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494
The multivariate Tutte polynomial (alias Potts model) for graphs and matroids
 In Surveys in combinatorics 2005, volume 327 of London Math. Soc. Lecture Note Ser
, 2005
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The Bergman complex of a matroid and phylogenetic trees
 THE JOURNAL OF COMBINATORIAL THEORY, SERIES B
, 2005
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Lowdegree tests at large distances
 In Proceedings of the 39th Annual ACM Symposium on Theory of Computing
, 2007
"... Abstract We define tests of boolean functions which distinguish between linear (or quadratic)polynomials, and functions which are very far, in an appropriate sense, from these polynomials. The tests have optimal or nearly optimal tradeoffs between soundness and thenumber of queries. In particular, ..."
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Cited by 37 (2 self)
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Abstract We define tests of boolean functions which distinguish between linear (or quadratic)polynomials, and functions which are very far, in an appropriate sense, from these polynomials. The tests have optimal or nearly optimal tradeoffs between soundness and thenumber of queries. In particular, we show that functions with small Gowers uniformity norms behave &quot;randomly &quot; with respect to hypergraph linearity tests. A central step in our analysis of quadraticity tests is the proof of an inverse theorem forthe third Gowers uniformity norm of boolean functions. The last result has also a coding theory application. It is possible to estimate efficientlythe distance from the secondorder ReedMuller code on inputs lying far beyond its listdecoding radius.
Submodular Maximization Over Multiple Matroids via Generalized Exchange Properties
, 2009
"... Submodularfunction maximization is a central problem in combinatorial optimization, generalizing many important NPhard problems including Max Cut in digraphs, graphs and hypergraphs, certain constraint satisfaction problems, maximumentropy sampling, and maximum facilitylocation problems. Our mai ..."
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Cited by 31 (6 self)
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Submodularfunction maximization is a central problem in combinatorial optimization, generalizing many important NPhard problems including Max Cut in digraphs, graphs and hypergraphs, certain constraint satisfaction problems, maximumentropy sampling, and maximum facilitylocation problems. Our main result is that for any k ≥ 2 and any ε> 0, there is a natural localsearch algorithm which has approximation guarantee of 1/(k + ε) for the problem of maximizing a monotone submodular function subject to k matroid constraints. This improves a 1/(k + 1)approximation of Nemhauser, Wolsey and Fisher, obtained more than 30 years ago. Also, our analysis can be applied to the problem of maximizing a linear objective function and even a general nonmonotone submodular function subject to k matroid constraints. We show that in these cases the approximation guarantees of our algorithms are 1/(k − 1 + ε) and 1/(k + 1 + 1/k + ε), respectively.
The excluded minors for GF(4)representable matroids
, 1997
"... There are exactly seven excluded minors for the class of GF(4)representable matroids. 1 Introduction We prove the following theorem. Theorem 1.1 A matroid M is GF(4)representable if and only if M has no minor isomorphic to any of U 2;6 , U 4;6 , P 6 , F \Gamma 7 , F \Gamma 7 , P 8 , and ..."
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Cited by 31 (9 self)
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There are exactly seven excluded minors for the class of GF(4)representable matroids. 1 Introduction We prove the following theorem. Theorem 1.1 A matroid M is GF(4)representable if and only if M has no minor isomorphic to any of U 2;6 , U 4;6 , P 6 , F \Gamma 7 , F \Gamma 7 , P 8 , and P 00 8 . The definitions for these matroids, with a summary of their interesting properties, can be found in the Appendix. Other than P 00 8 , they were all known to be excluded minors for GF(4) representability (see Oxley [13,15]). The matroid P 00 8 is obtained by relaxing the unique pair of disjoint circuithyperplanes of P 8 . Ever since Whitney's introductory paper [24] on matroid theory, researchers have sought ways to distinguish the representable matroids. For any field F, the class of Frepresentable matroids is closed under taking minors. Thus, it is natural to characterize the minorminimal matroids that are not Frepresentable; we refer to such matroids as excluded ...
Discrete polymatroids
 J. ALGEBRAIC COMBIN
, 2003
"... The discrete polymatroid is a multiset analogue of the matroid. Based on the polyhedral theory on integral polymatroids developed in late 1960’s and in early 1970’s, in the present paper the combinatorics and algebra on discrete polymatroids will be studied. ..."
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Cited by 29 (3 self)
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The discrete polymatroid is a multiset analogue of the matroid. Based on the polyhedral theory on integral polymatroids developed in late 1960’s and in early 1970’s, in the present paper the combinatorics and algebra on discrete polymatroids will be studied.
Finding branchdecompositions and rankdecompositions
, 2007
"... Abstract. We present a new algorithm that can output the rankdecomposition of width at most k of a graph if such exists. For that we use an algorithm that, for an input matroid represented over a fixed finite field, outputs its branchdecomposition of width at most k if such exists. This algorithm w ..."
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Cited by 28 (1 self)
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Abstract. We present a new algorithm that can output the rankdecomposition of width at most k of a graph if such exists. For that we use an algorithm that, for an input matroid represented over a fixed finite field, outputs its branchdecomposition of width at most k if such exists. This algorithm works also for partitioned matroids. Both these algorithms are fixedparameter tractable, that is, they run in time O(n 3) for each fixed value of k where n is the number of vertices / elements of the input. (The previous best algorithm for construction of a branchdecomposition or a rankdecomposition of optimal width due to Oum and Seymour [Testing branchwidth. J. Combin. Theory Ser. B, 97(3) (2007) 385–393] is not fixedparameter tractable.)
Triangulations Of Oriented Matroids
, 1997
"... We consider the concept of triangulation of an oriented matroid. We provide a definition which generalizes the previous ones by Billera Munson and by Anderson and which specializes to the usual notion of triangulation (or simplicial fan) in the realizable case. Then we study the relation existing ..."
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Cited by 25 (12 self)
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We consider the concept of triangulation of an oriented matroid. We provide a definition which generalizes the previous ones by Billera Munson and by Anderson and which specializes to the usual notion of triangulation (or simplicial fan) in the realizable case. Then we study the relation existing between triangulations of an oriented matroid M and extensions of its dual M , via the socalled lifting triangulations. We show that this duality behaves particularly well in the class of Lawrence matroid polytopes. In particular, that the extension space conjecture for realizable oriented matroids posed by Sturmfels and Ziegler is equivalent to the restriction to Lawrence polytopes of the Generalized Baues problem for subdivisions of polytopes. We finish showing examples and a combinatorial characterization of lifting triangulations. Introduction Matroids (see [23]) and oriented matroids (see [8]) are axiomatic abstract models for combinatorial geometry over general fields and ordere...
Lattice path matroids: enumerative aspects and Tutte polynomials
 J. Combin. Theory Ser. A
, 2003
"... Abstract. Fix two lattice paths P and Q from (0, 0) to (m, r) that use East and North steps with P never going above Q. We show that the lattice paths that go from (0, 0) to (m, r) and that remain in the region bounded by P and Q can be identified with the bases of a particular type of transversal m ..."
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Cited by 24 (9 self)
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Abstract. Fix two lattice paths P and Q from (0, 0) to (m, r) that use East and North steps with P never going above Q. We show that the lattice paths that go from (0, 0) to (m, r) and that remain in the region bounded by P and Q can be identified with the bases of a particular type of transversal matroid, which we call a lattice path matroid. We consider a variety of enumerative aspects of these matroids and we study three important matroid invariants, namely the Tutte polynomial and, for special types of lattice path matroids, the characteristic polynomial and the β invariant. In particular, we show that the Tutte polynomial is the generating function for two basic lattice path statistics and we show that certain sequences of lattice path matroids give rise to sequences of Tutte polynomials for which there are relatively simple generating functions. We show that Tutte polynomials of lattice path matroids can be computed in polynomial time. Also, we obtain a new result about lattice paths from an analysis of the β invariant of certain lattice path matroids. 1.
On the rank of a tropical matrix
"... Abstract. This is a foundational paper in tropical linear algebra, which is linear algebra over the minplus semiring. We introduce and compare three natural definitions of the rank of a matrix, called the Barvinok rank, the Kapranov rank and the tropical rank. We demonstrate how these notions arise ..."
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Cited by 24 (5 self)
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Abstract. This is a foundational paper in tropical linear algebra, which is linear algebra over the minplus semiring. We introduce and compare three natural definitions of the rank of a matrix, called the Barvinok rank, the Kapranov rank and the tropical rank. We demonstrate how these notions arise naturally in polyhedral and algebraic geometry, and we show that they differ in general. Realizability of matroids plays a crucial role here. Connections to optimization are also discussed. 1.