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Turing kernelization for finding long paths and cycles in restricted graph classes
 In Proc. 22nd ESA
, 2014
"... Abstract. We analyze the potential for provably effective preprocessing for the problems of finding paths and cycles with at least k edges. Several years ago, the question was raised whether the existing superpolynomial kernelization lower bounds for kPath and kCycle can be circumvented by relaxin ..."
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Abstract. We analyze the potential for provably effective preprocessing for the problems of finding paths and cycles with at least k edges. Several years ago, the question was raised whether the existing superpolynomial kernelization lower bounds for kPath and kCycle can be circumvented by relaxing the requirement that the preprocessing algorithm outputs a single instance. To this date, very few examples are known where the relaxation to Turing kernelization is fruitful. We provide a novel example by giving polynomialsize Turing kernels for kPath and kCycle on planar graphs, graphs of maximum degree t, clawfree graphs, and K3,tminorfree graphs, for each constant t ≥ 3. Concretely, we present algorithms for kPath (kCycle) on these restricted graph families that run in polynomial time when they are allowed to query an external oracle for the answers to kPath (kCycle) instances of size and parameter bounded polynomially in k. Our kernelization schemes are based on a new methodology called DecomposeQueryReduce. 1
Preprocessing Subgraph and Minor Problems: When Does a Small Vertex Cover Help?
, 2014
"... We prove a number of results around kernelization of problems parameterized by the size of a given vertex cover of the input graph. We provide three sets of simple general conditions characterizing problems admitting kernels of polynomial size. Our characterizations not only give generic explanatio ..."
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We prove a number of results around kernelization of problems parameterized by the size of a given vertex cover of the input graph. We provide three sets of simple general conditions characterizing problems admitting kernels of polynomial size. Our characterizations not only give generic explanations for the existence of many known polynomial kernels for problems like qColoring, Odd Cycle Transversal, Chordal Deletion, ηTransversal, or Long Path, parameterized by the size of a vertex cover, but also imply new polynomial kernels for problems like FMinorFree Deletion, which is to delete at most k vertices to obtain a graph with no minor from a fixed finite set F. While our characterization captures many interesting problems, the kernelization complexity landscape of parameterizations by vertex cover is much more involved. We demonstrate this by several results about induced subgraph and minor containment testing, which we find surprising. While it was known that testing for an induced complete subgraph has no polynomial kernel unless NP ⊆ coNP/poly, we show that the problem of testing if a graph contains a complete graph on t vertices as a minor admits a polynomial kernel. On the other hand, it was known that testing for a path on t vertices as a minor admits a polynomial kernel, but we show that testing for containment of an induced path on t vertices is unlikely to admit a polynomial kernel.
Dynamic Dominating Set and TurboCharging Greedy Heuristics
, 2014
"... The main purpose of this paper is to exposit two very different, but very general, motivational schemes in the art of parameterization and a concrete example connecting them. We introduce a dynamic version of the DOMINATING SET problem and prove that it is fixedparameter tractable (FPT). The probl ..."
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The main purpose of this paper is to exposit two very different, but very general, motivational schemes in the art of parameterization and a concrete example connecting them. We introduce a dynamic version of the DOMINATING SET problem and prove that it is fixedparameter tractable (FPT). The problem is motivated by settings where problem instances evolve. It also arises in the quest to improve a natural greedy heuristic for the DOMINATING SET problem.
Finding Highly Connected Subgraphs
"... A popular way of formalizing clusters in networks are highly connected subgraphs, that is, subgraphs of k vertices that have edge connectivity larger than k/2 (equivalently, minimum degree larger than k/2). We examine the computational complexity of finding highly connected subgraphs. We show tha ..."
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A popular way of formalizing clusters in networks are highly connected subgraphs, that is, subgraphs of k vertices that have edge connectivity larger than k/2 (equivalently, minimum degree larger than k/2). We examine the computational complexity of finding highly connected subgraphs. We show that the problem is NPhard. Thus, we explore possible parameterizations, such as the solution size, number of vertices in the input, the size of a vertex cover in the input, and the number of edges outgoing from the solution (edge isolation). For some parameters, we find strong intractability results; among the parameters yielding tractability, the edge isolation seems to provide the best tradeoff between running time bounds and a small value of the parameter in relevant instances.