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Approximation Results for the Optimum Cost Chromatic Partition Problem
 J. Algorithms
"... . In this paper, we study the optimum cost chromatic partition (OCCP) problem for several graph classes. The OCCP problem is the problem of coloring the vertices of a graph such that adjacent vertices get different colors and that the total coloring costs are minimum. We prove several approximation ..."
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. In this paper, we study the optimum cost chromatic partition (OCCP) problem for several graph classes. The OCCP problem is the problem of coloring the vertices of a graph such that adjacent vertices get different colors and that the total coloring costs are minimum. We prove several approximation results for the OCCP problem restricted to bipartite, chordal, comparability, interval, permutation, split and unimodular graphs. We prove that there exists no polynomial approximation algorithm with ratio O(jV j 0:5 ) for the OCCP problem restricted to bipartite and interval graphs, unless P = NP . Furthermore, we propose approximation algorithms with ratio O(jV j 0:5 ) for bipartite, interval and unimodular graphs. Finally, we prove that there exists no polynomial approximation algorithm with ratio O(jV j 1 ) for the OCCP problem restricted to split, chordal, permutation and comparability graphs, unless P = NP .
k–regular matroids
 In Combinatorics, Complexity and Logic, Proceedings of the First International Conference on Discrete Mathematics and Theoretical Computer Science
, 1996
"... Abstract. Let k be an integer exceeding one. The class of k–regular matroids is a generalization of the classes of regular and nearregular matroids. A simple rank–r regular matroid has the maximum number of points if and only if it is isomorphic to M(Kr+1), the cycle matroid of the complete graph o ..."
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Abstract. Let k be an integer exceeding one. The class of k–regular matroids is a generalization of the classes of regular and nearregular matroids. A simple rank–r regular matroid has the maximum number of points if and only if it is isomorphic to M(Kr+1), the cycle matroid of the complete graph on r + 1 vertices. A simple rank–r nearregular matroid has the maximum number of points if and only if it is isomorphic to the simplification of T M(K3)(M(Kr+2)), that is, the simplification of the matroid obtained, geometrically, by freely adding a point to a 3–point line of M(Kr+2) and then contracting this point. This paper determines the maximum number of points that a simple rank– r k–regular matroid can have and determines all such matroids having this number. With one exception, there is exactly one such matroid. This matroid is isomorphic to the simplification of T M(Kk+2)(M(Kr+k+1)), that is, the simplification of the matroid obtained, geometrically, by freely adding k independent points to a flat of M(Kr+k+1) isomorphic to M(Kk+2) and then contracting each of these points. 1.
Edge Isoperimetry and Rapid Mixing on Matroids and Geometric Markov Chains
 PROC. 33RD ANNUAL ACM SYMPOSIUM ON THEORY OF COMPUTING
, 2001
"... We show how to bound the mixing time and logSobolev constants of Markov chains by bounding the edgeisoperimetry of their underlying graphs. To do this we use two recent techniques, one involving Average Conductance and the other logSobolev constants. We show a sort of strong conductance bound on ..."
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Cited by 7 (2 self)
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We show how to bound the mixing time and logSobolev constants of Markov chains by bounding the edgeisoperimetry of their underlying graphs. To do this we use two recent techniques, one involving Average Conductance and the other logSobolev constants. We show a sort of strong conductance bound on a family of geometric Markov chains, give improved bounds for the mixing time of a Markov chain on balanced matroids, and in both cases nd lower bounds on the logSobolev constants of these chains.
Linear Discrepancy of Totally Unimodular Matrices
 Combinatorica
, 2001
"... We show that the linear discrepancy of a totally unimodular mn matrix A is at most lindisc(A) 1 1 n+1 : This bound is sharp. In particular, this result proves Spencer's conjecture lindisc(A) (1 1 n+1 ) herdisc(A) in the special case of totally unimodular matrices. If m 2, we also show lin ..."
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Cited by 5 (3 self)
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We show that the linear discrepancy of a totally unimodular mn matrix A is at most lindisc(A) 1 1 n+1 : This bound is sharp. In particular, this result proves Spencer's conjecture lindisc(A) (1 1 n+1 ) herdisc(A) in the special case of totally unimodular matrices. If m 2, we also show lindisc(A) 1 1 m . Finally we give a characterization of those totally unimodular matrices which have linear discrepancy 1 1 n+1 : Besides m 1 matrices containing a single nonzero entry, they are exactly the ones which contain n + 1 rows such that each n thereof are linearly independent. A central proof idea is the use of linear programs. A preliminary version of this result appeared at SODA 2001. This work was partially supported by the graduate school `Eziente Algorithmen und Multiskalenmethoden', Deutsche Forschungsgemeinschaft y A similar result has been independently obtained by T. Bohman and R. Holzman and presented at the Conference on Hypergraphs (Gyula O.H. Katona is 60), Budapest, in June 2001. Mathematics Subject Classication (2000): Primary 11K38, 90C05. Secondary 05C65. Proposed abbreviated title: Linear Discrepancy. 2 1
HIGHER ORDER INDEPENDENCE IN MATROIDS
"... One may regard vectors in a finite dimensional vector space as being linear forms in a polynomial ring in an obvious way. A collection of linear forms satisfying various linear dependence relations can become independent when each of the forms is raised to the kth power. In this paper we prove that ..."
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One may regard vectors in a finite dimensional vector space as being linear forms in a polynomial ring in an obvious way. A collection of linear forms satisfying various linear dependence relations can become independent when each of the forms is raised to the kth power. In this paper we prove that a certain class of matroids satisfies a " higher order " independence property of this kind. The case k = 2 is of particular importance, and we mention a number of applications to topology, algebraic geometry, electrical networks and chemical kinetics. 1. Power independence We assume that the reader is familiar with the terminology and elementary theory of matroids (combinatorial geometries) as found in, for example, CrapoRota [5]. We say that a matroid is simple if the empty set as well as every point is closed. If we can coordinatize a matroid with the columns of a nonzero matrix A, then the matroid is simple if and only if no column of A is a multiple of another. If G(S) is a nonsimple matroid, its simplification is the simple matroid obtained by removing the points in 0 and identifying all multiple points of G(S).
Faster Mixing for Conductance Methods
"... We consider isoperimetric (geometric) methods for showing rapid convergence of Markov chains. ..."
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Cited by 1 (1 self)
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We consider isoperimetric (geometric) methods for showing rapid convergence of Markov chains.
On Extremal NearRegular and 6√1Matroids
 GRAPHS COMBIN. 14 (1998), 163–179
, 1998
"... The classes of nearregular and 6 p 1{matroids arise in the study of matroids representable over GF (3) and other elds. For example, a matroid is representable over all elds except possibly GF (2) if and only if it is nearregular, and a matroid is representable over GF (3) and GF (4) if and only i ..."
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The classes of nearregular and 6 p 1{matroids arise in the study of matroids representable over GF (3) and other elds. For example, a matroid is representable over all elds except possibly GF (2) if and only if it is nearregular, and a matroid is representable over GF (3) and GF (4) if and only if it is a 6 p 1{matroid. This paper determines the maximum sizes of a simple rank{r nearregular and a simple rank{r 6 p 1matroid and determines all such matroids having these sizes.
Structure in MinorClosed Classes of Matroids
"... This paper gives an informal introduction to structure theory for minorclosed classes of matroids representable over a fixed finite field. The early sections describe some historical results that give evidence that welldefined structure exists for members of such classes. In later sections we descr ..."
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This paper gives an informal introduction to structure theory for minorclosed classes of matroids representable over a fixed finite field. The early sections describe some historical results that give evidence that welldefined structure exists for members of such classes. In later sections we describe the fundamental classes and other features that necessarily appear in structure theory for minorclosed classes of matroids. We conclude with an informal statement of the structure theorem itself. This theorem generalises the Graph Minors Structure Theorem of Robertson and Seymour. 1
Assortment Planning under the Multinomial Logit Model with Totally Unimodular Constraint Structures
, 2013
"... We consider constrained assortment problems assuming that customers select according to the multinomial logit model (MNL). The objective is to find an assortment that maximizes the expected revenue per customer and satisfies a set of totally unimodular constraints. We show that this fractional binar ..."
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We consider constrained assortment problems assuming that customers select according to the multinomial logit model (MNL). The objective is to find an assortment that maximizes the expected revenue per customer and satisfies a set of totally unimodular constraints. We show that this fractional binary problem can be solved as an equivalent linear program. We use this result to solve five classes of practical assortment optimization and pricing models under MNL, including (1) assortment models with various bounds on the cardinality of the assortment, (2) assortment models where we need to decide the display location of the selected products, (3) pricing models with a finite menu of possible prices, (4) quality consistent pricing models where the prices of the products have to follow a specified quality ordering, (5) assortment models with precedence constraints. We show that all of these classes of problems can be solved as linear programs. In some instances, constraints can be combined as long as total unimodularity is preserved. In addition, we show how the results extend to a larger class of attraction choice models that avoid some of the shortcomings of MNL. The problem of selecting assortments to maximize expected profits or expected welfare arises in a variety of industries ranging from transportation and retailing to travel and leisure. There is a
unknown title
, 1998
"... The class of matroids representable over all fields is the class of regular matroids. The class of matroids representable over all fields except perhaps GF(2) is the class of nearregular matroids. Let k be a nonnegative integer. This thesis considers the class of k–regular matroids, a generalizati ..."
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The class of matroids representable over all fields is the class of regular matroids. The class of matroids representable over all fields except perhaps GF(2) is the class of nearregular matroids. Let k be a nonnegative integer. This thesis considers the class of k–regular matroids, a generalization of the last two classes. Indeed, the classes of regular and nearregular matroids coincide with the classes of 0–regular and 1–regular matroids, respectively. This thesis extends many results for regular and nearregular matroids. In particular, for all k, the class of k–regular matroids is precisely the class of matroids representable over a particular partial field. Every 3–connected member of the classes of either regular or nearregular matroids has a unique representability property. This thesis extends this property to the 3–connected members of the class of k–regular matroids for all k. A matroid is ω–regular if it is k–regular for some k. It is shown that, for all k ≥ 0, every 3–connected k–regular matroid is uniquely representable over the partial field canonically associated with the