Deleting, Eliminating and Decomposing to Hereditary Classes Are All FPT-Equivalent

Agrawal, Akanksha; Kanesh, Lawqueen; Lokshtanov, Daniel; Panolan, Fahad; Ramanujan, M. S.; Saurabh, Saket; Zehavi, Meirav

doi:10.1137/1.9781611977073.79

Cited by 3 publications

(8 citation statements)

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“…This yields a (constructive) quadratic-time algorithm. We stress that our approach avoids techniques that have been recently used for this type of problems such as recursive understanding (in [4]) or the use of important separators (in [73]) that give worst running times in n. For a more detailed discussion on the results of [4,73] and their relation to our results, see the applications section (Section 3).…”

Section: Our Resultsmentioning

confidence: 92%

“…The development of parameterized algorithms for cut problems is a popular trend in parameterized algorithms [17,34,70,81,83,97,99]. More involved modification measures of vertex set removals, related to treewidth or treedepth, have been considered very recently [4,19,20,41,73].…”

Section: State Of the Art And Our Contributionmentioning

confidence: 99%

“…We provide these precise definitions in Subsection A. 4, including the concepts of flatness pair, homogeneity, regularity, tilt, and influence, and we stress that they are not critical in order to understand the main technical contributions of the current article (however, they are critical for their formal correctness). In what follows, when considering a flat wall W with an apex set A in a graph G, for simplicity we refer to W by using indistinguishably the terms "wall" and "compass of a wall", which can be roughly described as the component containing W in the graph obtained from G by removing A and the "boundary" of W (see Subsection A.…”

Section: General Scheme Of the Algorithmmentioning

confidence: 99%

“…In what follows, when considering a flat wall W with an apex set A in a graph G, for simplicity we refer to W by using indistinguishably the terms "wall" and "compass of a wall", which can be roughly described as the component containing W in the graph obtained from G by removing A and the "boundary" of W (see Subsection A. 4…”

Section: General Scheme Of the Algorithmmentioning

confidence: 99%

“…Very recently, in [4], Agrawal, Kanesh, Lokshtanov, Panolan, Ramanujan, Saurabh, and Zehavi proved that, under certain assumptions on G, G-treewidth, elimination distance to G, and vertex deletion to G are all FPT-equivalent. The techniques used in [4] are heavily based on the metaalgorithmic result of [96] (based on the recursive understanding technique) that is non-constructive and implies FPT-algorithms whose running time is worse than quadratic.…”

Section: Classic Modification Problemsmentioning

confidence: 99%

See 4 more Smart Citations

A Compound Logic for Modification Problems: Big Kingdoms Fall from Within

Fomin¹,

Golovach²,

Sau³

et al. 2021

Preprint

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We introduce a novel model-theoretic framework inspired from graph modification and based on the interplay between model theory and algorithmic graph minors. We propose a new compound logic operating with two types of sentences, expressing graph modification: the modulator sentence, defining some property of the modified part of the graph, and the target sentence, defining some property of the resulting graph. In our framework, modulator sentences are in monadic second-order logic and have models of bounded treewidth, while target sentences express firstorder logic properties along with minor-exclusion. Our logic captures problems that are not definable in first-order logic and, moreover, may have instances of unbounded treewidth. Also, it permits the modeling of wide families of problems involving vertex/edge removals, alternative modulator measures (such as elimination distance or G-treewidth), multistage modifications, and various cut problems. Our main result is that, for this compound logic, model checking can be done in quadratic time. This algorithmic meta-theorem encompasses, unifies, and extends all known meta-algorithmic results on minor-closed graph classes. Moreover, all derived algorithms are constructive and this, as a byproduct, extends the constructibility horizon of the algorithmic applications of the Graph Minors theorem of Robertson and Seymour. The proposed logic can be seen as a general framework to capitalize on the potential of the irrelevant vertex technique. It gives a way to deal with problem instances of unbounded treewidth, for which Courcelle's theorem does not apply. Our insight is that, for problems definable in our logic, such "big kingdoms" can be "corrupted internally": way deep inside instances of big treewidth, we can find "irrelevant territories" that can be safely discarded towards creating simpler equivalent instances. To prove this, we combine novel combinatorial results related to the Flat Wall theorem along with elements of the proof of Courcelle's theorem and Gaifman's theorem.

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Section: Our Resultsmentioning

confidence: 92%