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In this paper, we study the Harmless Set problem from a parameterized complexity perspective. Given a graph G = (V , E) , a threshold function t : V → N and an integer k, we study Harmless Set, where the goal is to find a subset of vertices S ⊆ V of size at least k such that every vertex v ∈ V has fewer than t(v) neighbours in S. On the positive side, we obtain fixed-parameter algorithms for the problem when parameterized by the neighbourhood diversity, the twin-cover number and the vertex integrity of the input graph. We complement two of these results from the negative side. On dense graphs, we show that the problem is W[1]-hard parameterized by cluster vertex deletion number—a natural generalization of the twin-cover number. We show that the problem is W[1]-hard parameterized by a wide range of fairly restrictive structural parameters such as the feedback vertex set number, pathwidth, and treedepth—a natural generalization of the vertex integrity. We thereby resolve one open question stated by Bazgan and Chopin (Discrete Optim 14(C):170–182, 2014) concerning the complexity of Harmless Set parameterized by the treewidth of the input graph. We also show that Harmless Set for a special case where each vertex has the threshold set to half of its degree (the so-called Majority Harmless Set problem) is W[1]-hard when parameterized by the treewidth of the input graph. Given a graph G and an irredundant c-expression of G, we prove that Harmless Set can be solved in XP-time when parameterized by clique-width. [ABSTRACT FROM AUTHOR]

The Maximum Covering Location Problem (MCLP) is a well-studied problem in the field of operations research. Given a network with positive or negative demands on the nodes, a positive integer k, the MCLP seeks to find k potential facility centers in the network such that the neighborhood coverage is maximized. We study the variant of MCLP where edges of the network are subject to random failures due to some disruptive events. One of the popular models capturing the unreliable nature of the facility location is the linear reliability ordering (LRO) model. In this model, with every edge e of the network, we associate its survival probability 0 ≤ pe ≤ 1, or equivalently, its failure probability 1 − pe. The failure correlation in LRO is the following: If an edge e fails then every edge e ′ with p e ′ ≤ p e surely fails. The task is to identify the positions of k facilities that maximize the expected coverage. We refer to this problem as Expected Coverage problem. We study the Expected Coverage problem from the parameterized complexity perspective and obtain the following results. 1. For the parameter pathwidth, we show that the Expected Coverage problem is W[1]-hard. We find this result a bit surprising, because the variant of the problem with non-negative demands is fixed-parameter tractable (FPT) parameterized by the treewidth of the input graph. 2. We complement the lower bound by the proof that Expected Coverage is FPT being parameterized by the treewidth and the maximum vertex degree. We give an algorithm that solves the problem in time 2 O (tw log Δ) n O (1) , where tw is the treewidth, Δ is the maximum vertex degree, and n the number of vertices of the input graph. In particular, since Δ ≤ n, it means the problem is solvable in time n O (tw) , that is, is in XP parameterized by treewidth. [ABSTRACT FROM AUTHOR]

An input to a conflict-free variant of a classical problem Γ , called Conflict-Free Γ , consists of an instance I of Γ coupled with a graph H, called the conflict graph. A solution to Conflict-Free Γ in (I, H) is a solution to I in Γ , which is also an independent set in H. In this paper, we study conflict-free variants of Maximum Matching and Shortest Path, which we call Conflict-Free Maximum Matching (CF-MM) and Conflict-Free Shortest Path (CF-SP), respectively. We show that both CF-MM and CF-SP are W[1]-hard, when parameterized by the solution size. Moreover, W[1]-hardness for CF-MM holds even when the input graph where we want to find a matching is itself a matching, and W[1]-hardness for CF-SP holds for conflict graph being a unit-interval graph. Next, we study these problems with restriction on the conflict graphs. We give FPT algorithms for CF-MM when the conflict graph is chordal. Also, we give FPT algorithms for both CF-MM and CF-SP, when the conflict graph is d-degenerate. Finally, we design FPT algorithms for variants of CF-MM and CF-SP, where the conflicting conditions are given by a (representable) matroid. [ABSTRACT FROM AUTHOR]