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14
A Shorter Proof of the Graph Minor Algorithm  The Unique Linkage Theorem
"... At the core of the seminal Graph Minor Theory of Robertson and Seymour is a powerful theorem which describes the structure of graphs excluding a fixed minor. This result is used to prove Wagner’s conjecture and provide a polynomial time algorithm for the disjoint paths problem when the number of the ..."
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At the core of the seminal Graph Minor Theory of Robertson and Seymour is a powerful theorem which describes the structure of graphs excluding a fixed minor. This result is used to prove Wagner’s conjecture and provide a polynomial time algorithm for the disjoint paths problem when the number of the terminals is fixed (i.e, the Graph Minor Algorithm). However, both results require the full power of the Graph Minor Theory, i.e, the structure theorem. In this paper, we show that this is not true in the latter case. Namely, we provide a new and much simpler proof of the correctness of the Graph Minor Algorithm. Specifically, we prove the “Unique Linkage Theorem ” without using Graph Minors structure theorem. The new argument, in addition to being simpler, is much shorter, cutting the proof by at least 200 pages. We also give a new full proof of correctness of an algorithm for the wellknown edgedisjoint paths problem when the number of the terminals is fixed, which is at most 25 pages long.
Subexponential Algorithms for Partial Cover Problems
"... Partial Cover problems are optimization versions of fundamental and well studied problems like Vertex Cover and Dominating Set. Here one is interested in covering (or dominating) the maximum number of edges (or vertices) using a given number (k) of vertices, rather than covering all edges (or vertic ..."
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Cited by 9 (3 self)
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Partial Cover problems are optimization versions of fundamental and well studied problems like Vertex Cover and Dominating Set. Here one is interested in covering (or dominating) the maximum number of edges (or vertices) using a given number (k) of vertices, rather than covering all edges (or vertices). In general graphs, these problems are hard for parameterized complexity classes when parameterized by k. It was recently shown by Amini et. al. [FSTTCS 08] that Partial Vertex Cover and Partial Dominating Set are fixed parameter tractable on large classes of sparse graphs, namely Hminor free graphs, which include planar graphs and graphs of bounded genus. In particular, it was shown that on planar graphs both problems can be solved in time 2 O(k) n O(1). During the last decade there has been an extensive study on parameterized subexponential algorithms. In particular, it was shown that the classical Vertex Cover and Dominating Set problems can be solved in subexponential time on Hminor free graphs. The techniques developed to obtain subexponential algorithms for classical problems do not apply to partial cover problems. It was left as an open problem by Amini et al. [FSTTCS 08] whether there is a subexponential algorithm for Partial Vertex Cover and Partial Dominating Set. In this paper, we answer the question affirmatively by solving both problems in time 2 O( √ k) n O(1) not only on planar graphs but also on much larger classes of graphs, namely, apexminor free graphs. Compared to previously known algorithms for these problems our algorithms are significantly faster and simpler. 1
OutputSensitive Algorithm for the EdgeWidth of an Embedded Graph
, 2010
"... Let G be an unweighted graph of complexity n cellularly embedded in a surface (orientable or not) of genus g. We describe improved algorithms to compute (the length of) a shortest noncontractible and a shortest nonseparating cycle of G. If k is an integer, we can compute such a nontrivial cycle w ..."
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Cited by 8 (2 self)
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Let G be an unweighted graph of complexity n cellularly embedded in a surface (orientable or not) of genus g. We describe improved algorithms to compute (the length of) a shortest noncontractible and a shortest nonseparating cycle of G. If k is an integer, we can compute such a nontrivial cycle with length at most k in O(gnk) time, or correctly report that no such cycle exists. In particular, on a fixed surface, we can test in linear time whether the edgewidth or facewidth of a graph is bounded from above by a constant. This also implies an outputsensitive algorithm to compute a shortest nontrivial cycle that runs in O(gnk) time, where k is the length of the cycle.
kChordal Graphs: from Cops and Robber to Compact Routing via Treewidth
, 2012
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Tight bounds for linkages in planar graphs
"... The DisjointPaths Problem asks, given a graph G and a set of pairs of terminals (s1,t1),...,(sk,tk), whether there is a collection of k pairwise vertexdisjoint paths linking si and ti, for i =1,...,k. In their f(k) · n 3 algorithm for this problem, Robertson and Seymour introduced the irrelevant ..."
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Cited by 8 (1 self)
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The DisjointPaths Problem asks, given a graph G and a set of pairs of terminals (s1,t1),...,(sk,tk), whether there is a collection of k pairwise vertexdisjoint paths linking si and ti, for i =1,...,k. In their f(k) · n 3 algorithm for this problem, Robertson and Seymour introduced the irrelevant vertex technique according to which in every instance of treewidth greater than g(k) there is an “irrelevant” vertex whose removal creates an equivalent instance of the problem. This fact is based on the celebrated Unique Linkage Theorem, whose – very technical – proof gives a function g(k) that is responsible for an immense parameter dependence in the running time of the algorithm. In this paper we prove this result for planar graphs achieving g(k) =2 O(k). Our bound is radically better than the bounds known for general graphs. Moreover, our proof is new and selfcontained, and it strongly exploits the combinatorial properties of planar graphs. We also prove that our result is optimal, in the sense that the function g(k) cannot become better than exponential. Our results suggest that any algorithm for the DisjointPaths Problem that runs in time better than 2 2o(k) · n O(1) will probably require drastically different ideas from those in the irrelevant vertex technique.
Linear Kernels for (Connected) Dominating Set on Hminorfree graphs
"... We give the first linear kernels for DOMINATING SET and CONNECTED DOMINATING SET problems on graphs excluding a fixed graph H as a minor. In other words, we give polynomial time algorithms that, for a given Hminor free graph G and positive integer k, output an Hminor free graph G ′ on O(k) vertice ..."
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Cited by 7 (3 self)
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We give the first linear kernels for DOMINATING SET and CONNECTED DOMINATING SET problems on graphs excluding a fixed graph H as a minor. In other words, we give polynomial time algorithms that, for a given Hminor free graph G and positive integer k, output an Hminor free graph G ′ on O(k) vertices such that G has a (connected) dominating set of size k if and only if G ′ has. Prior to our work, the only polynomial kernel for DOMINATING SET on graphs excluding a fixed graph H as a minor was due to Alon and Gutner [ECCC 2008, IWPEC 2009] and to Philip, Raman, and Sikdar [ESA 2009] but the size of their kernel is k c(H) , where c(H) is a constant depending on the size of H. Alon and Gutner asked explicitly, whether one can obtain a linear kernel for DOMINATING SET on Hminor free graphs. We answer this question in affirmative. For CONNECTED DOMINATING SET no polynomial kernel on Hminor free graphs was known prior to our work. Our results are based on a novel generic reduction rule producing an equivalent instance of the problem with treewidth O ( √ k). The application of this rule in a divideandconquer fashion together with protrusion techniques brings us to linear kernels. As a byproduct of our results we obtain the first subexponential time algorithms for CONNECTED DOMINATING SET, a deterministic algorithm solving the problem on an nvertex Hminor free graph in time 2 O( √ k log k) + n O(1) and a Monte Carlo algorithm of running time 2 O ( √ k) + n
The kinapath problem for clawfree graphs
, 2010
"... Testing whether there is an induced path in a graph spanning k given vertices is already NPcomplete in general graphs when k = 3. We show how to solve this problem in polynomial time on clawfree graphs, when k is not part of the input but an arbitrarily fixed integer. ..."
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Cited by 4 (1 self)
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Testing whether there is an induced path in a graph spanning k given vertices is already NPcomplete in general graphs when k = 3. We show how to solve this problem in polynomial time on clawfree graphs, when k is not part of the input but an arbitrarily fixed integer.
A Faster Algorithm to Recognize EvenHoleFree Graphs
, 2011
"... We study the problem of determining whether an nnode medge graph has an even hole, i.e., an induced simple cycle consisting of an even number of nodes. Conforti, Cornuéjols, Kapoor, and Vuˇsković gave the first polynomialtime algorithm for the problem, which runs in O(n40) time. Later, Chudnovsky ..."
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Cited by 3 (0 self)
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We study the problem of determining whether an nnode medge graph has an even hole, i.e., an induced simple cycle consisting of an even number of nodes. Conforti, Cornuéjols, Kapoor, and Vuˇsković gave the first polynomialtime algorithm for the problem, which runs in O(n40) time. Later, Chudnovsky, Kawarabayashi, and Seymour reduced the running time to O(n31). The best previously known algorithm for the problem, due to da Silva and Vuˇsković, runs in O(n19) time. In this paper, we solve the problem in time O(n11). 1