Dichotomy Results on the Hardness of $H$-free Edge Modification Problems

Aravind, N. R.; Sandeep, R. B.; Sivadasan, Naveen

doi:10.1137/16m1055797

Cited by 24 publications

(55 citation statements)

References 10 publications

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“…The number of triangles of is denoted by T . All triangles of can be computed in 2 ) time [1]. The trussness of a triangle is the minimum among the trussness of its edges.…”

Section: Preliminaries and Techniquesmentioning

confidence: 99%

“…It is NP-hard to find a smallest set of edges to delete to make triangle-free, which is exactly the MIN-3-TBS problem. Assuming the Exponential Time Hypothesis (ETH), we cannot even solve this problem in 2 ( | ′ |) • (1) time [2], where = | |.…”

Section: A Omitted Proofsmentioning

confidence: 99%

“…In A3 and A4, it allows less effort from users and more accurate analysis of the sanitized graph, respectively. The subgraph induced by the edges (0, 1), (0, 2), (0, 3), (1,2), (1,3), (2, 3) is a 4-truss because every edge of the subgraph is contained in at least 4 − 2 = 2 triangles of the subgraph. (b) The subgraph induced by all edges except the (dashed) edge (0, 4) is the maximal 4-truss of the graph.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On Breaking Truss-Based Communities

Chen

Conte

Grossi

et al. 2021

Proceedings of the 27th ACM SIGKDD Conference on Knowledge Discovery &Amp; Data Mining

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A -truss is a graph such that each edge is contained in at least − 2 triangles. This notion has attracted much attention, because it models meaningful cohesive subgraphs of a graph. We introduce the problem of identifying a smallest edge subset of a given graph whose removal makes the graph -truss-free. We also introduce a problem variant where the identified subset contains only edges incident to a given set of nodes and ensures that these nodes are not contained in any -truss. These problems are directly applicable in communication networks: the identified edges correspond to vital network connections; or in social networks: the identified edges can be hidden by users or sanitized from the output graph. We show that these problems are NP-hard. We thus develop exact exponentialtime algorithms to solve them. To process large networks, we also develop heuristics sped up by an efficient data structure for updating the truss decomposition under edge deletions. We complement our heuristics with a lower bound on the size of an optimal solution to rigorously evaluate their effectiveness. Extensive experiments on 10 real-world graphs show that our heuristics are effective (close to the optimal or to the lower bound) and also efficient (up to two orders of magnitude faster than a natural baseline).

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“…The number of triangles of is denoted by T . All triangles of can be computed in 2 ) time [1]. The trussness of a triangle is the minimum among the trussness of its edges.…”

Section: Preliminaries and Techniquesmentioning

confidence: 99%

Section: A Omitted Proofsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On Breaking Truss-Based Communities

Chen

Conte

Grossi

et al. 2021

Proceedings of the 27th ACM SIGKDD Conference on Knowledge Discovery &Amp; Data Mining

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show abstract

“…Note that it is possible to improve the branching rules on LO-Diamonds, IIZ-Diamonds, and CC-Hourglasses to obtain a branching vector (2,2,3,3), but then, branching on LC-Diamonds still needs a branching vector of (1,2,3), which is the bottleneck. To put the running time of Theorem 5.10 into perspective note that Cluster Deletion, which can be viewed as the uncolored version of BPD, can be solved in O(1.42 k + m) time [5].…”

Section: Parameterized Complexitymentioning

confidence: 99%

“…Besides the application, there is a more theoretical reason why graph modification problems are very important in computer science: Often these problems are NPhard [19,26] and thus represent interesting case studies for algorithmic approaches to NP-hard problems. For example, by systematically categorizing graph properties based on their forbidden subgraphs one may outline the border between tractable and hard graph modification problems [2,18,26].…”

Section: Introductionmentioning

confidence: 99%

Destroying Bicolored $$P_3$$s by Deleting Few Edges

Grüttemeier

Komusiewicz

Schestag

et al. 2019

Computing With Foresight and Industry

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We introduce and study the Bicolored P 3 Deletion problem defined as follows. The input is a graph G = (V, E) where the edge set E is partitioned into a set E b of blue edges and a set E r of red edges. The question is whether we can delete at most k edges such that G does not contain a bicolored P 3 as an induced subgraph. Here, a bicolored P 3 is a path on three vertices with one blue and one red edge. We show that Bicolored P 3 Deletion is NP-hard and cannot be solved in 2 o(|V |+|E|) time on bounded-degree graphs if the ETH is true. Then, we show that Bicolored P 3 Deletion is polynomial-time solvable when G does not contain a bicolored K 3 , that is, a triangle with edges of both colors. Moreover, we provide a polynomial-time algorithm for the case if G contains no blue P 3 , red P 3 , blue K 3 , and red K 3 . Finally, we show that Bicolored P 3 Deletion can be solved in O(1.85 k · |V | 5 ) time and that it admits a kernel with O(∆k 2 ) vertices, where ∆ is the maximum degree of G. * Some of the results of this work are contained in the third author's Bachelor thesis [21]. † FS was supported by the DFG, project MAGZ (KO 3669/4-1).

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Cutting a tree with subgraph complementation is hard, except for some small trees

Antony,

Pal,

Sandeep

et al. 2024

Journal of Graph Theory

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For a graph property , Subgraph Complementation to is the problem to find whether there is a subset of vertices of the input graph such that modifying by complementing the subgraph induced by results in a graph satisfying the property . We prove that the problem of Subgraph Complementation to ‐free graphs is NP‐Complete, for being a tree, except for 41 trees of at most 13 vertices (a graph is ‐free if it does not contain any induced copies of ). This result, along with the four known polynomial‐time solvable cases (when is a path on at most four vertices), leaves behind 37 open cases. Further, we prove that these hard problems do not admit any subexponential‐time algorithms, assuming the Exponential‐Time Hypothesis. As an additional result, we obtain that Subgraph Complementation to paw‐free graphs can be solved in polynomial‐time.

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Dichotomy Results on the Hardness of $H$-free Edge Modification Problems

Cited by 24 publications

References 10 publications

On Breaking Truss-Based Communities

On Breaking Truss-Based Communities

Destroying Bicolored $$P_3$$s by Deleting Few Edges

Cutting a tree with subgraph complementation is hard, except for some small trees

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