On the Complexity of Finding a Largest Common Subtree of Bounded Degree

Akutsu, Tatsuya; Tamura, Takeyuki; Melkman, Avraham A.; Takasu, Atsuhiro

doi:10.1007/978-3-642-40164-0_4

Cited by 6 publications

(7 citation statements)

References 21 publications

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“…From an instance G of k-Clique with k > 3, we build the equivalent instance Let us now say some words about the complexity of the connected version. First we note that MCIS is NP-hard on forests while MCCIS is solvable in polynomial-time in this case: given two forests G 1 and G 2 , run the polynomial-time MCIS algorithm of Akutsu on every pair of trees from G 1 and G 2 [3]. From the parameterized complexity standpoint, Maximum Common Connected Induced Subgraph is FPT whenever Induced Subgraph Isomorphism is FPT since the enumeration of all O(2 k 2 ) possible induced connected subgraphs can be used as described in the introduction.…”

Section: Theorem 7 Isi (Hence Mcis) and Icsi (Hence Mccis) Remain W[mentioning

confidence: 99%

On the Complexity of Various Parameterizations of Common Induced Subgraph Isomorphism

Abu-Khzam

Bonnet

Sikora

2015

Lecture Notes in Computer Science

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In the Maximum Common Induced Subgraph problem (henceforth MCIS), given two graphs G1 and G2, one looks for a graph with the maximum number of vertices being both an induced subgraph of G1 and G2. MCIS is among the most studied classical NP-hard problems. It remains NP-hard on many graph classes including forests. In this paper, we study the parameterized complexity of MCIS. As a generalization of Clique, it is W[1]-hard parameterized by the size of the solution. Being NP-hard even on forests, most structural parameterizations are intractable. One has to go as far as parameterizing by the size of the minimum vertex cover to get some tractability. Indeed, when parameterized by k := vc(G1) + vc(G2) the sum of the vertex cover number of the two input graphs, the problem was shown to be fixed-parameter tractable, with an algorithm running in time 2 O(k log k) . We complement this result by showing that, unless the ETH fails, it cannot be solved in time 2 o(k log k) . This kind of tight lower bound has been shown for a few problems and parameters but, to the best of our knowledge, not for the vertex cover number. We also show that MCIS does not have a polynomial kernel when parameterized by k, unless NP ⊆ coNP/poly. Finally, we study MCIS and its connected variant MCCIS on some special graph classes and with respect to other structural parameters.

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Section: Theorem 7 Isi (Hence Mcis) and Icsi (Hence Mccis) Remain W[mentioning

confidence: 99%

On the Complexity of Various Parameterizations of Common Induced Subgraph Isomorphism

Abu-Khzam

Bonnet

Sikora

2015

Lecture Notes in Computer Science

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“…The problem is also of practical relevance, since it can model important applications in a wide variety of areas. Subtree Isomorphism is at the core of many more expressive problems, such as Largest Common Subtree [35,6,7], which generally ask: how "similar" are two trees? Application areas include computational biology [58], structured text databases [36], and compiler optimization [52].…”

Section: Introductionmentioning

confidence: 99%

“…Our reduction is easily modified to obtain similar lower bounds for related problems such as Largest Common Subtree on two trees (LCST). This problem is NP-hard when the number of trees is a parameter or when the two trees are labelled (and unrooted) [59,57], while some approximation and parameterized algorithms are known [35,7,6]. When the two trees are binary and unlabeled, the problem can be solved in quadratic time, and an adaptation of Theorem 1 shows that even when the height is (1 + o(1)) log n, a truly subquadratic algorithm refutes SETH.…”

Section: Introductionmentioning

confidence: 99%

Subtree Isomorphism Revisited

Abboud¹,

Bačkurs²,

Hansen³

et al. 2015

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

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The Subtree Isomorphism problem asks whether a given tree is contained in another given tree. The problem is of fundamental importance and has been studied since the 1960s. For some variants, e.g., ordered trees, near-linear time algorithms are known, but for the general case truly subquadratic algorithms remain elusive.Our first result is a reduction from the Orthogonal Vectors problem to Subtree Isomorphism, showing that a truly subquadratic algorithm for the latter refutes the Strong Exponential Time Hypothesis (SETH).In light of this conditional lower bound, we focus on natural special cases for which no truly subquadratic algorithms are known. We classify these cases against the quadratic barrier, showing in particular that:• Even for binary, rooted trees, a truly subquadratic algorithm refutes SETH.• Even for rooted trees of depth O(log log n), where n is the total number of vertices, a truly subquadratic algorithm refutes SETH.• For every constant d, there is a constant ε d > 0 and a randomized, truly subquadratic algorithm for degree-d rooted trees of depth at most (1 + ε d ) log d n. In particular, there is an O(min{2.85 h , n 2 }) algorithm for binary trees of depth h.Our reductions utilize new "tree gadgets" that are likely useful for future SETH-based lower bounds for problems on trees. Our upper bounds apply a folklore result from randomized decision tree complexity.

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“…It is to be noted that the number of MCS can be exponential even for trees [19]. Therefore, our proposed algorithm and all polynomial-time algorithms mentioned above are focusing on finding the one of MCS's.…”

Section: Introductionmentioning

confidence: 99%

A Polynomial-Time Algorithm for Computing the Maximum Common Connected Edge Subgraph of Outerplanar Graphs of Bounded Degree

Akutsu

Tamura

2013

Algorithms

Self Cite

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The maximum common connected edge subgraph problem is to find a connected graph with the maximum number of edges that is isomorphic to a subgraph of each of the two input graphs, where it has applications in pattern recognition and chemistry. This paper presents a dynamic programming algorithm for the problem when the two input graphs are outerplanar graphs of a bounded vertex degree, where it is known that the problem is NP-hard, even for outerplanar graphs of an unbounded degree. Although the algorithm repeatedly modifies input graphs, it is shown that the number of relevant subproblems is polynomially bounded, and thus, the algorithm works in polynomial time

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On the Complexity of Finding a Largest Common Subtree of Bounded Degree

Cited by 6 publications

References 21 publications

On the Complexity of Various Parameterizations of Common Induced Subgraph Isomorphism

On the Complexity of Various Parameterizations of Common Induced Subgraph Isomorphism

Subtree Isomorphism Revisited

A Polynomial-Time Algorithm for Computing the Maximum Common Connected Edge Subgraph of Outerplanar Graphs of Bounded Degree

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