Polynomial fixed-parameter algorithms: A case study for longest path on interval graphs

Giannopoulou, Archontia C.; Mertzios, George B.; Niedermeier, Rolf

doi:10.1016/j.tcs.2017.05.017

“…Although several results of this kind are scattered throughout the literature [2,13,14,15,53], mostly concerning shortest path problems, no systematic investigations have been made so far. This direction fits into the general concept of "FPT within P" that was coined very recently by Giannopoulou et al [32] and by Abboud et al [1].…”

Section: Introductionsupporting

confidence: 70%

Fully polynomial-time parameterized computations for graphs and matrices of low treewidth

Fomin

¹

,

Lokshtanov

²

,

Pilipczuk

³

et al. 2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

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We investigate the complexity of several fundamental polynomial-time solvable problems on graphs and on matrices, when the given instance has low treewidth; in the case of matrices, we consider the treewidth of the graph formed by non-zero entries. In each of the considered cases, the best known algorithms working on general graphs run in polynomial, but far from linear, time. Thus, our goal is to construct algorithms with running time of the form poly(k)·n or poly(k) · n log n, where k is the width of the tree decomposition given on the input. Such procedures would outperform the best known algorithms for the considered problems already for moderate values of the treewidth, like O(n 1/c ) for some small constant c. Our results include:-an algorithm for computing the determinant and the rank of an n × n matrix using O(k 3 · n) time and arithmetic operations; -an algorithm for solving a system of linear equations using O(k 3 · n) time and arithmetic operations;-an O(k 3 · n log n)-time randomized algorithm for finding the cardinality of a maximum matching in a graph; -an O(k 4 · n log 2 n)-time randomized algorithm for constructing a maximum matching in a graph; -an O(k 2 · n log n)-time algorithm for finding a maximum vertex flow in a directed graph. * D.

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“…Related work: Fully polynomial parameterized algorithms. FPT algorithms for polynomial-time solvable problems were first considered by Giannopoulou et al [55]. Such a parameterized approach makes sense for any problem in P for which a conditional hardness result is proved, or simply no linear-time algorithms are known.…”

Section: Introductionmentioning

confidence: 99%

“…Such a parameterized approach makes sense for any problem in P for which a conditional hardness result is proved, or simply no linear-time algorithms are known. Interestingly, the authors of [55] proved that a matching of cardinality at least k in a graph can be computed in O(kn + k 3 )-time. We stress that Maximum Matching is a classical and intensively studied problem in computer science [37,45,46,49,66,72,71,86].…”

Section: Introductionmentioning

confidence: 99%

Fully polynomial FPT algorithms for some classes of bounded clique-width graphs

Coudert¹,

Ducoffe²,

Popa³

2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

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Recently, hardness results for problems in P were achieved using reasonable complexity theoretic assumptions such as the Strong Exponential Time Hypothesis. According to these assumptions, many graph theoretic problems do not admit truly subquadratic algorithms. A central technique used to tackle the difficulty of the above mentioned problems is fixed-parameter algorithms with polynomial dependency in the fixed parameter (P-FPT). Applying this technique to clique-width, an important graph parameter, remained to be done. In this paper we study several graph theoretic problems for which hardness results exist such as cycle problems, distance problems and maximum matching. We give hardness results and P-FPT algorithms, using cliquewidth and some of its upper-bounds as parameters. We believe that our most important result is an O(k 4 · n + m)-time algorithm for computing a maximum matching where k is either the modular-width or the P 4 -sparseness. The latter generalizes many algorithms that have been introduced so far for specific subclasses such as cographs. Our algorithms are based on preprocessing methods using modular decomposition and split decomposition. Thus they can also be generalized to some graph classes with unbounded clique-width.

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“…This direction fits into the general concept of "FPT within P" that was coined very recently by Giannopoulou et al [31] and by Abboud et al [1]. In particular, Abboud et al [1] made already the first step into investigating the treewidth parameterizations by considering the Diameter and Radius problems.…”

Section: Introductionmentioning

confidence: 59%

Fully Polynomial-Time Parameterized Computations for Graphs and Matrices of Low Treewidth

Fomin

¹

,

Lokshtanov

²

,

Saurabh

³

et al. 2018

ACM Trans. Algorithms

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We investigate the complexity of several fundamental polynomial-time solvable problems on graphs and on matrices, when the given instance has low treewidth; in the case of matrices, we consider the treewidth of the graph formed by non-zero entries. In each of the considered cases, the best known algorithms working on general graphs run in polynomial time, however the exponent of the polynomial is large. Therefore, our main goal is to construct algorithms with running time of the form poly(k)·n or poly(k)·n log n, where k is the width of the tree decomposition given on the input. Such procedures would outperform the best known algorithms for the considered problems already for moderate values of the treewidth, like O(n 1/c ) for some small constant c.Our results include:-an algorithm for computing the determinant and the rank of an n×n matrix using O(k 3 ·n) time and arithmetic operations; -an algorithm for solving a system of linear equations using O(k 3 · n) time and arithmetic operations; -an O(k 3 · n log n)-time randomized algorithm for finding the cardinality of a maximum matching in a graph; -an O(k 4 · n log 2 n)-time randomized algorithm for constructing a maximum matching in a graph; -an O(k 2 · n log n)-time algorithm for finding a maximum vertex flow in a directed graph.Moreover, we give an approximation algorithm for treewidth with time complexity suited to the running times as above. Namely, the algorithm, when given a graph G and integer k, runs in time O(k 7 · n log n) and either correctly reports that the treewidth of G is larger than k, or constructs a tree decomposition of G of width O(k 2 ).The above results stand in contrast with the recent work of Abboud et al.[to appear at SODA 2016], which shows that the existence of algorithms with similar running times is unlikely for the problems of finding the diameter and the radius of a graph of low treewidth. * D.Recently M ֒ adry [48] gave an O(m 10/7 ) algorithm for the unweighted bipartite case.

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Polynomial fixed-parameter algorithms: A case study for longest path on interval graphs

Cited by 38 publications

References 47 publications

Fully polynomial-time parameterized computations for graphs and matrices of low treewidth

Fully polynomial-time parameterized computations for graphs and matrices of low treewidth

Fully polynomial FPT algorithms for some classes of bounded clique-width graphs

Fully Polynomial-Time Parameterized Computations for Graphs and Matrices of Low Treewidth

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