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22
Eigenvalue bounds, spectral partitioning, and metrical deformations via flows
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A decomposition theory for binary linear codes
, 2008
"... The decomposition theory of matroids initiated by Paul Seymour in the 1980’s has had an enormous impact on research in matroid theory. This theory, when applied to matrices over the binary field, yields a powerful decomposition theory for binary linear codes. In this paper, we give an overview of ..."
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Cited by 17 (3 self)
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The decomposition theory of matroids initiated by Paul Seymour in the 1980’s has had an enormous impact on research in matroid theory. This theory, when applied to matrices over the binary field, yields a powerful decomposition theory for binary linear codes. In this paper, we give an overview of this code decomposition theory, and discuss some of its implications in the context of the recently discovered formulation of maximumlikelihood (ML) decoding of a binary linear code over a discrete memoryless channel as a linear programming problem. We translate matroidtheoretic results of Grötschel and Truemper from the combinatorial optimization literature to give examples of nontrivial families of codes for which the ML decoding problem can be solved in time polynomial in the length of the code. One such family is that consisting of codes C for which the codeword polytope is identical to the KoetterVontobel fundamental polytope derived from the entire dual code C ⊥. However, we also show that such families of codes are not good in a codingtheoretic sense — either their dimension or their minimum distance must grow sublinearly with codelength.
A simpler algorithm and shorter proof for the graph minor decomposition
 In Proceedings of the 43rd ACM Symposium on Theory of Computing
, 2011
"... At the core of the RobertsonSeymour theory of graph minors lies a powerful decomposition theorem which captures, for any fixed graph H, the common structural features of all the graphs which do not contain H as a minor. Robertson and Seymour used this result to prove Wagner’s Conjecture that finit ..."
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Cited by 12 (3 self)
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At the core of the RobertsonSeymour theory of graph minors lies a powerful decomposition theorem which captures, for any fixed graph H, the common structural features of all the graphs which do not contain H as a minor. Robertson and Seymour used this result to prove Wagner’s Conjecture that finite graphs are wellquasiordered under the graph minor relation, as well as give a polynomial time algorithm for the disjoint paths problem when the number of the terminals is fixed. The theorem has since found numerous applications, both in graph theory and theoretical computer science. The original proof runs more than 400 pages and the techniques used are highly nontrivial. In this paper, we give a simplified algorithm for finding the decomposition based on a new constructive proof of the decomposition theorem for graphs excluding a fixed minor H. The new proof is both dramatically simpler and shorter, making these results and techniques more accessible. The algorithm runs in time O(n3), as does the original algorithm of Robertson and Seymour. Moreover, our proof gives an explicit bound on the constants in the O notation, whereas the original proof of Robertson and Seymour does not. Categories and Subject Descriptors G.2.2 [Discrete Mathematics]: Graph Theory—graph algorithms, path and circuit problems
On the Geometry of Graphs with a Forbidden Minor
"... We study the topological simplification of graphs via random embeddings, leading ultimately to a reduction of the GuptaNewmanRabinovichSinclair (GNRS) L1 embedding conjecture to a pair of manifestly simpler conjectures. The GNRS conjecture characterizes all graphs that have an O(1)approximate mul ..."
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We study the topological simplification of graphs via random embeddings, leading ultimately to a reduction of the GuptaNewmanRabinovichSinclair (GNRS) L1 embedding conjecture to a pair of manifestly simpler conjectures. The GNRS conjecture characterizes all graphs that have an O(1)approximate multicommodity maxflow/mincut theorem. In particular, its resolution would imply a constant factor approximation for the general Sparsest Cut problem in every family of graphs which forbids some minor. In the course of our study, we prove a number of results of independent interest. • Every metric on a graph of pathwidth k embeds into a distribution over trees with distortion depending only on k. This is equivalent to the statement that any family of graphs excluding a fixed tree embeds into a distribution over trees with O(1) distortion. For graphs of treewidth k, GNRS showed that this is impossible even for k = 2. In particular, our result implies that pathwidthk metrics embed into L1 with bounded distortion, which resolves an open question even for k = 3. • We prove a generic peeling lemma which uses random retractions to peel simple structures like handles and apices off of graphs. This allows a number of new topological reductions. For example, if X is any metric space in which the removal of O(1) points leaves a bounded genus metric, then X embeds into a distribution over planar graphs. • Using these techniques, we show that the GNRS embedding conjecture is equivalent to two simpler conjectures: (1) The wellknown planar embedding conjecture, and (2) a conjecture about embeddings of ksums of graphs.
Minors in random and expanding hypergraphs
 in 27th SoCG’, ACM
, 2011
"... We introduce a new notion of minors for simplicial complexes (hypergraphs), socalled homological minors. Our motivation is to propose a general approach to attack certain extremal problems for sparse simplicial complexes and the corresponding threshold problems for random complexes. In this paper, ..."
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We introduce a new notion of minors for simplicial complexes (hypergraphs), socalled homological minors. Our motivation is to propose a general approach to attack certain extremal problems for sparse simplicial complexes and the corresponding threshold problems for random complexes. In this paper, we focus on threshold problems. The basic model for random complexes is the LinialMeshulam model X k (n, p). By definition, such a complex has n vertices, a complete (k − 1)dimensional skeleton, and every possible kdimensional simplex is chosen independently with probability p. We show that for every k, t ≥ 1, there is a constant C = C(k, t) such that for p ≥ C/n, the random complex X k (n, p) asymptotically almost surely contains K k t (the complete kdimensional complex on t vertices) as a homological minor. As corollary, the threshold for (topological) embeddability of X k (n, p) into R 2k is at p = Θ(1/n). The method can be extended to other models of random complexes (for which the lower skeleta are not necessarily complete) and also to more general Tverbergtype problems, where instead of continuous maps without doubly covered image points (embeddings), we consider maps without qfold covered image points.
Maximum matching in graphs with an excluded minor
 Proceedings of the Eighteenth Annual ACMSIAM Symposium on Discrete Algorithms (SODA) 108–117
, 2007
"... Abstract We present a new randomized algorithm for findinga maximum matching in Hminor free graphs. Forevery fixed H, our algorithm runs in O(n3!/(!+3)) < O(n1.326) time, where n is the number of verticesof the input graph and! < 2.376 is the exponentof matrix multiplication. This improves up ..."
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Cited by 8 (5 self)
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Abstract We present a new randomized algorithm for findinga maximum matching in Hminor free graphs. Forevery fixed H, our algorithm runs in O(n3!/(!+3)) < O(n1.326) time, where n is the number of verticesof the input graph and! < 2.376 is the exponentof matrix multiplication. This improves upon the previous O(n1.5) time bound obtained by applying the O(mn1/2)time algorithm of Micali and Vazirani on thisimportant class of graphs. For graphs with bounded genus, which are special cases of Hminor free graphs, we present a randomized algorithm for finding a maximum matching in O(n!/2) < O(n1.19) time. This extends a previous randomized algorithm of Mucha and Sankowski, having the same running time, that finds a maximum matching ina planar graphs. We also present a deterministic algorithm with arunning time of O(n1+!/2) < O(n2.19) for counting thenumber of perfect matchings in graphs with bounded genus. This algorithm combines the techniques usedby the algorithms above with the counting technique of Kasteleyn. Using this algorithm we can also count,within the same running time, the number of Tjoinsin planar graphs. As special cases, we get algorithms for counting Eulerian subgraphs (T = OE) and oddsubgraphs ( T = V) of planar graphs. 1 Introduction A matching in a graph is a set of pairwise disjointedges. A perfect matching in a graph with n verticesis a matching of size n/2, and a maximum matchingis a matching of largest possible size. The problems
On Tractable Parameterizations of Graph Isomorphism
"... Abstract. The fixedparameter tractability of graph isomorphism is an open problem with respect to a number of natural parameters, such as treewidth, genus and maximum degree. We show that graph isomorphism is fixedparameter tractable when parameterized by the treedepth of the graph. We also exte ..."
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Cited by 6 (0 self)
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Abstract. The fixedparameter tractability of graph isomorphism is an open problem with respect to a number of natural parameters, such as treewidth, genus and maximum degree. We show that graph isomorphism is fixedparameter tractable when parameterized by the treedepth of the graph. We also extend this result to a parameter generalizing both treedepth and maxleafnumber by deploying new variants of copsandrobbers games. 1
Matroid Secretary for Regular and Decomposable Matroids
"... In the matroid secretary problem we are given a stream of elements and asked to choose a set of elements that maximizes the total value of the set, subject to being an independent set of a matroid given in advance. The difficulty comes from the assumption that decisions are irrevocable: if we choose ..."
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Cited by 5 (0 self)
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In the matroid secretary problem we are given a stream of elements and asked to choose a set of elements that maximizes the total value of the set, subject to being an independent set of a matroid given in advance. The difficulty comes from the assumption that decisions are irrevocable: if we choose to accept an element when it is presented by the stream then we can never get rid of it, and if we choose not to accept it then we cannot later add it. Babaioff, Immorlica, and Kleinberg [SODA 2007] introduced this problem, gave O(1)competitive algorithms for certain classes of matroids, and conjectured that every matroid admits an O(1)competitive algorithm. However, most matroids that are known to admit an O(1)competitive algorithm can be easily represented using graphs (e.g. graphic, cographic, and transversal matroids). In particular, there is very little known about Frepresentable matroids (the class of matroids that can be represented as elements of a vector space over a field F), which are one of the foundational types of matroids. Moreover, most of the known techniques are as dependent on graph theory as they are on matroid theory. We go beyond graphs by giving O(1)competitive algorithms for regular matroids (the class of matroids that are representable over any field), and use techniques
Combinatorial optimization with 2joins
 Journal of Combinatorial Theory, Series B
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FPTalgorithms for connected feedback vertex set
, 2009
"... We study the recently introduced CONNECTED FEEDBACK VERTEX SET (CFVS) problem from the viewpoint of parameterized algorithms. CFVS is the connected variant of the classical FEEDBACK VERTEX SET problem and is defined as follows: given a graph G = (V, E) and an integer k, decide whether there exists ..."
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Cited by 5 (3 self)
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We study the recently introduced CONNECTED FEEDBACK VERTEX SET (CFVS) problem from the viewpoint of parameterized algorithms. CFVS is the connected variant of the classical FEEDBACK VERTEX SET problem and is defined as follows: given a graph G = (V, E) and an integer k, decide whether there exists F ⊆ V, F  ≤ k, such that G[V \ F] is a forest and G[F] is connected. We show that CONNECTED FEEDBACK VERTEX SET can be solved in time O(2 O(k) n O(1)) on general graphs and in time O(2 O( √ k log k) n O(1)) on graphs excluding a fixed graph H as a minor. Our result on general undirected graphs uses, as a subroutine, a parameterized algorithm for GROUP STEINER TREE, a well studied variant of STEINER TREE. We find the algorithm for GROUP STEINER TREE of independent interest and believe that it could be useful for obtaining parameterized algorithms for other connectivity problems.