Danil Sagunov — Page 2

arXiv2019

Lower Bounds for the Happy Coloring Problems

In this paper, we study the Maximum Happy Vertices and the Maximum Happy Edges problems (MHV and MHE for short). Very recently, the problems attracted a lot of attention and were studied in Agrawal '17, Aravind et al. '16, Choudhari and Reddy '18, Misra and Reddy '17. Main focus of our work is lower bounds on the computational complexity of these problems. Established lower bounds can be divided into the following groups: NP-hardness of the above guarantee parameterization, kernelization lower bounds (answering questions of Misra and Reddy '17), exponential lower bounds under the Set Cover Conjecture and the Exponential Time Hypothesis, and inapproximability results. Moreover, we present an $\mathcal{O}^*(\ell^k)$ randomized algorithm for MHV and an $\mathcal{O}^*(2^k)$ algorithm for MHE, where $\ell$ is the number of colors used and $k$ is the number of required happy vertices or edges. These algorithms cannot be improved to subexponential taking proved lower bounds into account.

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arXiv2018

Solving Target Set Selection with Bounded Thresholds Faster than $2^n$

In this paper we consider the Target Set Selection problem. The problem naturally arises in many fields like economy, sociology, medicine. In the Target Set Selection problem one is given a graph $G$ with a function $\operatorname{thr}: V(G) \to \mathbb{N} \cup \{0\}$ and integers $k, \ell$. The goal of the problem is to activate at most $k$ vertices initially so that at the end of the activation process there is at least $\ell$ activated vertices. The activation process occurs in the following way: (i) once activated, a vertex stays activated forever; (ii) vertex $v$ becomes activated if at least $\operatorname{thr}(v)$ of its neighbours are activated. The problem and its different special cases were extensively studied from approximation and parameterized points of view. For example, parameterizations by the following parameters were studied: treewidth, feedback vertex set, diameter, size of target set, vertex cover, cluster editing number and others. Despite the extensive study of the problem it is still unknown whether the problem can be solved in $\mathcal{O}^*((2-ε)^n)$ time for some $ε>0$. We partially answer this question by presenting several faster-than-trivial algorithms that work in cases of constant thresholds, constant dual thresholds or when the threshold value of each vertex is bounded by one-third of its degree. Also, we show that the problem parameterized by $\ell$ is W[1]-hard even when all thresholds are constant.

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