(Meta) Kernelization

Bodlaender, Hans L.; Fomin, Fedor V.; Lokshtanov, Daniel; Penninkx, Eelko; Saurabh, Saket; Thilikos, Dimitrios M.

doi:10.1145/2973749

Cited by 114 publications

(158 citation statements)

References 71 publications

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“…(Neither σ nor G X depend on the formula.) For 0 ≤ j ≤ m, let ξ j (x 1 ) be the formula obtained from ξ j by Lemma 7.2 (2). We recursively evaluate the formulas ξ 0 , .…”

Section: The Main Algorithmmentioning

confidence: 99%

“…Another well-known example is Papadimitriou and Yannakakis's [31] result that all optimisation problems in the class MAXSNP, which is defined in terms of a fragment of existential second-order logic, admit constant-ratio polynomial time approximation algorithms. By now, there is a rich literature on algorithmic meta theorems (see, for example, [2,5,6,7,8,13,17,26,27,33] and the surveys [19,21,25]). While the main motivation for proving such meta theorems may be to understand the "essence" and the scope of certain algorithmic techniques by abstracting from problem-specific details, sometimes meta theorems are also crucial for obtaining new algorithmic results.…”

Section: Introductionmentioning

confidence: 99%

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Deciding first-order properties of nowhere dense graphs

Grohe

Kreutzer

Siebertz

2014

Proceedings of the Forty-Sixth Annual ACM Symposium on Theory of Computing

129

217

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Nowhere dense graph classes, introduced by Nešetřil and Ossona de Mendez [30], form a large variety of classes of "sparse graphs" including the class of planar graphs, actually all classes with excluded minors, and also bounded degree graphs and graph classes of bounded expansion.We show that deciding properties of graphs definable in first-order logic is fixed-parameter tractable on nowhere dense graph classes. At least for graph classes closed under taking subgraphs, this result is optimal: it was known before that for all classes C of graphs closed under taking subgraphs, if deciding first-order properties of graphs in C is fixed-parameter tractable, then C must be nowhere dense (under a reasonable complexity theoretic assumption).As a by-product, we give an algorithmic construction of sparse neighbourhood covers for nowhere dense graphs. This extends and improves previous constructions of neighbourhood covers for graph classes with excluded minors. At the same time, our construction is considerably simpler than those.Our proofs are based on a new game-theoretic characterisation of nowhere dense graphs that allows for a recursive version of localitybased algorithms on these classes. On the logical side, we prove a "rank-preserving" version of Gaifman's locality theorem.

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“…(Neither σ nor G X depend on the formula.) For 0 ≤ j ≤ m, let ξ j (x 1 ) be the formula obtained from ξ j by Lemma 7.2 (2). We recursively evaluate the formulas ξ 0 , .…”

Section: The Main Algorithmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Deciding first-order properties of nowhere dense graphs

Grohe

Kreutzer

Siebertz

2014

Proceedings of the Forty-Sixth Annual ACM Symposium on Theory of Computing

129

217

View full text Add to dashboard Cite

show abstract

“…Given the amount of attention the planar version of Steiner-type problems received from the viewpoint of approximation (see, e.g., [2,3,11,26,32]) and the availability of techniques for parameterized algorithms on planar graphs (see, e.g., [6,27,40,50,59]), it is natural to explore SCSS and DSN restricted to planar graphs 1 . In general, one can have the expectation that the problems restricted to planar graphs become easier, but sophisticated techniques might be needed to exploit planarity.…”

Section: Our Results and Techniquesmentioning

confidence: 99%

“…• For every set S i, j in the GRID TILING instance, we construct a main gadget MG i, j using Lemma 3. 6 for the subset S i, j .…”

Section: Construction Of the Scss Instancementioning

confidence: 99%

Tight Bounds for Planar Strongly Connected Steiner Subgraph with Fixed Number of Terminals (and Extensions)

Chitnis¹,

Feldmann²,

Hajiaghayi³

et al. 2020

SIAM J. Comput.

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Given a vertex-weighted directed graph G = (V, E) and a set T = {t 1 ,t 2 , . . .t k } of k terminals, the objective of the STRONGLY CONNECTED STEINER SUBGRAPH (SCSS) problem is to find a vertex set H ⊆ V of minimum weight such that G[H] contains a t i → t j path for each i = j. The problem is NP-hard, but Feldman and Ruhl [FOCS '99; SICOMP '06] gave a novel n O(k) algorithm for the SCSS problem, where n is the number of vertices in the graph and k is the number of terminals. We explore how much easier the problem becomes on planar directed graphs.• Feldman and Ruhl generalized their n O(k) algorithm to the more general DIRECTED STEINER NETWORK (DSN) problem; here the task is to find a subgraph of minimum weight such that for every source s i there is a path to the corresponding terminal t i . We show that, assuming ETH, there is no f (k) · n o(k) time algorithm for DSN on acyclic planar graphs.All our lower bounds hold for the edge-unweighted version, while the algorithm works for the more general vertex-(un)weighted version. IntroductionThe STEINER TREE (ST) problem is one of the earliest and most fundamental problems in combinatorial optimization: given an undirected graph G = (V, E) and a set T ⊆ V of terminals, the objective is to find a tree of minimum size which connects all the terminals. The ST problem is believed to have been first formally defined by Gauss in a letter in 1836, and the first combinatorial formulation is attributed independently to Hakimi [43] and Levin [52] in 1971. The ST problem is known to be NP-complete, and was in fact part of Karp's original list [49] of 21 NP-complete problems. In the directed version of the ST problem, called DIRECTED STEINER TREE (DST), we are also given a root vertex r and the objective is to find a minimum size arborescence which connects the root r to each terminal from T . An easy reduction from SET COVER shows that the DST problem is also NP-complete.Steiner-type problems arise in the design of networks. Since many networks are symmetric, the directed versions of Steiner-type problems were mostly of theoretical interest. However, in recent years, it has been observed [65, 67] that the connection cost in various networks such as satellite or radio networks are not symmetric. Therefore, directed graphs form the most suitable model for such networks. In addition, Ramanathan [65] also used the DST problem to find low-cost multicast trees, which have applications in point-to-multipoint communication in high bandwidth networks. We refer the interested reader to Winter [69] for a survey on applications of Steiner problems in networks.In this paper we consider two well-studied Steiner-type problems in directed graphs, namely the STRONGLY CONNECTED STEINER SUBGRAPH and the DIRECTED STEINER NETWORK problems. In the (vertex-unweighted) STRONGLY CONNECTED STEINER SUBGRAPH (SCSS) problem, given a directed graph G = (V, E) and a set T = {t 1 ,t 2 , . . . ,t k } of k terminals, the objective is to find a set S ⊆ V of minimum size such that G[S] contains a t i → t j path...

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“…An important subclass of FPT is formed by those problems that allow kernelizations with size guarantee polynomial in k, so-called polynomial kernelizations. This includes plenty of results with linear or quadratic size kernels (e.g., [Thomassé 2010;Bodlaender et al 2009b;) and also benefits of the good closure properties of polynomials.…”

Section: Introductionmentioning

confidence: 99%

Co-Nondeterminism in Compositions

Kratsch

2014

ACM Trans. Algorithms

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The field of kernelization offers a rigorous way of studying the ubiquitous technique of data reduction and preprocessing for combinatorially hard problems. A widely accepted definition of useful data reduction is that of a polynomial kernelization where the output instance is guaranteed to be of size polynomial in some parameter of the input. The fairly recent development of a framework for kernelization lower bounds has made this notion even more attractive as we can now classify many problems into admitting or not admitting polynomial kernelizations. The central notion of the framework is that of a polynomial-time composition algorithm due to Bodlaender et al. (ICALP 2008, JSCC 2009): given t input instances, an OR-composition algorithm returns a single-output instance with bounded parameter value that is yes if and only if one of t input instances is yes; it encodes the logical OR of the input instances. Based on a result of Fortnow and Santhanam (STOC 2008, JSCC 2011 show that an OR-composition for an NP-hard problem rules out polynomial kernelizations for it unless NP ⊆ coNP/poly (which is known to imply a collapse of the polynomial hierarchy). It is implicit in the work of Fortnow and Santhanam that even co-nondeterministic composition algorithms suffice to rule out polynomial kernelizations. This was first observed in unpublished work of Chen and Müller, and it is an explicit conclusion of recent results by Dell and van Melkebeek (STOC 2010). However, in contrast to the numerous applications of deterministic composition, the added power of co-nondeterminism has not yet been harnessed to obtain kernelization lower bounds.In this work, we present the first example of how co-nondeterminism can help to make a composition algorithm. We study the existence of polynomial kernels for a Ramsey-type problem where, given a graph G and an integer k, the question is whether G contains an independent set or a clique of size at least k. It was asked by Rod Downey whether this problem admits a polynomial kernelization with respect to k; such a result would greatly speed up the computation of Ramsey numbers. We provide a co-nondeterministic composition based on embedding t instances into a single host graph H. The crux is that the host graph H needs to observe a bound of ∈ O(log t) on both its maximum independent set and maximum clique size, while also having a cover of its vertex set by independent sets and cliques all of size ; the co-nondeterministic composition is built around the search for such graphs. Thus, we show that, unless NP ⊆ coNP/poly, the problem does not admit a kernelization with polynomial size guarantee. ACM Reference Format:Stefan Kratsch. 2014. Co-nondeterminism in compositions: A kernelization lower bound for a Ramsey-type problem. ACM Trans.

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(Meta) Kernelization

Cited by 114 publications

References 71 publications

Deciding first-order properties of nowhere dense graphs

Deciding first-order properties of nowhere dense graphs

Tight Bounds for Planar Strongly Connected Steiner Subgraph with Fixed Number of Terminals (and Extensions)

Co-Nondeterminism in Compositions

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