arXiv subjects

Tomohiro Koana

Publications and source records attributed to Tomohiro Koana.

42 records · Page 3Linked to original sources

arXiv2021

Essentially Tight Kernels for (Weakly) Closed Graphs

We study kernelization of classic hard graph problems when the input graphs fulfill triadic closure properties. More precisely, we consider the recently introduced parameters closure number $c$ and the weak closure number $γ$ [Fox et al., SICOMP 2020] in addition to the standard parameter solution size $k$. For Capacitated Vertex Cover, Connected Vertex Cover, and Induced Matching we obtain the first kernels of size $k^{\mathcal{O}(γ)}$ and $(γk)^{\mathcal{O}(γ)}$, respectively, thus extending previous kernelization results on degenerate graphs. The kernels are essentially tight, since these problems are unlikely to admit kernels of size $k^{o(γ)}$ by previous results on their kernelization complexity in degenerate graphs [Cygan et al., ACM TALG 2017]. In addition, we provide lower bounds for the kernelization of Independent Set on graphs with constant closure number~$c$ and kernels for Dominating Set on weakly closed split graphs and weakly closed bipartite graphs.

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arXiv2021

The Complexity of Gerrymandering Over Graphs: Paths and Trees

Roughly speaking, gerrymandering is the systematic manipulation of the boundaries of electoral districts to make a specific (political) party win as many districts as possible. While typically studied from a geographical point of view, addressing social network structures, the investigation of gerrymandering over graphs was recently initiated by Cohen-Zemach et al. [AAMAS 2018]. Settling three open questions of Ito et al. [AAMAS 2019], we classify the computational complexity of the NP-hard problem Gerrymandering over Graphs when restricted to paths and trees. Our results, which are mostly of negative nature (that is, worst-case hardness), in particular yield two complexity dichotomies for trees. For instance, the problem is polynomial-time solvable for two parties but becomes weakly NP-hard for three. Moreover, we show that the problem remains NP-hard even when the input graph is a path.

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arXiv2020

Parameterized Complexity of Geodetic Set

A vertex set $S$ of a graph $G$ is geodetic if every vertex of $G$ lies on a shortest path between two vertices in $S$. Given a graph $G$ and $k \in \mathbb N$, the NP-hard Geodetic Set problem asks whether there is a geodetic set of size at most $k$. Complementing various works on Geodetic Set restricted to special graph classes, we initiate a parameterized complexity study of Geodetic Set and show, on the negative side, that Geodetic Set is W[1]-hard when parameterized by feedback vertex number, path-width, and solution size, combined. On the positive side, we develop fixed-parameter algorithms with respect to the feedback edge number, the tree-depth, and the modular-width of the input graph.

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arXiv2020

Detecting and Enumerating Small Induced Subgraphs in $c$-Closed Graphs

Fox et al. [SIAM J. Comp. 2020] introduced a new parameter, called $c$-closure, for a parameterized study of clique enumeration problems. A graph $G$ is $c$-closed if every pair of vertices with at least $c$ common neighbors is adjacent. The $c$-closure of $G$ is the smallest $c$ such that $G$ is $c$-closed. We systematically explore the impact of $c$-closure on the computational complexity of detecting and enumerating small induced subgraphs. More precisely, for each graph $H$ on three or four vertices, we investigate parameterized polynomial-time algorithms for detecting $H$ and for enumerating all occurrences of $H$ in a given $c$-closed graph.

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arXiv2020

Parameterized Complexity of Min-Power Asymmetric Connectivity

We investigate parameterized algorithms for the NP-hard problem Min-Power Asymmetric Connectivity (MinPAC) that has applications in wireless sensor networks. Given a directed arc-weighted graph, MinPAC asks for a strongly connected spanning subgraph minimizing the summed vertex costs. Here, the cost of each vertex is the weight of its heaviest outgoing arc in the chosen subgraph. We present linear-time algorithms for the cases where the number of strongly connected components in a so-called obligatory subgraph or the feedback edge number in the underlying undirected graph is constant. Complementing these results, we prove that the problem is W[2]-hard with respect to the solution cost, even on restricted graphs with one feedback arc and binary arc weights.

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arXiv2020

Parameterized Algorithms for Matrix Completion With Radius Constraints

Considering matrices with missing entries, we study NP-hard matrix completion problems where the resulting completed matrix shall have limited (local) radius. In the pure radius version, this means that the goal is to fill in the entries such that there exists a 'center string' which has Hamming distance to all matrix rows as small as possible. In stringology, this problem is also known as Closest String with Wildcards. In the local radius version, the requested center string must be one of the rows of the completed matrix. Hermelin and Rozenberg [CPM 2014, TCS 2016] performed parameterized complexity studies for Closest String with Wildcards. We answer one of their open questions, fix a bug concerning a fixed-parameter tractability result in their work, and improve some upper running time bounds. For the local radius case, we reveal a computational complexity dichotomy. In general, our results indicate that, although being NP-hard as well, this variant often allows for faster (fixed-parameter) algorithms.

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