Results 1  10
of
13
Subexponential Parameterized Algorithms on Graphs of Bounded Genus and HMinorFree Graphs
, 2003
"... We introduce a new framework for designing fixedparameter algorithms with subexponential running time2 . Our results apply to a broad family of graph problems, called bidimensional problems, which includes many domination and covering problems such as vertex cover, feedback vertex set, minimum m ..."
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Cited by 41 (13 self)
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We introduce a new framework for designing fixedparameter algorithms with subexponential running time2 . Our results apply to a broad family of graph problems, called bidimensional problems, which includes many domination and covering problems such as vertex cover, feedback vertex set, minimum maximal matching, dominating set, edge dominating set, cliquetransversal set, and many others restricted to bounded genus graphs. Furthermore, it is fairly straightforward to prove that a problem is bidimensional. In particular, our framework includes as special cases all previously known problems to have such subexponential algorithms. Previously, these algorithms applied to planar graphs, singlecrossingminorfree graphs, and/or map graphs; we extend these results to apply to boundedgenus graphs as well. In a parallel development of combinatorial results, we establish an upper bound on the treewidth (or branchwidth) of a boundedgenus graph that excludes some planar graph H as a minor. This bound depends linearly on the size (H) of the excluded graph H and the genus g(G) of the graph G, and applies and extends the graphminors work of Robertson and Seymour. Building on these results...
Bidimensionality: New Connections between FPT Algorithms and PTASs
"... We demonstrate a new connection between fixedparameter tractability and approximation algorithms for combinatorial optimization problems on planar graphs and their generalizations. Specifically, we extend the theory of socalled “bidimensional” problems to show that essentially all such problems ha ..."
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Cited by 36 (5 self)
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We demonstrate a new connection between fixedparameter tractability and approximation algorithms for combinatorial optimization problems on planar graphs and their generalizations. Specifically, we extend the theory of socalled “bidimensional” problems to show that essentially all such problems have both subexponential fixedparameter algorithms and PTASs. Bidimensional problems include e.g. feedback vertex set, vertex cover, minimum maximal matching, face cover, a series of vertexremoval problems, dominating set, edge dominating set, rdominating set, diameter, connected dominating set, connected edge dominating set, and connected rdominating set. We obtain PTASs for all of these problems in planar graphs and certain generalizations; of particular interest are our results for the two wellknown problems of connected dominating set and general feedback vertex set for planar graphs and their generalizations, for which PTASs were not known to exist. Our techniques generalize and in some sense unify the two main previous approaches for designing PTASs in planar graphs, namely, the LiptonTarjan separator approach [FOCS’77] and the Baker layerwise decomposition approach [FOCS’83]. In particular, we replace the notion of separators with a more powerful tool from the bidimensionality theory, enabling the first approach to apply to a much broader class of minimization problems than previously possible; and through the use of a structural backbone and thickening of layers we demonstrate how the second approach can be applied to problems with a “nonlocal” structure.
Graphs Excluding a Fixed Minor have Grids as Large as Treewidth, with Combinatorial and Algorithmic Applications through Bidimensionality
, 2005
"... We prove that any Hminorfree graph, for a fixed graph H, of treewidth w has an Ω(w) × Ω(w) grid graph as a minor. Thus grid minors suffice to certify that Hminorfree graphs have large treewidth, up to constant factors. This strong relationship was previously known for the special cases of plana ..."
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Cited by 21 (7 self)
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We prove that any Hminorfree graph, for a fixed graph H, of treewidth w has an Ω(w) × Ω(w) grid graph as a minor. Thus grid minors suffice to certify that Hminorfree graphs have large treewidth, up to constant factors. This strong relationship was previously known for the special cases of planar graphs and boundedgenus graphs, and is known not to hold for general graphs. The approach of this paper can be viewed more generally as a framework for extending combinatorial results on planar graphs to hold on Hminorfree graphs for any fixed H. Our result has many combinatorial consequences on bidimensionality theory, parametertreewidth bounds, separator theorems, and bounded local treewidth; each of these combinatorial results has several algorithmic consequences including subexponential fixedparameter algorithms and approximation algorithms.
Linearity of Grid Minors in Treewidth with Applications through Bidimensionality
, 2005
"... We prove that any Hminorfree graph, for a fixed graph H, of treewidth w has an \Omega (w) *\Omega ( w) grid graph as a minor. Thus grid minors suffice to certify that Hminorfree graphs havelarge treewidth, up to constant factors. This strong relationship was previously known for the special cas ..."
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Cited by 21 (1 self)
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We prove that any Hminorfree graph, for a fixed graph H, of treewidth w has an \Omega (w) *\Omega ( w) grid graph as a minor. Thus grid minors suffice to certify that Hminorfree graphs havelarge treewidth, up to constant factors. This strong relationship was previously known for the special cases of planar graphs and boundedgenus graphs, and is known not to hold for generalgraphs. The approach of this paper can be viewed more generally as a framework for extending combinatorial results on planar graphs to hold on Hminorfree graphs for any fixed H. Ourresult has many combinatorial consequences on bidimensionality theory, parametertreewidth bounds, separator theorems, and bounded local treewidth; each of these combinatorial resultshas several algorithmic consequences including subexponential fixedparameter algorithms and approximation algorithms.
Subexponential parameterized algorithms
 Computer Science Review
"... We give a review of a series of techniques and results on the design of subexponential parameterized algorithms for graph problems. The design of such algorithms usually consists of two main steps: first find a branch (or tree) decomposition of the input graph whose width is bounded by a sublinear ..."
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Cited by 18 (8 self)
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We give a review of a series of techniques and results on the design of subexponential parameterized algorithms for graph problems. The design of such algorithms usually consists of two main steps: first find a branch (or tree) decomposition of the input graph whose width is bounded by a sublinear function of the parameter and, second, use this decomposition to solve the problem in time that is single exponential to this bound. The main tool for the first step is Bidimensionality Theory. Here we present the potential, but also the boundaries, of this theory. For the second step, we describe recent techniques, associating the analysis of subexponential algorithms to combinatorial bounds related to Catalan numbers. As a result, we have 2 O( √ k) · n O(1) time algorithms for a wide variety of parameterized problems on graphs, where n is the size of the graph and k is the parameter. 1
A Simple and Fast Approach for Solving Problems on Planar Graphs
 In Proc. 21st STACS, volume 2996 of LNCS
, 2004
"... It is well known that the celebrated LiptonTarjan planar separation theorem, in a combination with a divideandconquer strategy leads to many complexity results for planar graph problems. For example, by using this approach, many planar graph problems can be solved in time 2 , where n is th ..."
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Cited by 15 (1 self)
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It is well known that the celebrated LiptonTarjan planar separation theorem, in a combination with a divideandconquer strategy leads to many complexity results for planar graph problems. For example, by using this approach, many planar graph problems can be solved in time 2 , where n is the number of vertices. However, the constants hidden in bigOh, usually are too large to claim the algorithms to be practical even on graphs of moderate size. Here we introduce a new algorithm design paradigm for solving problems on planar graphs. The paradigm is so simple that it can be explained in any textbook on graph algorithms: Compute tree or branch decomposition of a planar graph and do dynamic programming. Surprisingly such a simple approach provides faster algorithms for many problems. For example, Independent Set on planar graphs can be solved in time O(2 ) and Dominating Set in time O(2 ). In addition, significantly broader class of problems can be attacked by this method. Thus with our approach, Longest cycle on planar graphs is solved in time O(2 and Bisection is solved in time O(2 ). The proof of these results is based on complicated combinatorial arguments that make strong use of results derived by the Graph Minors Theory. In particular we prove that branchwidth of a planar graph is at most 2.122 # n. In addition we observe how a similar approach can be used for solving di#erent fixed parameter problems on planar graphs. We prove that our method provides the best so far exponential speedup for fundamental problems on planar graphs like Vertex Cover, (Weighted) Dominating Set, and many others.
The bidimensional theory of boundedgenus graphs
 SIAM Journal on Discrete Mathematics
, 2004
"... 1 Introduction The recent theory of fixedparameter algorithms and parameterized complexity [13] has attracted much attention in its less than 10 years of existence. In general the goal is to understand when NPhard problems have algorithms thatare exponential only in a parameter k of the problem i ..."
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Cited by 11 (6 self)
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1 Introduction The recent theory of fixedparameter algorithms and parameterized complexity [13] has attracted much attention in its less than 10 years of existence. In general the goal is to understand when NPhard problems have algorithms thatare exponential only in a parameter k of the problem instead of the problemsize n. Fixedparameter algorithms whose running time is polynomial for fixedparameter valuesor more precisely
Fast algorithms for hard graph problems: Bidimensionality, minors, and local treewidth
 In Proceedings of the 12th International Symposium on Graph Drawing, volume 3383 of Lecture Notes in Computer Science
, 2004
"... Abstract. This paper surveys the theory of bidimensional graph problems. We summarize the known combinatorial and algorithmic results of this theory, the foundational Graph Minor results on which this theory is based, and the remaining open problems. 1 ..."
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Cited by 10 (3 self)
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Abstract. This paper surveys the theory of bidimensional graph problems. We summarize the known combinatorial and algorithmic results of this theory, the foundational Graph Minor results on which this theory is based, and the remaining open problems. 1
Some Parameterized Problems on Digraphs
 The Computer Journal
"... We survey known results on parameterized complexity of the feedback set and induced subdigraph problems for digraphs. We prove new results on some parameterizations of the paired comparison problems on digraphs. One of our theorems implies a new result for a parameterized version of the linear arran ..."
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Cited by 6 (2 self)
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We survey known results on parameterized complexity of the feedback set and induced subdigraph problems for digraphs. We prove new results on some parameterizations of the paired comparison problems on digraphs. One of our theorems implies a new result for a parameterized version of the linear arrangement problem for undirected graphs. We state several open problems. 1
Algorithmic graph minor theory: Improved grid minor bounds and wagner’s contraction
 Proceedings of the Third International Conference on Distributed Computing and Internet Technology
, 2006
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