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110
Combinatorial Optimization
, 1998
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Cited by 226 (1 self)
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(All rights reserved by the authors.) C o n t e n t s
Matching is as Easy as Matrix Inversion
, 1987
"... A new algorithm for finding a maximum matching in a general graph is presented; its special feature being that the only computationally nontrivial step required in its execution is the inversion of a single integer matrix. Since this step can be parallelized, we get a simple parallel (RNC2) algorit ..."
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Cited by 166 (5 self)
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A new algorithm for finding a maximum matching in a general graph is presented; its special feature being that the only computationally nontrivial step required in its execution is the inversion of a single integer matrix. Since this step can be parallelized, we get a simple parallel (RNC2) algorithm. At the heart of our algorithm lies a probabilistic lemma, the isolating lemma. We show applications of this lemma to parallel computation and randomized reductions.
Edmonds polytopes and a hierarchy of combinatorial problems
, 2006
"... Let S be a set of linear inequalities that determine a bounded polyhedron P. The closure of S is the smallest set of inequalities that contains S and is closed under two operations: (i) taking linear combinations of inequalities, (ii) replacing an inequality Σaj xj ≤ a0, where a1,a2,...,an are integ ..."
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Cited by 146 (0 self)
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Let S be a set of linear inequalities that determine a bounded polyhedron P. The closure of S is the smallest set of inequalities that contains S and is closed under two operations: (i) taking linear combinations of inequalities, (ii) replacing an inequality Σaj xj ≤ a0, where a1,a2,...,an are integers, by the inequality Σaj xj ≤ a with a ≥[a0]. Obviously, if integers x1,x2,...,xn satisfy all the inequalities in S, then they satisfy also all inequalities in the closure of S. Conversely, let Σcj xj ≤ c0 hold for all choices of integers x1,x2,...,xn, that satisfy all the inequalities in S. Then we prove that Σcj xj ≤ c0 belongs to the closure of S. To each integer linear programming problem, we assign a nonnegative integer, called its rank. (The rank is the minimum number of iterations of the operation (ii) that are required in order to eliminate the integrality constraint.) We prove that there is no upper bound on the rank of problems arising from the search for largest independent sets in graphs.
Matching: a wellsolved class of integer linear programs
 in: Combinatorial structures and their applications (Gordon and Breach
, 1970
"... A main purpose of this work is to give a good algorithm for a certain welldescribed class of integer linear programming problems, called matching problems (or the matching problem). Methods developed for simple matching [2,3], a special case to which these problems can be reduced [4], are applied di ..."
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Cited by 60 (1 self)
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A main purpose of this work is to give a good algorithm for a certain welldescribed class of integer linear programming problems, called matching problems (or the matching problem). Methods developed for simple matching [2,3], a special case to which these problems can be reduced [4], are applied directly to the larger class. In the process, we derive a description of a system of linear inequalities whose polyhedron is the convex hull of the admissible solution vectors to the given matching problem. At the same time, various combinatorial results about matchings are derived and discussed in terms of graphs. The general integer linear programming problem can be stated as: (1) Minimize z = ∑ j∈E cjxj, where cj is a given real number, subject to (2) xj an integer for each j ∈ E; (3) 0 ≤ xj ≤ αj, j ∈ E, where αj is a given positive integer or +∞; (4) ∑ j∈E aijxj = bi, i ∈ V, where aij and bi are given integers; V and E are index sets having cardinalities V  and E. (5) The integer program (1) is called a matching problem whenever i∈V aij  ≤ 2 holds for all j ∈ E. (6) A solution to the integer program (1) is a vector [xj], j ∈ E, satisfying (2), (3), and (4), and an optimum solution is a solution which minimizes z among all solutions. When the integer program is a matching problem, a solution is called a matching and an optimum solution is an optimum matching. If the integer restriction (2) is omitted, the problem becomes a linear program. An optimum solution to that linear program will typically have fractional values. There is an important class of linear programs, called transportation or network flow problems, which have the property that for any integer righthand side bi, i ∈ V, and any cost vector cj, j ∈ E, there is an optimum solution which has all integer xj, j ∈ E. The class of matching probems includes that class of linear programs, but, in addition, includes problems for which omitting
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
Combinatorial Optimization: Packing and Covering
, 2000
"... The integer programming models known as set packing and set covering have a wide range of applications, such as pattern recognition, plant location and airline crew scheduling. Sometimes, due to the special structure of the constraint matrix, the natural linear programming relaxation yields an optim ..."
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Cited by 40 (1 self)
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The integer programming models known as set packing and set covering have a wide range of applications, such as pattern recognition, plant location and airline crew scheduling. Sometimes, due to the special structure of the constraint matrix, the natural linear programming relaxation yields an optimal solution that is integer, thus solving the problem. Sometimes, both the linear programming relaxation and its dual have integer optimal solutions. Under which conditions do such integrality properties hold? This question is of both theoretical and practical interest. Minmax theorems, polyhedral combinatorics and graph theory all come together in this rich area of discrete mathematics. In addition to minmax and polyhedral results, some of the deepest results in this area come in two flavors: “excluded minor” results and “decomposition ” results. In these notes, we present several of these beautiful results. Three chapters cover minmax and polyhedral results. The next four cover excluded minor results. In the last three, we
GEELEN: The optimal pathmatching problem
 Proceedings o.f thirtyseventh Symposium on the Foundations of Computing, IEEE Computer
, 1996
"... We describe a common generalization of the weighted matching problem and the weighted matroid intersection problem. In this context we establish common generalizations of the main results on those two problemspolynomialtime solvability, rainmax theorems, and totally dual integral polyhedral desc ..."
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Cited by 26 (2 self)
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We describe a common generalization of the weighted matching problem and the weighted matroid intersection problem. In this context we establish common generalizations of the main results on those two problemspolynomialtime solvability, rainmax theorems, and totally dual integral polyhedral descriptions. New applications of these results include a strongly polynomial separation algorithm for the convex hull of matchable sets of a graph, and a polynomialtime algorithm to compute the rank of a certain matrix of indeterminates. 1.
Algorithmic Theory of Random graphs
, 1997
"... The theory of random graphs has been mainly concerned with structural properties, in particular the most likely values of various graph invariants  see Bollob`as [21]. There has been increasing interest in using random graphs as models for the average case analysis of graph algorithms. In this pap ..."
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Cited by 24 (1 self)
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The theory of random graphs has been mainly concerned with structural properties, in particular the most likely values of various graph invariants  see Bollob`as [21]. There has been increasing interest in using random graphs as models for the average case analysis of graph algorithms. In this paper we survey some of the results in this area. 1 Introduction The theory of random graphs as initiated by Erdos and R'enyi [52] and developed along with others, has been mainly concerned with structural properties, in particular the most likely values of various graph invariantss  see Bollob`as [21]. There has been increasing interest in using random graphs as models for the average case analysis of graph algorithms. We would like in this paper to survey some of the results in this area. We hope to be fairly comprehensive in terms of the areas we tackle and so depth will be sacrificed in favour of breadth. One attractive feature of average case analysis is that it banishes the pessimism o...
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.
Zeros of chromatic and flow polynomials of graphs
 J. Geometry
, 2003
"... We survey results and conjectures concerning the zero distribution of chromatic and flow polynomials of graphs, and characteristic polynomials of matroids. 1 ..."
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Cited by 21 (4 self)
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We survey results and conjectures concerning the zero distribution of chromatic and flow polynomials of graphs, and characteristic polynomials of matroids. 1