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The Complexity Ecology of Parameters: An Illustration Using Bounded Max Leaf Number
"... In the framework of parameterized complexity, exploring how one parameter affects the complexity of a different parameterized (or unparameterized problem) is of general interest. A well-developed example is the investigation of how the parameter treewidth influences the complexity of (other) graph ..."
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Cited by 14 (7 self)
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In the framework of parameterized complexity, exploring how one parameter affects the complexity of a different parameterized (or unparameterized problem) is of general interest. A well-developed example is the investigation of how the parameter treewidth influences the complexity of (other) graph problems. The reason why such investigations are of general interest is that real-world input distributions for computational problems often inherit structure from the natural computational processes that produce the problem instances (not necessarily in obvious, or well-understood ways). The max leaf number ml(G) of a connected graph G is the maximum number of leaves in a spanning tree for G. Exploring questions analogous to the well-studied case of treewidth, we can ask: how hard is it to solve 3-Coloring, Hamilton Path, Minimum Dominating Set, Minimum Bandwidth or many other problems, for graphs of bounded max leaf number? What optimization problems are W [1]-hard under this parameterization? We do two things: (1) We describe much improved FPT algorithms for a large number of graph problems, for input graphs G for which ml(G) ≤ k, based on the polynomial-time extremal structure theory canonically associated to this parameter. We consider improved algorithms both from the point of view of kernelization bounds, and in terms of improved fixed-parameter tractable (FPT) runtimes O ∗ (f(k)). (2) The way that we obtain these concrete algorithmic results is general and systematic. We describe the approach, and raise programmatic questions.
Some recent progress and applications in graph minor theory
, 2006
"... In the core of the seminal Graph Minor Theory of Robertson and Seymour lies a powerful theorem capturing the “rough ” structure of graphs excluding a fixed minor. This result was used to prove Wagner’s Conjecture that finite graphs are well-quasi-ordered under the graph minor relation. Recently, a n ..."
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Cited by 13 (4 self)
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In the core of the seminal Graph Minor Theory of Robertson and Seymour lies a powerful theorem capturing the “rough ” structure of graphs excluding a fixed minor. This result was used to prove Wagner’s Conjecture that finite graphs are well-quasi-ordered under the graph minor relation. Recently, a number of beautiful results that use this structural result have appeared. Some of these along with some other recent advances on graph minors are surveyed.
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 13 (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
Fast subexponential algorithm for non-local problems on graphs of bounded genus, in
- Proc. of the 10th Scandinavian Workshop on Algorithm Theory, SWAT, in: LNCS
"... Abstract We give a general technique for designing fast subexponential algorithms for several graph problems whose instances are restricted to graphs of bounded genus. We use it to obtain time 2 O( √ n) algorithms for a wide family of problems such as Hamiltonian Cycle, Σ-embedded Graph Travelling ..."
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Cited by 12 (6 self)
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Abstract We give a general technique for designing fast subexponential algorithms for several graph problems whose instances are restricted to graphs of bounded genus. We use it to obtain time 2 O( √ n) algorithms for a wide family of problems such as Hamiltonian Cycle, Σ-embedded Graph Travelling Salesman Problem, Longest Cycle, and Max Leaf Tree. For our results, we combine planarizing techniques with dynamic programming on special type branch decompositions. Our techniques can also be used to solve parameterized problems. Thus, for example, we show how to find a cycle of length p (or to conclude that there is no such a cycle) on graphs of bounded genus in time 2 O( √ p) · n O(1) .
Solving dominating set in larger classes of graphs: FPT algorithms and polynomial kernels
- in Proceedings of the 17th Annual European Symposium on Algorithms (ESA 2009
"... Abstract. We show that the k-Dominating Set problem is fixed parameter tractable (FPT) and has a polynomial kernel for any class of graphs that exclude Ki,j as a subgraph, for any fixed i, j ≥ 1. This strictly includes every class of graphs for which this problem has been previously shown to have FP ..."
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Cited by 12 (5 self)
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Abstract. We show that the k-Dominating Set problem is fixed parameter tractable (FPT) and has a polynomial kernel for any class of graphs that exclude Ki,j as a subgraph, for any fixed i, j ≥ 1. This strictly includes every class of graphs for which this problem has been previously shown to have FPT algorithms and/or polynomial kernels. In particular, our result implies that the problem restricted to boundeddegenerate graphs has a polynomial kernel, solving an open problem posed by Alon and Gutner in [3]. 1
Bidimensionality and EPTAS
"... Bidimensionality theory appears to be a powerful framework for the development of meta-algorithmic techniques. It was introduced by Demaine et al. [J. ACM 2005] as a tool to obtain sub-exponential time parameterized algorithms for problems on H-minor free graphs. Demaine and Hajiaghayi [SODA 2005] e ..."
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Cited by 12 (5 self)
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Bidimensionality theory appears to be a powerful framework for the development of meta-algorithmic techniques. It was introduced by Demaine et al. [J. ACM 2005] as a tool to obtain sub-exponential time parameterized algorithms for problems on H-minor free graphs. Demaine and Hajiaghayi [SODA 2005] extended the theory to obtain polynomial time approximation schemes (PTASs) for bidimensional problems, and subsequently improved these results to EP-TASs. Fomin et. al [SODA 2010] established a third metaalgorithmic direction for bidimensionality theory by relating it to the existence of linear kernels for parameterized problems. In this paper we revisit bidimensionality theory from the perspective of approximation algorithms and redesign the framework for obtaining EPTASs to be more powerful, easier to apply and easier to understand. One of the important conditions required in the framework developed by Demaine and Hajiaghayi [SODA 2005] is that to obtain an EPTAS for a graph optimization problem Π, we have to know a constant-factor approximation algorithm for Π. Our approach eliminates this strong requirement, which makes it amenable to more problems. At the heart of our framework is a decomposition lemma which states that for “most ” bidimensional problems, there is a polynomial time algorithm which given an H-minor-free graph G as input and an ɛ> 0 outputs a vertex set X of size ɛ · OP T such that the treewidth of G \ X is O(1/ɛ). Here, OP T is the objective function value of the problem in question This allows us to obtain EPTASs on (apex)-minor-free graphs for all problems covered by the previous framework, as well as for a wide range of packing problems, partial covering problems and problems that are neither closed under taking minors, nor contractions. To the best of our knowledge for many of these problems including Cycle Packing, Vertex-H-
Logic, Graphs, and Algorithms
, 2007
"... Algorithmic meta theorems are algorithmic results that apply to whole families of combinatorial problems, instead of just specific problems. These families are usually defined in terms of logic and graph theory. An archetypal algorithmic meta theorem is Courcelle’s Theorem [9], which states that all ..."
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Cited by 11 (0 self)
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Algorithmic meta theorems are algorithmic results that apply to whole families of combinatorial problems, instead of just specific problems. These families are usually defined in terms of logic and graph theory. An archetypal algorithmic meta theorem is Courcelle’s Theorem [9], which states that all graph properties definable in monadic second-order logic can be decided in linear time on graphs of bounded tree width. This article is an introduction into the theory underlying such meta theorems and a survey of the most important results in this area.
PLANAR SUBGRAPH ISOMORPHISM REVISITED
, 2009
"... Ten years after Eppstein’s results on planar subgraph isomorphism for k-sized patterns, we improve the exponential term of the running time 2 O(k log k) · n of Eppstein’s algorithm to 2 O(k) (keeping the term in n linear!) Next to deciding subgraph isomorphism, we can construct a solution and enume ..."
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Cited by 11 (2 self)
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Ten years after Eppstein’s results on planar subgraph isomorphism for k-sized patterns, we improve the exponential term of the running time 2 O(k log k) · n of Eppstein’s algorithm to 2 O(k) (keeping the term in n linear!) Next to deciding subgraph isomorphism, we can construct a solution and enumerate all solutions in the same asymptotic running time. We may list ω subgraphs with an additive term O(ωn) in the running time of our algorithm. For exact algorithms, this means we obtain a truly subexponential algorithm for patterns of size O ( √ n) of running time 2 O( √ n) imrpoving the former bound of 2 O( √ n log n)
Contraction Bidimensionality: The Accurate Picture
- Proceedings of the 17th Annual European Symposium on Algorithms, Lecture Notes in Computer Science
, 2009
"... Abstract. We provide new combinatorial theorems on the structure of graphs that are contained as contractions in graphs of large treewidth. As a consequence of our combinatorial results we unify and significantly sim-plify contraction bidimensionality theory—the meta algorithmic frame-work to design ..."
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Cited by 9 (5 self)
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Abstract. We provide new combinatorial theorems on the structure of graphs that are contained as contractions in graphs of large treewidth. As a consequence of our combinatorial results we unify and significantly sim-plify contraction bidimensionality theory—the meta algorithmic frame-work to design efficient parameterized and approximation algorithms for contraction closed parameters. 1
