrained Paths
, 1998
"... New emerging distributed multimedia applications provide guaranteed endtoend quality of service (QoS) and have stringent constraints on delay, delayjitter, cost, etc. The task of QoS routing is to find a route in the network which has sufficient resources to satisfy the constraints. The delaycost ..."
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Cited by 133 (6 self)
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New emerging distributed multimedia applications provide guaranteed endtoend quality of service (QoS) and have stringent constraints on delay, delayjitter, cost, etc. The task of QoS routing is to find a route in the network which has sufficient resources to satisfy the constraints. The delaycost constrained routing problem is NPcomplete. We propose a heuristic algorithm for this problem. The idea is to first reduce the NPcomplete problem to a simpler one which can be solved in polynomial time, and then solve the new problem by either an extended Dijkstra's algorithm or an extended BellmanFord algorithm. We prove the correctness of our algorithm by showing that a solution for the simpler problem must also be a solution for the original problem. The performance of the algorithm is studied by both theoretical analysis and simulation. 1 Introduction Quality of Service (QoS) routing has been attracting considerable attention in the research community recently [6, 10, 11, 12, 13]. T...
Solving lowdensity subset sum problems
 in Proceedings of 24rd Annu. Symp. Foundations of comput. Sci
, 1983
"... Abstract. The subset sum problem is to decide whether or not the O1 integer programming problem C aixi = M, Vi,x,=O or 1, il has a solution, where the ai and M are given positive integers. This problem is NPcomplete, and the difficulty of solving it is the basis of publickey cryptosystems of kna ..."
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Cited by 124 (3 self)
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Abstract. The subset sum problem is to decide whether or not the O1 integer programming problem C aixi = M, Vi,x,=O or 1, il has a solution, where the ai and M are given positive integers. This problem is NPcomplete, and the difficulty of solving it is the basis of publickey cryptosystems of knapsack type. An algorithm is proposed that searches for a solution when given an instance of the subset sum problem. This algorithm always halts in polynomial time but does not always find a solution when one exists. It converts the problem to one of finding a particular short vector v in a lattice, and then uses a lattice basis reduction algorithm due to A. K. Lenstra, H. W. Lenstra, Jr., and L. Lovasz to attempt to find v. The performance of the proposed algorithm is analyzed. Let the density d of a subset sum problem be defined by d = n/log2(maxi ai). Then for “almost all ” problems of density d c 0.645, the vector v we searched for is the shortest nonzero vector in the lattice. For “almost all ” problems of density d < l/a it is proved that the lattice basis reduction algorithm locates v. Extensive computational tests of the algorithm suggest that it works for densities d < de(n), where d=(n) is a cutoff value that is substantially larger than I/n. This method gives a polynomial time attack on knapsack publickey cryptosystems that can be expected to break them if they transmit information at rates below d=(n), as n+ 01.
Reachability is harder for directed than for undirected finite graphs
 Journal of Symbolic Logic
, 1990
"... Abstract. Although it is known that reachability in undirected finite graphs can be expressed by an existential monadic secondorder sentence, our main result is that this is not the case for directed finite graphs (even in the presence of certain “builtin ” relations, such as the successor relatio ..."
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Cited by 76 (6 self)
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Abstract. Although it is known that reachability in undirected finite graphs can be expressed by an existential monadic secondorder sentence, our main result is that this is not the case for directed finite graphs (even in the presence of certain “builtin ” relations, such as the successor relation). The proof makes use of EhrenfeuchtFrai’sse games, along with probabilistic arguments. However, we show that for directed finite graphs with degree at most k, reachability is expressible by an existential monadic secondorder sentence. $1. Introduction. If s and t denote distinguished points in a directed (resp. undirected) graph, then we say that a graph is (s, t)connected if there is a directed (undirected) path from s to t. We sometimes refer to the problem of deciding whether a given directed (undirected) graph with two given points sand t is (s, t)connected as the directed (undirected) reachability problem.
A simplified npcomplete satisfiability problem,”
 Discrete Appl. Math.,
, 1984
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Approximation Algorithms for Hitting Objects with Straight Lines
 Discrete Applied Mathematics
, 1989
"... In the hitting set problem one is given subsets of a finite set N and one has to find an X ae N of minimum cardinality that "hits" (intersects) all of them. The problem is NPhard. It is not known whether there exists a polynomialtime approximation algorithm for the hitting set prob ..."
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Cited by 31 (1 self)
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In the hitting set problem one is given subsets of a finite set N and one has to find an X ae N of minimum cardinality that "hits" (intersects) all of them. The problem is NPhard. It is not known whether there exists a polynomialtime approximation algorithm for the hitting set problem with a finite performance ratio. Special cases of the hitting set problem are described for which finite performance ratios are guaranteed. These problems arise in a geometric setting. We consider special cases of the following problem: Given n compact subsets of , find a set of straight lines of minimum cardinality so that each of the given subsets is hit by at least one line. The algorithms are based on several techniques of representing objects bypoints, not necessarily points on the objects, and solving (in some cases, only approximately) the problem of hitting the representative points. Finite performance ratios are obtained when the dimension, the number of types of sets to be hit and the number of directions of the hitting lines are bounded.
Scheduling with incompatible jobs
 DISCRETE APPLIED MATHEMATICS
, 1994
"... We consider scheduling problems in a multiprocessor system with incompatible jobs (two incompatible jobs cannot be processed by the same machine). We consider the problem to minimize the maximum job completion time. the makespan. This problem is NPcomplete. We present a number of polynomial time ap ..."
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Cited by 25 (2 self)
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We consider scheduling problems in a multiprocessor system with incompatible jobs (two incompatible jobs cannot be processed by the same machine). We consider the problem to minimize the maximum job completion time. the makespan. This problem is NPcomplete. We present a number of polynomial time approximation algorithms for this problem where the job incompatibilities possess a special structure. As the incompatibilities form a graph on the set ofjobs, our algorithms strongly rely on graph theoretic methods. We also solve an open problem by Birb et al. [l] on coloring precolored bipartite graphs.
Shortest path problems with node failures
 Networks
, 1992
"... Consider the problem of finding the shortest paths from a node source s to a node sink t in a complete network. On any given instance of the problem, only a subset of the intermediate nodes can be used to go from s to t, the subset being chosen according to a given probability law. We wish to find ..."
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Cited by 15 (0 self)
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Consider the problem of finding the shortest paths from a node source s to a node sink t in a complete network. On any given instance of the problem, only a subset of the intermediate nodes can be used to go from s to t, the subset being chosen according to a given probability law. We wish to find an a priori path from s to f such that, on any given instance of the problem, the sequence of nodes defining the path is preserved but only the permissible nodes are traversed, the others being skipped. The problem of finding an a priori path of minimum expected length is defined as the Probabilistic Shortest Path Problem (PSPP). Note that if the network is not originally complete, the PSPP methodology can still be used if we first add each missing edge, together with a deterministic length (being defined by an alternative path using nodes that have no probability of failure). In this paper, after discussing potential applications of the PSPP, we study the complexity of this class of problems. We first show that the problem is, in general, NPhard and then we develop polynomial time procedures for special cases of it. We also consider the complexity of a related problem: the Probabilistic Minimum Spanning Tree Problem (PMSTP). Finally, we provide a discussion of the implications of the results.