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Linear Assignment Problems and Extensions
"... This paper aims at describing the state of the art on linear assignment problems (LAPs). Besides sum LAPs it discusses also problems with other objective functions like the bottleneck LAP, the lexicographic LAP, and the more general algebraic LAP. We consider different aspects of assignment problems ..."
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Cited by 42 (0 self)
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This paper aims at describing the state of the art on linear assignment problems (LAPs). Besides sum LAPs it discusses also problems with other objective functions like the bottleneck LAP, the lexicographic LAP, and the more general algebraic LAP. We consider different aspects of assignment problems, starting with the assignment polytope and the relationship between assignment and matching problems, and focusing then on deterministic and randomized algorithms, parallel approaches, and the asymptotic behaviour. Further, we describe different applications of assignment problems, ranging from the well know personnel assignment or assignment of jobs to parallel machines, to less known applications, e.g. tracking of moving objects in the space. Finally, planar and axial threedimensional assignment problems are considered, and polyhedral results, as well as algorithms for these problems or their special cases are discussed. The paper will appear in the Handbook of Combinatorial Optimization to be published
The Capacitated KCenter Problem
 In Proceedings of the 4th Annual European Symposium on Algorithms, Lecture Notes in Computer Science 1136
, 1996
"... The capacitated Kcenter problem is a fundamental facility location problem, where we are asked to locate K facilities in a graph, and to assign vertices to facilities, so as to minimize the maximum distance from a vertex to the facility to which it is assigned. Moreover, each facility may be assign ..."
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Cited by 34 (5 self)
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The capacitated Kcenter problem is a fundamental facility location problem, where we are asked to locate K facilities in a graph, and to assign vertices to facilities, so as to minimize the maximum distance from a vertex to the facility to which it is assigned. Moreover, each facility may be assigned at most L vertices. This problem is known to be NPhard. We give polynomial time approximation algorithms for two different versions of this problem that achieve approximation factors of 5 and 6. We also study some generalizations of this problem. 1. Introduction The basic Kcenter problem is a fundamental facility location problem [17] and is defined as follows: given an edgeweighted graph G = (V; E) find a subset S ` V of size at most K such that each vertex in V is "close" to some vertex in S. More formally, the objective function is defined as follows: min S`V max u2V min v2S d(u; v) where d is the distance function. For example, one may wish to install K fire stations and mi...
Selected Topics on Assignment Problems
, 1999
"... We survey recent developments in the fields of bipartite matchings, linear sum assignment and bottleneck assignment problems and applications, multidimensional assignment problems, quadratic assignment problems, in particular lower bounds, special cases and asymptotic results, biquadratic and co ..."
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Cited by 22 (1 self)
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We survey recent developments in the fields of bipartite matchings, linear sum assignment and bottleneck assignment problems and applications, multidimensional assignment problems, quadratic assignment problems, in particular lower bounds, special cases and asymptotic results, biquadratic and communication assignment problems.
Fault Tolerant KCenter Problems
, 1997
"... The basic Kcenter problem is a fundamental facility location problem, where we are asked to locate K facilities in a graph, and to assign vertices to facilities, so as to minimize the maximum distance from a vertex to the facility to which it is assigned. This problem is known to be NPhard, and se ..."
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Cited by 15 (1 self)
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The basic Kcenter problem is a fundamental facility location problem, where we are asked to locate K facilities in a graph, and to assign vertices to facilities, so as to minimize the maximum distance from a vertex to the facility to which it is assigned. This problem is known to be NPhard, and several optimal approximation algorithms that achieve a factor of 2 have been developed for it. We focus our attention on a generalization of this problem, where each vertex is required to have a set of ff (ff K) centers close to it. In particular, we study two different versions of this problem. In the first version, each vertex is required to have at least ff centers close to it. In the second version, each vertex that does not have a center placed on it is required to have at least ff centers close to it. For both these versions we are able to provide polynomial time approximation algorithms that achieve constant approximation factors for any ff. For the first version we give an algorithm ...
Applying Lehman's Theorems to Packing Problems
, 1995
"... A 01 matrix A is ideal if the polyhedron Q(A) = convfx 2 Q : A \Delta x 1; x 0g (V denotes the column index set of A), is integral. Similarly a matrix is perfect if P (A) = convfx 2 Q : A \Delta x 1; x 0g is integral. Little is known about the relationship between these two classes of mat ..."
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Cited by 9 (1 self)
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A 01 matrix A is ideal if the polyhedron Q(A) = convfx 2 Q : A \Delta x 1; x 0g (V denotes the column index set of A), is integral. Similarly a matrix is perfect if P (A) = convfx 2 Q : A \Delta x 1; x 0g is integral. Little is known about the relationship between these two classes of matrices. We consider a transformation between the two classes which enables us to apply Lehman's modified theorem about deletionminimal nonideal matrices to obtain new results about packing polyhedra. This results in a polyhedral description for the stable set polytopes of nearbipartite graphs (the deletion of any neighbourhood produces a bipartite graph). Note that this class includes the complements of line graphs. To date, this is the only natural class, besides the perfect graphs, for which such a description is known for the graphs and their complements. Some remarks are also made on possible approaches to describing the stable set polyhedra of quasiline graphs, and more generally clawfree graphs. These results also yield a new class of tperfect graphs.
Twoway rounding
 SIAM J. Discrete Math
, 1995
"... Abstract. Given n real numbers 0 ≤ x1,..., xn < 1 and a permutation σ of {1,..., n}, we can always find ¯x1,..., ¯xn ∈ {0, 1} so that the partial sums ¯x1 + · · · + ¯xk and ¯xσ1 + · · · + ¯xσk differ from the unrounded values x1 + · · · + xk and xσ1 + · · · + xσk by at most n/(n + 1), for 1 ..."
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Cited by 7 (0 self)
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Abstract. Given n real numbers 0 ≤ x1,..., xn < 1 and a permutation σ of {1,..., n}, we can always find ¯x1,..., ¯xn ∈ {0, 1} so that the partial sums ¯x1 + · · · + ¯xk and ¯xσ1 + · · · + ¯xσk differ from the unrounded values x1 + · · · + xk and xσ1 + · · · + xσk by at most n/(n + 1), for 1 ≤ k ≤ n. The latter bound is best possible. The proof uses an elementary argument about flows in a certain network, and leads to a simple algorithm that finds an optimum way to round. Many combinatorial optimization problems in integers can be solved or approximately solved by first obtaining a realvalued solution and then rounding to integer values. Spencer [11] proved that it is always possible to do the rounding so that partial sums in two independent orderings are properly rounded. His proof was indirect—a corollary of more general results [7] about discrepancies of set systems—and it guaranteed only that the rounded partial sums would differ by at most 1 − 2−2n from the unrounded values. The purpose of this note is to give a more direct proof, which leads to a sharper result. Let x1,..., xn be real numbers and let σ be a permutation of {1,..., n}. We will write Sk = x1 + · · · + xk, Σk = xσ1 + · · · + xσk, 0 ≤ k ≤ n,
Operations Research
, 1987
"... Fear appeal as a tactic of persuasion has been studied primarily from a positivistic and nondiscursive perspective. This study examines the use of fear appeal in a natural discursive setting of fundamentalist rhetoric. More specifically, we examine the interactional problems facing Jewish fundament ..."
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Cited by 5 (0 self)
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Fear appeal as a tactic of persuasion has been studied primarily from a positivistic and nondiscursive perspective. This study examines the use of fear appeal in a natural discursive setting of fundamentalist rhetoric. More specifically, we examine the interactional problems facing Jewish fundamentalist preachers who attempt to manipulate fear and identify several discursive strategies that aim at solving these problems. General conclusions point to the power of a discursive perspective in examining fear appeal and its sophisticated application in rhetoric. Acknowledgement
On blockers in bounded posets
 Int. J. Math. Math. Sci
"... Antichains of a bounded poset are assigned antichains playing a role analogous to that played by blockers in the Boolean lattice of all subsets of a finite set. Some properties of lattices of generalized blockers and maps on them are discussed. 1 ..."
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Cited by 4 (2 self)
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Antichains of a bounded poset are assigned antichains playing a role analogous to that played by blockers in the Boolean lattice of all subsets of a finite set. Some properties of lattices of generalized blockers and maps on them are discussed. 1
Linear inequalities for flags in graded posets
 J. Comb. Theory Ser. A
"... Abstract. The closure of the convex cone generated by all flag fvectors of graded posets is shown to be polyhedral. In particular, we give the facet inequalities to the polar cone of all nonnegative chainenumeration functionals on this class of posets. These are in onetoone correspondence with a ..."
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Cited by 3 (3 self)
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Abstract. The closure of the convex cone generated by all flag fvectors of graded posets is shown to be polyhedral. In particular, we give the facet inequalities to the polar cone of all nonnegative chainenumeration functionals on this class of posets. These are in onetoone correspondence with antichains of intervals on the set of ranks and thus are counted by Catalan numbers. Furthermore, we prove that the convolution operation introduced by Kalai assigns extreme rays to pairs of extreme rays in most cases. We describe the strongest possible inequalities for graded posets of rank at most 5.