We study the NP-complete Minimum Shared Edges (MSE) problem. Given an undirected graph, a source and a sink vertex, and two integers p and k, the question is whether there are p paths in the graph connecting the source with the sink and sharing at most k edges. Herein, an edge is shared if it appears in at least two paths. We show that MSE is W[1]-hard when parameterized by the treewidth of the input graph and the number k of shared edges combined. We show that MSE is fixed-parameter tractable with respect to p, but does not admit a polynomial-size kernel (unless NP is contained in coNP/poly). In the proof of the fixed-parameter tractability of MSE parameterized by p, we employ the treewidth reduction technique due to Marx, O'Sullivan, and Razgon [ACM TALG 2013].
In this paper, we consider the parameterized complexity of the following scheduling problem. We must schedule a number of jobs on $m$ machines, where each job has unit length, and the graph of precedence constraints consists of a set of chains. Each precedence constraint is labelled with an integer that denotes the exact (or minimum) delay between the jobs. We study different cases; delays can be given in unary and in binary, and the case that we have a single machine is discussed separately. We consider the complexity of this problem parameterized by the number of chains, and by the thickness…
On the Approximability of Parameterized Minimum Monotone Satisfying Assignment
Simons Institute for the Theory of Computing（计算理论西蒙斯研究所）；Departments of EECS & Mathematics, UC Berkeley（加州大学伯克利分校电气工程和计算机科学系及数学系）；State Key Laboratory of Novel Software Technology, Nanjing University（南京大学软件新技术国家重点实验室）
In this paper, we study the parameterized complexity of a generalized domination problem called the [$σ, ρ$] Dominating Set problem. This problem generalizes a large number of problems including the Minimum Dominating Set problem and its many variants. The parameterized complexity of the [$σ, ρ$] Dominating Set problem parameterized by treewidth is well studied. Here the properties of the sets $σ$ and $ρ$ that make the problem tractable are identified [1]. We consider a larger parameter and investigate the existence of polynomial sized kernels. When $σ$ and $ρ$ are finite, we identify the exac…
The Minimum Eccentricity Shortest Path Problem consists in finding a shortest path with minimum eccentricity in a given undirected graph. The problem is known to be NP-complete and W[2]-hard with respect to the desired eccentricity. We present fpt-algorithms for the problem parameterized by the modular width, distance to cluster graph, the combination of treewidth with the desired eccentricity, and maximum leaf number.
In the Binary Networked Public Goods game, every player needs to decide if she participates in a public project whose utility is shared equally by the community. We study the problem of deciding if there exists a pure strategy Nash equilibrium (PSNE) in such games. The problem is already known to be NP-complete. We provide fine-grained analysis of this problem under the lens of parameterized complexity theory. We consider various natural graph parameters and show either W[1]-hardness or exhibit an FPT algorithm. We finally exhibit some special graph classes, for example path, cycle, bi-clique,…
Deciding whether a diagram of a knot can be untangled with a given number of moves (as a part of the input) is known to be NP-complete. In this paper we determine the parameterized complexity of this problem with respect to a natural parameter called defect. Roughly speaking, it measures the efficiency of the moves used in the shortest untangling sequence of Reidemeister moves. We show that the II- moves in a shortest untangling sequence can be essentially performed greedily. Using that, we show that this problem belongs to W[P] when parameterized by the defect. We also show that this problem …
Given a simple, undirected graph $G$ with a threshold function $τ:V(G) \rightarrow \mathbb{N}$, the \textsc{Target Set Selection} (TSS) problem is about choosing a minimum cardinality set, say $S \subseteq V(G)$, such that starting a diffusion process with $S$ as its seed set will eventually result in activating all the nodes in $G$. For any non-negative integer $i$, we say a set $T\subseteq V(G)$ is a "degree-$i$ modulator" of $G$ if the degree of any vertex in the graph $G-T$ is at most $i$. Degree-$0$ modulators of a graph are precisely its vertex covers. Consider a graph $G$ on $n$ vertice…
We consider the problem of assigning agents to resources under the two-sided preference list model where resources specify an upper-quota and a lower-quota, that is, respectively the maximum and minimum number of agents that can be assigned to it. Different notions of optimality including envy-freeness and relaxed stability are investigated for this setting and the goal is to compute a largest optimal matching. Krishnaa et al. [SAGT 2020] show that in this setting, the problem of computing a maximum size envy-free matching (MAXEFM) or a maximum size relaxed stable matching (MAXRSM) is not appr…
A set $S$ of vertices of a graph is a defensive alliance if, for each element of $S$, the majority of its neighbours are in $S$. We study the parameterized complexity of the Defensive Alliance problem, where the aim is to find a minimum size defensive alliance. Our main results are the following: (1) The Defensive Alliance problem has been studied extensively during the last twenty years, but the question whether it is FPT when parameterized by feedback vertex set has still remained open. We prove that the problem is W[1]-hard parameterized by a wide range of fairly restrictive structural para…
Indian Institute of Science Education and Research（印度科学教育与研究所）
The Offensive Alliance problem has been studied extensively during the last twenty years. A set $S\subseteq V$ of vertices is an offensive alliance in an undirected graph $G=(V,E)$ if each $v\in N(S)$ has at least as many neighbours in $S$ as it has neighbours (including itself) not in $S$. We study the parameterized complexity of the Offensive Alliance problem, where the aim is to find a minimum size offensive alliance. Our focus here lies on parameters that measure the structural properties of the input instance. We enhance our understanding of the problem from the viewpoint of parameterized…
We study a variant of Min Cost Flow in which the flow needs to be connected. Specifically, in the Connected Flow problem one is given a directed graph $G$, along with a set of demand vertices $D \subseteq V(G)$ with demands $\mathsf{dem}: D \rightarrow \mathbb{N}$, and costs and capacities for each edge. The goal is to find a minimum cost flow that satisfies the demands, respects the capacities and induces a (strongly) connected subgraph. This generalizes previously studied problems like the (Many Visits) TSP. We study the parameterized complexity of Connected Flow parameterized by $|D|$, the …
