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713
The multivariate tutte polynomial (alias potts model) for graphs and matroids
 Surveys in combinatorics
, 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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On the Index Coding Problem and its Relation to Network Coding and Matroid Theory
"... The index coding problem has recently attracted a significant attention from the research community due to its theoretical significance and applications in wireless adhoc networks. An instance of the index coding problem includes a sender that holds a set of information messages X = {x1,..., xk} an ..."
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Cited by 62 (5 self)
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The index coding problem has recently attracted a significant attention from the research community due to its theoretical significance and applications in wireless adhoc networks. An instance of the index coding problem includes a sender that holds a set of information messages X = {x1,..., xk} and a set of receivers R. Each receiver ρ = (x,H) in R needs to obtain a message x ∈ X and has prior side information consisting of a subset H of X. The sender uses a noiseless communication channel to broadcast encoding of messages in X to all clients. The objective is to find an encoding scheme that minimizes the number of transmissions required to satisfy the demands of all the receivers. In this paper, we analyze the relation between the index coding problem, the more general network coding problem, and the problem of finding a linear representation of a matroid. In particular, we show that any instance of the network coding and matroid representation problems can be efficiently reduced to an instance of the index coding problem. Our reduction implies that many important properties of the network coding and matroid representation problems carry over to the index coding problem. Specifically, we show that vector linear codes outperform scalar linear index codes and that vector linear codes are insufficient for achieving the optimum number of transmissions.
MATROID POLYTOPES, NESTED SETS AND BERGMAN FANS
, 2004
"... The tropical variety defined by linear equations with constant coefficients is the Bergman fan of the corresponding matroid. Building on a selfcontained introduction to matroid polytopes, we present a geometric construction of the Bergman fan, and we discuss its relationship with the simplicial com ..."
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Cited by 57 (6 self)
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The tropical variety defined by linear equations with constant coefficients is the Bergman fan of the corresponding matroid. Building on a selfcontained introduction to matroid polytopes, we present a geometric construction of the Bergman fan, and we discuss its relationship with the simplicial complex of nested sets in the lattice of flats. The Bergman complex is triangulated by the nested set complex, and the two complexes coincide if and only if every connected flat remains connected after contracting along any subflat. This sharpens a result of ArdilaKlivans who showed that the Bergman complex is triangulated by the order complex of the lattice of flats. The nested sets specify the De ConciniProcesi compactification of the complement of a hyperplane arrangement, while the Bergman fan specifies the tropical compactification. These two compactifications are almost equal, and we highlight the subtle differences.
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 54 (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.
Finding Branchdecomposition and Rankdecomposition
 SIAM J. Comput
"... 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 ..."
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Cited by 53 (4 self)
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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(n3) where n is the number of vertices / elements of the input, for each constant value of k and any fixed finite field. 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.
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 50 (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 47 (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.
Networks, matroids, and nonShannon information inequalities
 IEEE TRANSACTIONS ON INFORMATION THEORY
, 2007
"... We define a class of networks, called matroidal networks, which includes as special cases all scalarlinearly solvable networks, and in particular solvable multicast networks. We then present a method for constructing matroidal networks from known matroids. We specifically construct networks that p ..."
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Cited by 47 (7 self)
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We define a class of networks, called matroidal networks, which includes as special cases all scalarlinearly solvable networks, and in particular solvable multicast networks. We then present a method for constructing matroidal networks from known matroids. We specifically construct networks that play an important role in proving results in the literature, such as the insufficiency of linear network coding and the unachievability of network coding capacity. We also construct a new network, from the Vámos matroid, which we call the Vámos network, and use it to prove that Shannontype information inequalities are in general not sufficient for computing network coding capacities. To accomplish this, we obtain a capacity upper bound for the Vámos network using a nonShannontype information inequality discovered in 1998 by Zhang and Yeung, and then show that it is smaller than any such bound derived from Shannontype information inequalities. This is the first application of a nonShannontype inequality to network coding. We also compute the exact routing capacity and linear coding capacity of the Vámos network. Finally, using a variation of the Vámos network, we prove that Shannontype information inequalities are insufficient even for computing network coding capacities of multipleunicast networks.
Approximating Rankwidth and Cliquewidth Quickly
, 2006
"... Rankwidth was defined by Oum and Seymour [2006. Approximating cliquewidth and branchwidth. J. Combin. Theory Ser. B 96, 4, 514–528] to investigate cliquewidth. They constructed an algorithm that either outputs a rankdecomposition of width at most f(k) for some function f or confirms that rankw ..."
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Cited by 47 (4 self)
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Rankwidth was defined by Oum and Seymour [2006. Approximating cliquewidth and branchwidth. J. Combin. Theory Ser. B 96, 4, 514–528] to investigate cliquewidth. They constructed an algorithm that either outputs a rankdecomposition of width at most f(k) for some function f or confirms that rankwidth is larger than k in time O(V 9 log V ) for an input graph G = (V,E) and a fixed k. We develop three separate algorithms of this kind with faster running time. We construct an O(V 4)time algorithm with f(k) = 3k + 1 by constructing a subroutine for the previous algorithm; we avoid generic algorithms minimizing submodular functions used by Oum and Seymour. Another one is an O(V 3)time algorithm with f(k) = 24k by giving a reduction from graphs to binary matroids; then we use an approximation algorithm for matroid branchwidth by Hliněny ́ [2005. A parametrized algorithm for matroid branchwidth. SIAM J. Comput. 35, 2, 259–277]. Finally we construct an O(V 3)time algorithm with f(k) = 3k − 1 by combining the ideas of above two cited papers.