Results 1  10
of
36
Dominating Sets in Planar Graphs: BranchWidth and Exponential Speedup
, 2002
"... Graph minors theory, developed by Robertson & Seymour, provides a list of powerful theoretical results and tools. However, the wide spread opinion in Graph Algorithms community about this theory is that it is mainly of theoretical importance. ..."
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Cited by 69 (15 self)
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Graph minors theory, developed by Robertson & Seymour, provides a list of powerful theoretical results and tools. However, the wide spread opinion in Graph Algorithms community about this theory is that it is mainly of theoretical importance.
Parameterized Complexity: Exponential SpeedUp for Planar Graph Problems
 in Electronic Colloquium on Computational Complexity (ECCC
, 2001
"... A parameterized problem is xed parameter tractable if it admits a solving algorithm whose running time on input instance (I; k) is f(k) jIj , where f is an arbitrary function depending only on k. Typically, f is some exponential function, e.g., f(k) = c k for constant c. We describe general techniqu ..."
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Cited by 64 (21 self)
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A parameterized problem is xed parameter tractable if it admits a solving algorithm whose running time on input instance (I; k) is f(k) jIj , where f is an arbitrary function depending only on k. Typically, f is some exponential function, e.g., f(k) = c k for constant c. We describe general techniques to obtain growth of the form f(k) = c p k for a large variety of planar graph problems. The key to this type of algorithm is what we call the "Layerwise Separation Property" of a planar graph problem. Problems having this property include planar vertex cover, planar independent set, and planar dominating set.
Subexponential Parameterized Algorithms Collapse the Whierarchy (Extended Abstract)
, 2001
"... It is shown that for essentially all MAX SNPhard optimization problems finding exact solutions in subexponential time is not possible unless W [1] = FPT . In particular, we show that O(2 o(k) p(n)) parameterized algorithms do not exist for Vertex Cover, Max Cut, Max cSat, and a number of pr ..."
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Cited by 45 (2 self)
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It is shown that for essentially all MAX SNPhard optimization problems finding exact solutions in subexponential time is not possible unless W [1] = FPT . In particular, we show that O(2 o(k) p(n)) parameterized algorithms do not exist for Vertex Cover, Max Cut, Max cSat, and a number of problems on bounded degree graphs such as Dominating Set and Independent Set, unless W [1] = FPT . Our results are derived via an approach that uses an extended parameterization of optimization problems and associated techniques to relate the parameterized complexity of problems in FPT to the parameterized complexity of extended versions that are W [1]hard.
Improved Tree Decomposition Based Algorithms for Dominationlike Problems
 in LATIN’02: Theoretical informatics (Cancun
, 2001
"... We present an improved dynamic programming strategy for dominating set and related problems on graphs that are given together with a tree decomposition of width k. We obtain an O(4 n) algorithm for dominating set, where n is the number of nodes of the tree decomposition. ..."
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Cited by 34 (9 self)
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We present an improved dynamic programming strategy for dominating set and related problems on graphs that are given together with a tree decomposition of width k. We obtain an O(4 n) algorithm for dominating set, where n is the number of nodes of the tree decomposition.
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 34 (16 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
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
Equivalence of Local Treewidth and Linear Local Treewidth and its Algorithmic Applications
 In Proceedings of the 15th ACMSIAM Symposium on Discrete Algorithms (SODA’04
, 2003
"... We solve an open problem posed by Eppstein in 1995 [14, 15] and reenforced by Grohe [16, 17] concerning locally bounded treewidth in minorclosed families of graphs. A graph has bounded local treewidth if the subgraph induced by vertices within distance r of any vertex has treewidth bounded by a f ..."
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Cited by 31 (11 self)
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We solve an open problem posed by Eppstein in 1995 [14, 15] and reenforced by Grohe [16, 17] concerning locally bounded treewidth in minorclosed families of graphs. A graph has bounded local treewidth if the subgraph induced by vertices within distance r of any vertex has treewidth bounded by a function of r (not n). Eppstein characterized minorclosed families of graphs with bounded local treewidth as precisely minorclosed families that minorexclude an apex graph, where an apex graph has one vertex whose removal leaves a planar graph. In particular, Eppstein showed that all apexminorfree graphs have bounded local treewidth, but his bound is doubly exponential in r, leaving open whether a tighter bound could be obtained. We improve this doubly exponential bound to a linear bound, which is optimal. In particular, any minorclosed graph family with bounded local treewidth has linear local treewidth. Our bound generalizes previously known linear bounds for special classes of graphs proved by several authors. As a consequence of our result, we obtain substantially faster polynomialtime approximation schemes for a broad class of problems in apexminorfree graphs, improving the running time from .
Faster Exact Algorithms for Hard Problems: A Parameterized Point of View
 Discrete Mathematics
, 2000
"... Recent times have seen quite some progress in the development of "efficient" exponential time algorithms for NPhard problems. These results are also tightly related to the socalled theory of fixed parameter tractability. In this incomplete, personally biased survey, we reflect on some re ..."
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Cited by 30 (4 self)
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Recent times have seen quite some progress in the development of "efficient" exponential time algorithms for NPhard problems. These results are also tightly related to the socalled theory of fixed parameter tractability. In this incomplete, personally biased survey, we reflect on some recent developments and prospects in the field of fixed parameter algorithms.
Graph separators: a parameterized view
 Journal of Computer and System Sciences
, 2001
"... Graph separation is a wellknown tool to make (hard) graph problems accessible to a divide and conquer approach. We show how to use graph separator theorems in combination with (linear) problem kernels in order to develop xed parameter algorithms for many wellknown NPhard (planar) graph problems. ..."
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Cited by 30 (12 self)
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Graph separation is a wellknown tool to make (hard) graph problems accessible to a divide and conquer approach. We show how to use graph separator theorems in combination with (linear) problem kernels in order to develop xed parameter algorithms for many wellknown NPhard (planar) graph problems. We coin the key notion of glueable select&verify graph problems and derive from that a prospective way to easily check whether a planar graph problem will allow for a xed parameter algorithm of running time c p