Integer programming in parameterized complexity: Five miniatures

Gavenčiak, Tomáš; Koutecký, Martin; Knop, Dušan

doi:10.1016/j.disopt.2020.100596

Cited by 19 publications

(15 citation statements)

References 30 publications

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“…We start from two recently introduced parameters: modular-width [22] and neighborhood diversity [31]. Both parameters received much attention [1,2,5,7,12,17,18,21,24,25,29] also due to their property of being computable in polynomial time [22,31].…”

Section: Introductionmentioning

confidence: 99%

Iterated Type Partitions

Cordasco

Gargano

Rescigno

2020

Lecture Notes in Computer Science

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This paper introduces a novel parameter, called iterated type partition, that can be computed in polynomial time and nicely places between modular-width and neighborhood diversity. We prove that the Equitable Coloring problem is W[1]-hard when parametrized by the iterated type partition. This result extends to modular-width, answering an open question on the complexity of Equitable Coloring when parametrized by modular-width. On the contrary, we show that the Equitable Coloring problem is FPT when parameterized by neighborhood diversity. Furthermore, we present a scheme for devising FPT algorithms parameterized by iterated type partition, which enables us to find optimal solutions for several graph problems. While the considered problems are already known to be FPT with respect to modular-width, the novel algorithms are both simpler and more efficient. As an example, in this paper, we give an algorithm for the Dominating Set problem that outputs an optimal set in time O(2 t + poly(n)), where n and t are the size and the iterated type partition of the input graph, respectively.

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Section: Introductionmentioning

confidence: 99%

Iterated Type Partitions

Cordasco

Gargano

Rescigno

2020

Lecture Notes in Computer Science

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“…For example, neither Gera et al [34] nor Sorge and Weller [50] list it. This is surprising, since the Dilworth number is bounded from above by the neighborhood diversity (see Section 2.3), which is a well-established parameter in parameterized complexity studies [4,[31][32][33]39], and it seems a logical step to analyze which parameterized algorithms for the neighborhood diversity can be strengthened to use the Dilworth number instead.…”

Section: Problem 11 (Multiple Hitting Set)mentioning

confidence: 99%

“…A related frequently studied graph parameter is the neighborhood diversity [4,[31][32][33]39]. To introduce it, consider the relation ∼ on the vertices of a graph G = (V, E) such that…”

Section: Dilworth Number and Neighborhood Diversitymentioning

confidence: 99%

Serial and parallel kernelization of Multiple Hitting Set parameterized by the Dilworth number, implemented on the GPU

Bevern¹,

Kirilin²,

Skachkov³

et al. 2021

Preprint

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The NP-hard Multiple Hitting Set problem is finding a minimum-cardinality set intersecting each of the sets in a given input collection a given number of times. Generalizing a well-known data reduction algorithm due to Weihe, we show a problem kernel for Multiple Hitting Set parameterized by the Dilworth number, a graph parameter introduced by Foldes and Hammer in 1978 yet seemingly so far unexplored in the context of parameterized complexity theory. Using matrix multiplication, we speed up the algorithm to quadratic sequential time and logarithmic parallel time. We experimentally evaluate our algorithms. By implementing our algorithm on GPUs, we show the feasibility of realizing kernelization algorithms on SIMD (Single Instruction, Multiple Date) architectures.

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“…Let us compute the complexity. We are maintaining (22) and the fact that ρ ≤ д 1 (A), updating one convolution tree takes time (∥A∥ ∞ • д 1 (A)) O(td D (A)) σ log n with σ = д 1 (A). Because n ′ ≤ 3n log(2f max ) we update each tree at most 3n log(2f max ) times, in total taking time…”

Section: Lemma 76mentioning

confidence: 99%

“…Moreover, if u is an internal node, let its left and right child be v and w, respectively, and let k := max v be the rightmost leaf of T v . Consider the following auxiliary problem, which is intuitively problem (22) restricted to blocks i to j: simply append "[i, j]" to all relevant objects, namely f T , g, A, l T and u T :…”

Section: Lemma 76mentioning

confidence: 99%

An Algorithmic Theory of Integer Programming

Eisenbrand¹,

Hunkenschröder²,

Klein³

et al. 2019

Preprint

Self Cite

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We study the general integer programming problem where the number of variables n is a variable part of the input. We consider two natural parameters of the constraint matrix A: its numeric measure a and its sparsity measure d. We show that integer programming can be solved in time д(a, d)poly(n, L), where д is some computable function of the parameters a and d, and L is the binary encoding length of the input. In particular, integer programming is fixed-parameter tractable parameterized by a and d, and is solvable in polynomial time for every fixed a and d. Our results also extend to nonlinear separable convex objective functions. Moreover, for linear objectives, we derive a strongly-polynomial algorithm, that is, with running time д(a, d)poly(n), independent of the rest of the input data.We obtain these results by developing an algorithmic framework based on the idea of iterative augmentation: starting from an initial feasible solution, we show how to quickly find augmenting steps which rapidly converge to an optimum. A central notion in this framework is the Graver basis of the matrix A, which constitutes a set of fundamental augmenting steps. The iterative augmentation idea is then enhanced via the use of other techniques such as new and improved bounds on the Graver basis, rapid solution of integer programs with bounded variables, proximity theorems and a new proximity-scaling algorithm, the notion of a reduced objective function, and others.As a consequence of our work, we advance the state of the art of solving block-structured integer programs. In particular, we develop near-linear time algorithms for n-fold, tree-fold, and 2-stage stochastic integer programs. We also discuss some of the many applications of these classes.

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Integer programming in parameterized complexity: Five miniatures

Cited by 19 publications

References 30 publications

Iterated Type Partitions

Iterated Type Partitions

Serial and parallel kernelization of Multiple Hitting Set parameterized by the Dilworth number, implemented on the GPU

An Algorithmic Theory of Integer Programming

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