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Explicit linear kernels via dynamic programming
- IN STACS, VOLUME 25 OF LIPICS
, 2014
"... Several algorithmic meta-theorems on kernelization have appeared in the last years, starting with the result of Bodlaender et al. [FOCS 2009] on graphs of bounded genus, then generalized by Fomin et al. [SODA 2010] to graphs excluding a fixed minor, and by Kim et al. [ICALP 2013] to graphs excluding ..."
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Several algorithmic meta-theorems on kernelization have appeared in the last years, starting with the result of Bodlaender et al. [FOCS 2009] on graphs of bounded genus, then generalized by Fomin et al. [SODA 2010] to graphs excluding a fixed minor, and by Kim et al. [ICALP 2013] to graphs excluding a fixed topological minor. Typically, these results guarantee the existence of linear or polynomial kernels on sparse graph classes for problems satisfying some generic conditions but, manly due to their generality, it is not known how to derive from them constructive kernels with explicit constants. In this paper we make a step toward a fully constructive meta-kernelization theory on sparse graphs. Our approach is based on a more explicit protrusion replacement machinery that, instead of expressibility in CMSO logic, uses dynamic programming, which allows us to find an explicit upper bound on the size of the derived kernels. We demonstrate the usefulness of our techniques by providing the first explicit linear kernels for r-Dominating Set and r-Scattered Set on apex-minor-free graphs, and for Planar-F-Deletion and Planar-F-Packing on graphs excluding a fixed (topological) minor in the case where all the graphs in F are connected.
Complexity of counting subgraphs: Only the boundedness of the vertex-cover number counts
"... Abstract—For a class C of graphs, #Sub(C) is the counting problem that, given a graph H from C and an arbitrary graph G, asks for the number of subgraphs of G isomorphic to H. It is known that if C has bounded vertex-cover number (equiv-alently, the size of the maximum matching in C is bounded), the ..."
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Abstract—For a class C of graphs, #Sub(C) is the counting problem that, given a graph H from C and an arbitrary graph G, asks for the number of subgraphs of G isomorphic to H. It is known that if C has bounded vertex-cover number (equiv-alently, the size of the maximum matching in C is bounded), then #Sub(C) is polynomial-time solvable. We complement this result with a corresponding lower bound: if C is any recursively enumerable class of graphs with unbounded vertex-cover number, then #Sub(C) is #W[1]-hard parameterized by the size of H and hence not polynomial-time solvable and not even fixed-parameter tractable, unless FPT is equal to #W[1]. As a first step of the proof, we show that counting k-matchings in bipartite graphs is #W[1]-hard. Recently, Curt-icapean [ICALP 2013] proved the #W[1]-hardness of counting k-matchings in general graphs; our result strengthens this statement to bipartite graphs with a considerably simpler proof and even shows that, assuming the Exponential Time Hy-pothesis (ETH), there is no f(k)no(k / log k) time algorithm for counting k-matchings in bipartite graphs for any computable function f. As a consequence, we obtain an independent and somewhat simpler proof of the classical result of Flum and Grohe [SICOMP 2004] stating that counting paths of length k is #W[1]-hard, as well as a similar almost-tight ETH-based lower bound on the exponent. I.
The All-or-Nothing Flow Problem in Directed Graphs with Symmetric Demand Pairs
, 2014
"... We study the approximability of the All-or-Nothing multicommodity flow problem in directed graphs with symmetric demand pairs (SymANF). The input consists of a directed graph G = (V,E) and a collection of (unordered) pairs of nodesM = {s1t1, s2t2,..., sktk}. A subsetM ′ of the pairs is routable if t ..."
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We study the approximability of the All-or-Nothing multicommodity flow problem in directed graphs with symmetric demand pairs (SymANF). The input consists of a directed graph G = (V,E) and a collection of (unordered) pairs of nodesM = {s1t1, s2t2,..., sktk}. A subsetM ′ of the pairs is routable if there is a feasible multicommodity flow in G such that, for each pair siti ∈ M′, the amount of flow from si to ti is at least one and the amount of flow from ti to si is at least one. The goal is to find a maximum cardinality subset of the given pairs that can be routed. Our main result is a poly-logarithmic approximation with constant congestion for SymANF. We obtain this result by extending the well-linked decomposition framework of [11] to the directed graph setting with symmetric demand pairs. We point out the importance of studying routing problems in this setting and the relevance of our result to future work.
Solving d-SAT via Backdoors to Small Treewidth
"... A backdoor set of a CNF formula is a set of variables such that fixing the truth values of the variables from this set moves the formula into a polynomial-time decidable class. In this work we obtain several algorithmic results for solving d-SAT, by exploiting backdoors to d-CNF formulas whose incid ..."
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A backdoor set of a CNF formula is a set of variables such that fixing the truth values of the variables from this set moves the formula into a polynomial-time decidable class. In this work we obtain several algorithmic results for solving d-SAT, by exploiting backdoors to d-CNF formulas whose incidence graphs have small treewidth. For a CNF formula φ and integer t, a strong backdoor set to treewidth t is a set of variables such that each possible partial assignment τ to this set reduces φ to a formula whose incidence graph is of treewidth at most t. A weak backdoor set to treewidth t is a set of variables such that there is a partial assignment to this set that reduces φ to a satisfiable formula of treewidth at most t. Our main contribution is an algorithm that, given a d-CNF formula φ and an integer k, in time 2O(k)|φ|, • either finds a satisfying assignment of φ, or • reports correctly that φ is not satisfiable, or • concludes correctly that φ has no weak or strong backdoor set to treewidth t of size at most k. As a consequence of the above, we show that d-SAT parameterized by the size of a smallest weak/strong backdoor set to formulas of treewidth t, is fixed-parameter tractable. Prior to our work, such results were know only for the very special case of t = 1 (Gaspers and Szeider, ICALP 2012). Our result not only extends the previous work, it also improves the running time substantially. The running time of our algorithm is linear in the input size for every fixed k. Moreover, the exponential dependence on the parameter k is asymptotically optimal under Exponential Time Hypothesis (ETH). One of our main technical contributions is a linear time “protrusion replacer” improving over a O(n log2 n)-time procedure of Fomin et al. (FOCS 2012). The new deterministic linear time protrusion replacer has several applications in kernelization and parameterised algorithms.
Excluded Grid Theorem: Improved and Simplified
, 2015
"... We study the Excluded Grid Theorem of Robertson and Seymour. This is a fundamental result in graph theory, that states that there is some function f: Z+ → Z+, such that for any integer g> 0, any graph of treewidth at least f(g), contains the (g × g)-grid as a minor. Until recently, the best known ..."
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We study the Excluded Grid Theorem of Robertson and Seymour. This is a fundamental result in graph theory, that states that there is some function f: Z+ → Z+, such that for any integer g> 0, any graph of treewidth at least f(g), contains the (g × g)-grid as a minor. Until recently, the best known upper bounds on f were super-exponential in g. A recent work of Chekuri and Chuzhoy provided the first polynomial bound, by showing that treewidth f(g) = O(g98 poly log g) is sufficient to ensure the existence of the (g × g)-grid minor in any graph. In this paper we provide a much simpler proof of the Excluded Grid Theorem, achieving a bound of f(g) = O(g36 poly log g). Our proof is self-contained, except for using prior work to reduce the maxi-mum vertex degree of the input graph to a constant.
The Directed Grid Theorem
, 2014
"... The grid theorem, originally proved by Robertson and Seymour in Graph Minors V in 1986, is one of the most central results in the study of graph minors. It has found numerous applications in algorithmic graph structure theory, for instance in bidimensionality theory, and it is the basis for several ..."
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The grid theorem, originally proved by Robertson and Seymour in Graph Minors V in 1986, is one of the most central results in the study of graph minors. It has found numerous applications in algorithmic graph structure theory, for instance in bidimensionality theory, and it is the basis for several other structure theorems developed in the graph minors project. In the mid-90s, Reed and Johnson, Robertson, Seymour and Thomas (see [26, 18]), indepen-dently, conjectured an analogous theorem for directed graphs, i.e. the existence of a function f: N → N such that every digraph of directed tree-width at least f(k) contains a directed grid of order k. In an unpublished manuscript from 2001, Johnson, Robertson, Seymour and Thomas give a proof of this conjecture for planar digraphs. But for over a decade, this was the most general case proved for the Reed, Johnson, Robertson, Seymour and Thomas conjecture. Only very recently, this result has been extended to all classes of digraphs excluding a fixed undirected graph as a minor (see [22]). In this paper, nearly two decades after the conjecture was made, we are finally able to confirm the Reed, Johnson, Robertson, Seymour and Thomas conjecture in full generality and to prove the directed grid theorem. As consequence of our results we are able to improve results in Reed et al. in 1996 [28] (see also [25]) on disjoint cycles of length at least l and in [20] on quarter-integral disjoint paths. We expect many more algorithmic results to follow from the grid theorem.
