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67
Parameterized Complexity: A Framework for Systematically Confronting Computational Intractability
 DIMACS Series in Discrete Mathematics and Theoretical Computer Science
, 1997
"... In this paper we give a programmatic overview of parameterized computational complexity in the broad context of the problem of coping with computational intractability. We give some examples of how fixedparameter tractability techniques can deliver practical algorithms in two different ways: (1) by ..."
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Cited by 85 (16 self)
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In this paper we give a programmatic overview of parameterized computational complexity in the broad context of the problem of coping with computational intractability. We give some examples of how fixedparameter tractability techniques can deliver practical algorithms in two different ways: (1) by providing useful exact algorithms for small parameter ranges, and (2) by providing guidance in the design of heuristic algorithms. In particular, we describe an improved FPT kernelization algorithm for Vertex Cover, a practical FPT algorithm for the Maximum Agreement Subtree (MAST) problem parameterized by the number of species to be deleted, and new general heuristics for these problems based on FPT techniques. In the course of making this overview, we also investigate some structural and hardness issues. We prove that an important naturally parameterized problem in artificial intelligence, STRIPS Planning (where the parameter is the size of the plan) is complete for W [1]. As a corollary, this implies that kStep Reachability for Petri Nets is complete for W [1]. We describe how the concept of treewidth can be applied to STRIPS Planning and other problems of logic to obtain FPT results. We describe a surprising structural result concerning the top end of the parameterized complexity hierarchy: the naturally parameterized Graph kColoring problem cannot be resolved with respect to XP either by showing membership in XP, or by showing hardness for XP without settling the P = NP question one way or the other.
Algorithms For Vertex Partitioning Problems On Partial kTrees
, 1997
"... In this paper, we consider a large class of vertex partitioning problems and apply to those the theory of algorithm design for problems restricted to partial ktrees. We carefully describe the details of algorithms and analyze their complexity in an attempt to make the algorithms feasible as solutio ..."
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Cited by 56 (5 self)
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In this paper, we consider a large class of vertex partitioning problems and apply to those the theory of algorithm design for problems restricted to partial ktrees. We carefully describe the details of algorithms and analyze their complexity in an attempt to make the algorithms feasible as solutions for practical applications.
Combinatorial Optimization on Graphs of Bounded Treewidth
, 2007
"... There are many graph problems that can be solved in linear or polynomial time with a dynamic programming algorithm when the input graph has bounded treewidth. For combinatorial optimization problems, this is a useful approach for obtaining fixedparameter tractable algorithms. Starting from trees an ..."
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Cited by 51 (4 self)
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There are many graph problems that can be solved in linear or polynomial time with a dynamic programming algorithm when the input graph has bounded treewidth. For combinatorial optimization problems, this is a useful approach for obtaining fixedparameter tractable algorithms. Starting from trees and seriesparallel graphs, we introduce the concepts of treewidth and tree decompositions, and illustrate the technique with the Weighted Independent Set problem as an example. The paper surveys some of the latest developments, putting an emphasis on applicability, on algorithms that exploit tree decompositions, and on algorithms that determine or approximate treewidth and find tree decompositions with optimal or close to optimal treewidth. Directions for further research and suggestions for further reading are also given.
Approximation algorithms via contraction decomposition
 Proc. 18th Ann. ACMSIAM Symp. Discrete Algorithms ACMSIAM symposium on Discrete algorithms
, 2007
"... We prove that the edges of every graph of bounded (Euler) genus can be partitioned into any prescribed number k of pieces such that contracting any piece results in a graph of bounded treewidth (where the bound depends on k). This decomposition result parallels an analogous, simpler result for edge ..."
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Cited by 37 (8 self)
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We prove that the edges of every graph of bounded (Euler) genus can be partitioned into any prescribed number k of pieces such that contracting any piece results in a graph of bounded treewidth (where the bound depends on k). This decomposition result parallels an analogous, simpler result for edge deletions instead of contractions, obtained in [Bak94, Epp00, DDO + 04, DHK05], and it generalizes a similar result for “compression ” (a variant of contraction) in planar graphs [Kle05]. Our decomposition result is a powerful tool for obtaining PTASs for contractionclosed problems (whose optimal solution only improves under contraction), a much more general class than minorclosed problems. We prove that any contractionclosed problem satisfying just a few simple conditions has a PTAS in boundedgenus graphs. In particular, our framework yields PTASs for the weighted Traveling Salesman Problem and for minimumweight cedgeconnected submultigraph on boundedgenus graphs, improving and generalizing previous algorithms of [GKP95, AGK + 98, Kle05, Gri00, CGSZ04, BCGZ05]. We also highlight the only main difficulty in extending our results to general Hminorfree graphs.
Fast Mucalculus Model Checking when Treewidth is Bounded
, 2003
"... We show that the model checking problem for calculus on graphs of bounded treewidth can be solved in time linear in the size of the system. The result is presented by rst showing a related result: the winner in a parity game on a graph of bounded treewidth can be decided in polynomial time. ..."
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Cited by 24 (2 self)
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We show that the model checking problem for calculus on graphs of bounded treewidth can be solved in time linear in the size of the system. The result is presented by rst showing a related result: the winner in a parity game on a graph of bounded treewidth can be decided in polynomial time. The given algorithm is then modi ed to obtain a new algorithm for calculus model checking. One possible use of this algorithm may be software veri cation, since control ow graphs of programs written in highlevel languages are usually of bounded treewidth.
Pathwidth and ThreeDimensional StraightLine Grid Drawings of Graphs
"... We prove that every nvertex graph G with pathwidth pw(G) has a threedimensional straightline grid drawing with O(pw(G) n) volume. Thus for ..."
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Cited by 23 (12 self)
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We prove that every nvertex graph G with pathwidth pw(G) has a threedimensional straightline grid drawing with O(pw(G) n) volume. Thus for
Branch and Tree Decomposition Techniques for Discrete Optimization
, 2005
"... This chapter gives a general overview of two emerging techniques for discrete optimization that have footholds in mathematics, computer science, and operations research: branch decompositions and tree decompositions. Branch decompositions and tree decompositions along with their respective connectiv ..."
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Cited by 21 (3 self)
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This chapter gives a general overview of two emerging techniques for discrete optimization that have footholds in mathematics, computer science, and operations research: branch decompositions and tree decompositions. Branch decompositions and tree decompositions along with their respective connectivity invariants, branchwidth and treewidth, were first introduced to aid in proving the Graph Minors Theorem, a wellknown conjecture (Wagner’s conjecture) in graph theory. The algorithmic importance of branch decompositions and tree decompositions for solving NPhard problems modelled on graphs was first realized by computer scientists in relation to formulating graph problems in monadic second order logic. The dynamic programming techniques utilizing branch decompositions and tree decompositions, called branch decomposition and tree decomposition based algorithms, fall into a class of algorithms known as fixedparameter tractable algorithms and have been shown to be effective in a practical setting for NPhard problems such as minimum domination, the travelling salesman problem, general minor containment, and frequency assignment problems.
The Treewidth of Java Programs
 Proceedings ALENEX’02 — 4th Workshop on Algorithm Engineering and Experiments
, 2002
"... Intuitively, the treewidth of a graph G measures how close G is to being a tree. The lower the treewidth, the faster we can solve various optimization problems on G, by dynamic programming along the tree structure. In the paper M.Thorup, All Structured Programs have Small TreeWidth and Good Reg ..."
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Cited by 18 (1 self)
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Intuitively, the treewidth of a graph G measures how close G is to being a tree. The lower the treewidth, the faster we can solve various optimization problems on G, by dynamic programming along the tree structure. In the paper M.Thorup, All Structured Programs have Small TreeWidth and Good Register Allocation [8] it is shown that the controlflow graph of any gotofree C program is at most 6. This result opened for the possibility of applying the dynamic programming bounded treewidth algorithms to various compiler optimization tasks. In this paper we explore this possibility, in particular for Java programs. We first show that even if Java does not have a goto, the labelled break and continue statements are in a sense equally bad, and can be used to construct Java programs that are arbitrarily hard to understand and optimize. For Java programs lacking these labelled constructs Thorup's result for C still holds, and in the second part of the paper we analyze the treewid...