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Width parameters beyond treewidth and their applications
 Computer Journal
, 2007
"... Besides the very successful concept of treewidth (see [Bodlaender, H. and Koster, A. (2007) Combinatorial optimisation on graphs of bounded treewidth. These are special issues on Parameterized Complexity]), many concepts and parameters measuring the similarity or dissimilarity of structures compare ..."
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Besides the very successful concept of treewidth (see [Bodlaender, H. and Koster, A. (2007) Combinatorial optimisation on graphs of bounded treewidth. These are special issues on Parameterized Complexity]), many concepts and parameters measuring the similarity or dissimilarity of structures compared to trees have been born and studied over the past years. These concepts and parameters have proved to be useful tools in many applications, especially in the design of efficient algorithms. Our presented novel look at the contemporary developments of these ‘width ’ parameters in combinatorial structures delivers—besides traditional treewidth and derived dynamic programming schemes—also a number of other useful parameters like branchwidth, rankwidth (cliquewidth) or hypertreewidth. In this contribution, we demonstrate how ‘width ’ parameters of graphs and generalized structures (such as matroids or hypergraphs), can be used to improve the design of parameterized algorithms and the structural analysis in other applications on an abstract level.
A decomposition theory for binary linear codes,” submitted to
 IEEE Trans. Inform. Theory. ArXiv
"... ABSTRACT. 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 over ..."
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ABSTRACT. 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. 1.
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 upon the ..."
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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 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.
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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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.
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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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