Improved Bounds for the Excluded-Minor Approximation of Treedepth

Czerwiński, Wojciech; Nadara, Wojciech; Pilipczuk, Marcin

doi:10.1137/19m128819x

Cited by 12 publications

(19 citation statements)

References 6 publications

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“…For treedepth, Kawarabayashi and Rossman [31] first demonstrated a polynomial excluded-minor approximation for graphs with large treedepth which was later improved by Czerwiński, Nadara, and Pilipczuk [15]. Theorem 18 ([15, 31]).…”

Section: Pathwidth and Treedepthmentioning

confidence: 99%

Structural properties of bipartite subgraphs

Hickingbotham,

Wood

2021

Preprint

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This paper establishes sufficient conditions that force a graph to contain a bipartite subgraph with a given structural property. In particular, let β be any of the following graph parameters: Hadwiger number, Hajós number, treewidth, pathwidth, and treedepth. In each case, we show that there exists a function f such that every graph G with β(G) f (k) contains a bipartite subgraph Ĝ ⊆ G with β( Ĝ) k.

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Section: Pathwidth and Treedepthmentioning

confidence: 99%

Structural properties of bipartite subgraphs

Hickingbotham,

Wood

2021

Preprint

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“…The treedepth of a graph G is the least possible height of an elimination forest of G. Treedepth as a graph parameter plays a central role in the structural theory of sparse graphs, see [45,Chapters 6 and 7]. It also has several applications in parameterized complexity and algorithm design [12,22,28,44,46,47], as well as exhibits interesting combinatorial properties [12,17,21] and connections to descriptive complexity theory [23]. We refer to the introductory sections of the above works for a wider discussion.…”

Section: Preliminariesmentioning

confidence: 99%

Isolation schemes for problems on decomposable graphs

Nederlof,

Pilipczuk,

Swennenhuis

et al. 2021

Preprint

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The Isolation Lemma of Mulmuley, Vazirani and Vazirani [Combinatorica'87] provides a self-reduction scheme that allows one to assume that a given instance of a problem has a unique solution, provided a solution exists at all. Since its introduction, much e ort has been dedicated towards derandomization of the Isolation Lemma for speci c classes of problems. So far, the focus was mainly on problems solvable in polynomial time.In this paper, we study a setting that is more typical for NP-complete problems, and obtain partial derandomizations in the form of signi cantly decreasing the number of required random bits. In particular, motivated by the advances in parameterized algorithms, we focus on problems on decomposable graphs. For example, for the problem of detecting a Hamiltonian cycle, we build upon the rank-based approach from [Bodlaender et al., Inf. Comput. '15] and design isolation schemes that use• O(t log n + log 2 n) random bits on graphs of treewidth at most t;• O( √ n) random bits on planar or H-minor free graphs; and • O(n)-random bits on general graphs. In all these schemes, the weights are bounded exponentially in the number of random bits used. As a corollary, for every xed H we obtain an algorithm for detecting a Hamiltonian cycle in an Hminor-free graph that runs in deterministic time 2 O( √ n) and uses polynomial space; this is the rst algorithm to achieve such complexity guarantees. For problems of more local nature, such as nding an independent set of maximum size, we obtain isolation schemes on graphs of treedepth at most d that use O(d) random bits and assign polynomially-bounded weights.We also complement our ndings with several unconditional and conditional lower bounds, which show that many of the results cannot be signi cantly improved.

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“…Unfortunately, no such approximation algorithm is known for the treedepth. Namely, it is known that the treedepth can be computed exactly in time and space 2 O(d 2 ) • n [27] and approximated up to factor O(t log 3/2 t) in polynomial time [9], where d and t are the values of the treedepth and the treewidth of the input graph, respectively. A piece of the theory that seems particularly missing is a constant-factor approximation algorithm for treedepth running in time 2 O(d) • n O (1) ; polynomial space usage would be also desired.…”

Section: Approximation Of Treedepthmentioning

confidence: 99%

Hamiltonian Cycle Parameterized by Treedepth in Single Exponential Time and Polynomial Space

Nederlof¹,

Pilipczuk²,

Swennenhuis³

et al. 2020

Preprint

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For many algorithmic problems on graphs of treewidth t, a standard dynamic programming approach gives an algorithm with time and space complexity 2 O(t) • n O(1) . It turns out that when one considers a more restrictive parameter treedepth, it is often the case that a variation of this technique can be used to reduce the space complexity to polynomial, while retaining time complexity of the form 2 O(d) • n O(1) , where d is the treedepth. This transfer of methodology is, however, far from being automatic. For instance, for problems with connectivity constraints, standard dynamic programming techniques give algorithms with time and space complexity 2 O(t log t) • n O(1) on graphs of treewidth t, but it is not clear how to convert them into time-efficient polynomial space algorithms for graphs of low treedepth.Cygan et al. (FOCS'11) introduced the Cut&Count technique and showed that a certain class of problems with connectivity constraints can be solved in time and space complexity 2 O(t) • n O(1) . Recently, Hegerfeld and Kratsch (STACS'20) showed that, for some of those problems, the Cut&Count technique can be also applied in the setting of treedepth, and it gives algorithms with running time 2 O(d) • n O(1) and polynomial space usage. However, a number of important problems eluded such a treatment, with the most prominent examples being Hamiltonian Cycle and Longest Path.In this paper we clarify the situation by showing that Hamiltonian Cycle, Hamiltonian Path, Long Cycle, Long Path, and Min Cycle Cover all admit 5 d •n O(1) -time and polynomial space algorithms on graphs of treedepth d. The algorithms are randomized Monte Carlo with only false negatives.

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Improved Bounds for the Excluded-Minor Approximation of Treedepth

Cited by 12 publications

References 6 publications

Structural properties of bipartite subgraphs

Structural properties of bipartite subgraphs

Isolation schemes for problems on decomposable graphs

Hamiltonian Cycle Parameterized by Treedepth in Single Exponential Time and Polynomial Space

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