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63
Bidimensionality and Kernels
, 2010
"... Bidimensionality theory appears to be a powerful framework in the development of metaalgorithmic techniques. It was introduced by Demaine et al. [J. ACM 2005] as a tool to obtain subexponential time parameterized algorithms for bidimensional problems on Hminor free graphs. Demaine and Hajiaghayi ..."
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Cited by 58 (23 self)
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Bidimensionality theory appears to be a powerful framework in the development of metaalgorithmic techniques. It was introduced by Demaine et al. [J. ACM 2005] as a tool to obtain subexponential time parameterized algorithms for bidimensional problems on Hminor free graphs. Demaine and Hajiaghayi [SODA 2005] extended the theory to obtain polynomial time approximation schemes (PTASs) for bidimensional problems. In this paper, we establish a third metaalgorithmic direction for bidimensionality theory by relating it to the existence of linear kernels for parameterized problems. In parameterized complexity, each problem instance comes with a parameter k and the parameterized problem is said to admit a linear kernel if there is a polynomial time algorithm, called
Algorithmic Graph Minor Theory: Decomposition, Approximation, and Coloring
 In 46th Annual IEEE Symposium on Foundations of Computer Science
, 2005
"... At the core of the seminal Graph Minor Theory of Robertson and Seymour is a powerful structural theorem capturing the structure of graphs excluding a fixed minor. This result is used throughout graph theory and graph algorithms, but is existential. We develop a polynomialtime algorithm using topolog ..."
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Cited by 56 (15 self)
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At the core of the seminal Graph Minor Theory of Robertson and Seymour is a powerful structural theorem capturing the structure of graphs excluding a fixed minor. This result is used throughout graph theory and graph algorithms, but is existential. We develop a polynomialtime algorithm using topological graph theory to decompose a graph into the structure guaranteed by the theorem: a cliquesum of pieces almostembeddable into boundedgenus surfaces. This result has many applications. In particular, we show applications to developing many approximation algorithms, including a 2approximation to graph coloring, constantfactor approximations to treewidth and the largest grid minor, combinatorial polylogarithmicapproximation to halfintegral multicommodity flow, subexponential fixedparameter algorithms, and PTASs for many minimization and maximization problems, on graphs excluding a fixed minor. 1.
Efficient exact algorithms on planar graphs: Exploiting sphere cut branch decompositions
 IN PROCEEDINGS OF THE 13TH ANNUAL EUROPEAN SYMPOSIUM ON ALGORITHMS (ESA 2005
, 2005
"... A divideandconquer strategy based on variations of the LiptonTarjan planar separator theorem has been one of the most common approaches for solving planar graph problems for more than 20 years. We present a new framework for designing fast subexponential exact and parameterized algorithms on pla ..."
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Cited by 48 (18 self)
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A divideandconquer strategy based on variations of the LiptonTarjan planar separator theorem has been one of the most common approaches for solving planar graph problems for more than 20 years. We present a new framework for designing fast subexponential exact and parameterized algorithms on planar graphs. Our approach is based on geometric properties of planar branch decompositions obtained by Seymour & Thomas, combined with refined techniques of dynamic programming on planar graphs based on properties of noncrossing partitions. Compared to divideandconquer algorithms, the main advantages of our method are a) it is a generic method which allows to attack broad classes of problems; b) the obtained algorithms provide a better worst case analysis. To exemplify our approach we show how to obtain an O(26.903pn) time algorithm solving weighted Hamiltonian Cycle. We observe how our technique can be used to solve Planar Graph TSP in time O(29.8594pn). Our approach can be used to design parameterized algorithms as well. For example we introduce the first 2O(pk)nO(1) time algorithm for parameterized Planar kcycle by showing that for a given k we can decide if a planar graph on n vertices has a cycle of length at least k in time O(213.6pkn + n3).
Refined search tree technique for DOMINATING SET on planar graphs
 in Proceedings of the 26th International Symposium on Mathematical Foundations of Computer Science
"... Abstract We establish a refined search tree technique for the parameterized DOMINATING SET problem on planar graphs. Here, we are given an undirected graph and we ask for a set of at most k vertices such that every other vertex has at least one neighbor in this set. We describe algorithms with runn ..."
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Cited by 39 (11 self)
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Abstract We establish a refined search tree technique for the parameterized DOMINATING SET problem on planar graphs. Here, we are given an undirected graph and we ask for a set of at most k vertices such that every other vertex has at least one neighbor in this set. We describe algorithms with running times O(8 k n) and O(8 k k + n 3 ), where n is the number of vertices in the graph, based on bounded search trees. We describe a set of polynomial time datareduction rules for a more general "annotated" problem on black/white graphs that asks for a set of k vertices (black or white) ଁ An extended abstract of this paper appeared under nearly the same title in J. Sgall, A. Pultr, and P. Alber et al. / Journal of Computer and System Sciences 71 (2005) 385 405 that dominate all the black vertices. An intricate argument based on the Euler formula then establishes an efficient branching strategy for reduced inputs to this problem. In addition, we give a family examples showing that the bound of the branching theorem is optimal with respect to our reduction rules. Our final search tree algorithm is easy to implement; its analysis, however, is involved.
Bidimensional parameters and local treewidth
 SIAM Journal on Discrete Mathematics
, 2004
"... Abstract. For several graph theoretic parameters such as vertex cover and dominating set, it is known that if their values are bounded by k then the treewidth of the graph is bounded by some function of k. This fact is used as the main tool for the design of several fixedparameter algorithms on min ..."
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Cited by 31 (17 self)
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Abstract. For several graph theoretic parameters such as vertex cover and dominating set, it is known that if their values are bounded by k then the treewidth of the graph is bounded by some function of k. This fact is used as the main tool for the design of several fixedparameter algorithms on minorclosed graph classes such as planar graphs, singlecrossingminorfree graphs, and graphs of bounded genus. In this paper we examine the question whether similar bounds can be obtained for larger minorclosed graph classes, and for general families of parameters including all the parameters where such a behavior has been reported so far. Given a graph parameter P, we say that a graph family F has the parametertreewidth property for P if there is a function f(p) such that every graph G ∈ F with parameter at most p has treewidth at most f(p). We prove as our main result that, for a large family of parameters called contractionbidimensional parameters, a minorclosed graph family F has the parametertreewidth property if F has bounded local treewidth. We also show “if and only if ” for some parameters, and thus this result is in some sense tight. In addition we show that, for a slightly smaller family of parameters called minorbidimensional parameters, all minorclosed graph families F excluding some fixed graphs have the parametertreewidth property. The bidimensional parameters include many domination and covering parameters such as vertex cover, feedback vertex set, dominating set, edgedominating set, qdominating set (for fixed q). We use these theorems to develop new fixedparameter algorithms in these contexts. 1
Complexity of Inference in Graphical Models
"... It is wellknown that inference in graphical models is hard in the worst case, but tractable for models with bounded treewidth. We ask whether treewidth is the only structural criterion of the underlying graph that enables tractable inference. In other words, is there some class of structures with u ..."
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Cited by 23 (0 self)
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It is wellknown that inference in graphical models is hard in the worst case, but tractable for models with bounded treewidth. We ask whether treewidth is the only structural criterion of the underlying graph that enables tractable inference. In other words, is there some class of structures with unbounded treewidth in which inference is tractable? Subject to a combinatorial hypothesis due to Robertson et al. (1994), we show that low treewidth is indeed the only structural restriction that can ensure tractability. Thus, even for the “best case” graph structure, there is no inference algorithm with complexity polynomial in the treewidth. 1
Known Algorithms on Graphs of Bounded Treewidth are Probably Optimal
, 2010
"... We obtain a number of lower bounds on the running time of algorithms solving problems on graphs of bounded treewidth. We prove the results under the Strong Exponential Time Hypothesis of Impagliazzo and Paturi. In particular, assuming that SAT cannot be solved in (2−ǫ) n m O(1) time, we show that fo ..."
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Cited by 19 (5 self)
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We obtain a number of lower bounds on the running time of algorithms solving problems on graphs of bounded treewidth. We prove the results under the Strong Exponential Time Hypothesis of Impagliazzo and Paturi. In particular, assuming that SAT cannot be solved in (2−ǫ) n m O(1) time, we show that for any ǫ> 0; • INDEPENDENT SET cannot be solved in (2 − ǫ) tw(G) V (G)  O(1) time, • DOMINATING SET cannot be solved in (3 − ǫ) tw(G) V (G)  O(1) time, • MAX CUT cannot be solved in (2 − ǫ) tw(G) V (G)  O(1) time, • ODD CYCLE TRANSVERSAL cannot be solved in (3 − ǫ) tw(G) V (G)  O(1) time, • For any q ≥ 3, qCOLORING cannot be solved in (q − ǫ) tw(G) V (G)  O(1) time, • PARTITION INTO TRIANGLES cannot be solved in (2 − ǫ) tw(G) V (G)  O(1) time. Our lower bounds match the running times for the best known algorithms for the problems, up to the ǫ in the base.
Polynomialtime approximation schemes for subsetconnectivity problems in boundedgenus graphs
, 2009
"... We present the first polynomialtime approximation schemes (PTASes) for the following subsetconnectivity problems in edgeweighted graphs of bounded genus: Steiner tree, lowconnectivity survivablenetwork design, and subset TSP. The schemes run in O(n log n) time for graphs embedded on both orien ..."
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Cited by 18 (4 self)
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We present the first polynomialtime approximation schemes (PTASes) for the following subsetconnectivity problems in edgeweighted graphs of bounded genus: Steiner tree, lowconnectivity survivablenetwork design, and subset TSP. The schemes run in O(n log n) time for graphs embedded on both orientable and nonorientable surfaces. This work generalizes the PTAS frameworks of Borradaile, Klein, and Mathieu [BMK07, Kle06] from planar graphs to boundedgenus graphs: any future problems shown to admit the required structure theorem for planar graphs will similarly extend to boundedgenus graphs.
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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Cited by 17 (6 self)
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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.
Capacitated domination and covering: A parameterized perspective
 Proceedings 3rd International Workshop on Parameterized and Exact Computation, IWPEC 2008
"... Capacitated versions of Dominating Set and Vertex Cover have been studied intensively in terms of polynomial time approximation algorithms. Although the problems Dominating Set and Vertex Cover have been subjected to considerable scrutiny in the parameterized complexity world, this is not true for t ..."
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Cited by 16 (9 self)
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Capacitated versions of Dominating Set and Vertex Cover have been studied intensively in terms of polynomial time approximation algorithms. Although the problems Dominating Set and Vertex Cover have been subjected to considerable scrutiny in the parameterized complexity world, this is not true for the capacitated versions. Here we make an attempt to understand the behavior of the problems Capacitated Dominating Set and Capacitated Vertex Cover from the perspective of parameterized complexity. The original versions of these problems, Vertex Cover and Dominating Set, are known to be fixed parameter tractable when parameterized by a structure of the graph called the treewidth (tw). In this paper we show that the capacitated versions of these problems behave differently. Our results are: • Capacitated Dominating Set is W[1]hard when parameterized by treewidth. In fact, Capacitated Dominating Set is W[1]hard when parameterized by both treewidth and solution size k of the capacitated dominating set. • Capacitated Vertex Cover is W[1]hard when parameterized by treewidth. • Capacitated Vertex Cover can be solved in time 2O(tw log k) nO(1) where tw is the treewidth of the input graph and k is the solution size. As a corollary, we show that the weighted version of Capacitated Vertex Cover in general graphs can be solved in time 2O(k log k) nO(1). This improves the earlier algorithm of Guo et al. [15] running in time O(1.2k2 + n2). We would also like to point out that our W[1]hardness result for Capacitated Vertex Cover, when parameterized by treewidth, makes it (to the best of our knowledge) the first known “subset problem ” which has turned out to be fixed parameter tractable when parameterized by solution size but W[1]hard when parameterized by treewidth. 1