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Register a new userParameterized Complexity of Safe Set
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Added by
Ioannis Katsikarelis
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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In this paper we study the problem of finding a small safe set $S$ in a graph $G$, i.e. a non-empty set of vertices such that no connected component of $G[S]$ is adjacent to a larger component in $G - S$. We enhance our understanding of the problem from the viewpoint of parameterized complexity by showing that (1) the problem is W[2]-hard when parameterized by the pathwidth $pw$ and cannot be solved in time $n^{o(pw)}$ unless the ETH is false, (2) it admits no polynomial kernel parameterized by the vertex cover number $vc$ unless $mathrm{PH} = Sigma^{mathrm{p}}_{3}$, but (3) it is fixed-parameter tractable (FPT) when parameterized by the neighborhood diversity $nd$, and (4) it can be solved in time $n^{f(cw)}$ for some double exponential function $f$ where $cw$ is the clique-width. We also present (5) a faster FPT algorithm when parameterized by solution size.
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Let $G$ be a graph on $n$ vertices and $mathrm{STAB}_k(G)$ be the convex hull of characteristic vectors of its independent sets of size at most $k$. We study extension complexity of $mathrm{STAB}_k(G)$ with respect to a fixed parameter $k$ (analogously to, e.g., parameterized computational complexity of problems). We show that for graphs $G$ from a class of bounded expansion it holds that $mathrm{xc}(mathrm{STAB}_k(G))leqslant mathcal{O}(f(k)cdot n)$ where the function $f$ depends only on the class. This result can be extended in a simple way to a wide range of similarly defined graph polytopes. In case of general graphs we show that there is {em no function $f$} such that, for all values of the parameter $k$ and for all graphs on $n$ vertices, the extension complexity of $mathrm{STAB}_k(G)$ is at most $f(k)cdot n^{mathcal{O}(1)}.$ While such results are not surprising since it is known that optimizing over $mathrm{STAB}_k(G)$ is $FPT$ for graphs of bounded expansion and $W[1]$-hard in general, they are also not trivial and in both cases stronger than the corresponding computational complexity results.
We study the NP-hard textsc{$k$-Sparsest Cut} problem ($k$SC) in which, given an undirected graph $G = (V, E)$ and a parameter $k$, the objective is to partition vertex set into $k$ subsets whose maximum edge expansion is minimized. Herein, the edge expansion of a subset $S subseteq V$ is defined as the sum of the weights of edges exiting $S$ divided by the number of vertices in $S$. Another problem that has been investigated is textsc{$k$-Small-Set Expansion} problem ($k$SSE), which aims to find a subset with minimum edge expansion with a restriction on the size of the subset. We extend previous studies on $k$SC and $k$SSE by inspecting their parameterized complexity. On the positive side, we present two FPT algorithms for both $k$SSE and 2SC problems where in the first algorithm we consider the parameter treewidth of the input graph and uses exponential space, and in the second we consider the parameter vertex cover number of the input graph and uses polynomial space. Moreover, we consider the unweighted version of the $k$SC problem where $k geq 2$ is fixed and proposed two FPT algorithms with parameters treewidth and vertex cover number of the input graph. We also propose a randomized FPT algorithm for $k$SSE when parameterized by $k$ and the maximum degree of the input graph combined. Its derandomization is done efficiently. oindent On the negative side, first we prove that for every fixed integer $k,taugeq 3$, the problem $k$SC is NP-hard for graphs with vertex cover number at most $tau$. We also show that $k$SC is W[1]-hard when parameterized by the treewidth of the input graph and the number~$k$ of components combined using a reduction from textsc{Unary Bin Packing}. Furthermore, we prove that $k$SC remains NP-hard for graphs with maximum degree three and also graphs with degeneracy two. Finally, we prove that the unweighted $k$SSE is W[1]-hard for the parameter $k$.
In $d$-Scattered Set we are given an (edge-weighted) graph and are asked to select at least $k$ vertices, so that the distance between any pair is at least $d$, thus generalizing Independent Set. We provide upper and lower bounds on the complexity of this problem with respect to various standard graph parameters. In particular, we show the following: - For any $dge2$, an $O^*(d^{textrm{tw}})$-time algorithm, where $textrm{tw}$ is the treewidth of the input graph. - A tight SETH-based lower bound matching this algorithms performance. These generalize known results for Independent Set. - $d$-Scattered Set is W[1]-hard parameterized by vertex cover (for edge-weighted graphs), or feedback vertex set (for unweighted graphs), even if $k$ is an additional parameter. - A single-exponential algorithm parameterized by vertex cover for unweighted graphs, complementing the above-mentioned hardness. - A $2^{O(textrm{td}^2)}$-time algorithm parameterized by tree-depth ($textrm{td}$), as well as a matching ETH-based lower bound, both for unweighted graphs. We complement these mostly negative results by providing an FPT approximation scheme parameterized by treewidth. In particular, we give an algorithm which, for any error parameter $epsilon > 0$, runs in time $O^*((textrm{tw}/epsilon)^{O(textrm{tw})})$ and returns a $d/(1+epsilon)$-scattered set of size $k$, if a $d$-scattered set of the same size exists.
In this paper, we consider the Target Set Selection problem: given a graph and a threshold value $thr(v)$ for any vertex $v$ of the graph, find a minimum size vertex-subset to activate s.t. all the vertices of the graph are activated at the end of the propagation process. A vertex $v$ is activated during the propagation process if at least $thr(v)$ of its neighbors are activated. This problem models several practical issues like faults in distributed networks or word-to-mouth recommendations in social networks. We show that for any functions $f$ and $rho$ this problem cannot be approximated within a factor of $rho(k)$ in $f(k) cdot n^{O(1)}$ time, unless FPT = W[P], even for restricted thresholds (namely constant and majority thresholds). We also study the cardinality constraint maximization and minimizati
A matching is a set of edges in a graph with no common endpoint. A matching $M$ is called acyclic if the induced subgraph on the endpoints of the edges in $M$ is acyclic. Given a graph $G$ and an integer $k$, Acyclic Matching Problem seeks for an acyclic matching of size $k$ in $G$. The problem is known to be NP-complete. In this paper, we investigate the complexity of the problem in different aspects. First, we prove that the problem remains NP-complete for the class of planar bipartite graphs with maximum degree three and girth of arbitrary large. Also, the problem remains NP-complete for the class of planar line graphs with maximum degree four. Moreover, we study the parameterized complexity of the problem. In particular, we prove that the problem is W[1]-hard on bipartite graphs with respect to the parameter $k$. On the other hand, the problem is fixed parameter tractable with respect to $k$, for line graphs, $C_4$-free graphs and every proper minor-closed class of graphs (including bounded tree-width and planar graphs).
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