Given an undirected graph $G = (V, E)$ and a weight function $w:E \to \mathbb{R}$, the \textsc{Minimum Dominating Tree} problem asks to find a minimum weight sub-tree of $G$, $T = (U, F)$, such that every $v \in V \setminus U$ is adjacent to at least one vertex in $U$. The special case when the weight function is uniform is known as the \textsc{Minimum Connected Dominating Set} problem.
Given an undirected graph $G = (V, E)$ with some subsets of vertices called groups, and a weight function $w:E \to \mathbb{R}$, the \textsc{Group Steiner Tree} problem is to find a minimum weight sub-tree of $G$ which contains at least one vertex from each group.
In this paper we show that the two problems are equivalents from approximability perspective. This improves upon both the best known approximation algorithm and the best inapproximability result for the \textsc{Minimum Dominating Tree} problem. We also consider two extrema variants of the \textsc{Minimum Dominating Tree} problem, namely, the \textsc{Minimum Dominating Star} and the \textsc{Minimum Dominating Path} problems which ask to find a minimum dominating star and path respectively.
We prove that there is no fpt-algorithm that can approximate the dominating set problem with any constant ratio, unless FPT= W[1]. Our hardness reduction is built on the second author's recent W[1]-hardness proof of the biclique problem. This yields, among other things, a proof without the PCP machinery that the classical dominating set problem has no polynomial time constant approximation under the exponential time hypothesis.
A set $D \subseteq V$ of a graph $G = (V,E)$ is called an outer-connected dominating set of $G$ if every vertex $v$ not in $D$ is adjacent to at least one vertex in $D$, and the induced subgraph of $G$ on $V \setminus D$ is connected. The Minimum Outer-connected Domination problem is to find an outer-connected dominating set of minimum cardinality for the input graph $G$. Given a positive integer $k$ and a graph $G = (V, E)$, the Outer-connected Domination Decision problem is to decide whether $G$ has an outer-connected dominating set of cardinality at most $k$. The Outer-connected Domination …
A semitotal dominating set of a graph $G$ with no isolated vertex is a dominating set $D$ of $G$ such that every vertex in $D$ is within distance two of another vertex in $D$. The minimum size $γ_{t2}(G)$ of a semitotal dominating set of $G$ is squeezed between the domination number $γ(G)$ and the total domination number $γ_{t}(G)$. \textsc{Semitotal Dominating Set} is the problem of finding, given a graph $G$, a semitotal dominating set of $G$ of size $γ_{t2}(G)$. In this paper, we continue the systematic study on the computational complexity of this problem when restricted to special graph c…
Given a graph $G=(V,E)$, the dominating set problem asks for a minimum subset of vertices $D\subseteq V$ such that every vertex $u\in V\setminus D$ is adjacent to at least one vertex $v\in D$. That is, the set $D$ satisfies the condition that $|N[v]\cap D|\geq 1$ for each $v\in V$, where $N[v]$ is the closed neighborhood of $v$. In this paper, we study two variants of the classical dominating set problem: $\boldmath{k}$-tuple dominating set ($k$-DS) problem and Liar's dominating set (LDS) problem, and obtain several algorithmic and hardness results. On the algorithmic side, we present a consta…
The power dominating set (PDS) problem is the following extension of the well-known dominating set problem: find a smallest-size set of nodes $S$ that power dominates all the nodes, where a node $v$ is power dominated if (1) $v$ is in $S$ or $v$ has a neighbor in $S$, or (2) $v$ has a neighbor $w$ such that $w$ and all of its neighbors except $v$ are power dominated. We show a hardness of approximation threshold of $2^{\log^{1-ε}{n}}$ in contrast to the logarithmic hardness for the dominating set problem. We give an $O(\sqrt{n})$ approximation algorithm for planar graphs, and show that our met…
In this paper, we showcase the class XNLP as a natural place for many hard problems parameterized by linear width measures. This strengthens existing $W[1]$-hardness proofs for these problems, since XNLP-hardness implies $W[t]$-hardness for all $t$. It also indicates, via a conjecture by Pilipczuk and Wrochna [ToCT 2018], that any XP algorithm for such problems is likely to require XP space. In particular, we show XNLP-completeness for natural problems parameterized by pathwidth, linear clique-width, and linear mim-width. The problems we consider are Independent Set, Dominating Set, Odd Cycle …
In this paper, the author puts forward a variation of Feige's Hypothesis, which claims that it is hard on average refuting Unbalanced Max 3-XOR under biased assignments on a natural distribution. Under this hypothesis, the author strengthens the previous known hardness for approximating Minimum Unique Game, $5/4-ε$, by proving that Min 2-Lin-2 is hard to within $3/2-ε$ and strengthens the previous known hardness for approximating Small Set Expansion, $4/3-ε$, by proving that Min Bisection is hard to approximate within $3-ε$. In addition, the author discusses the limitation of this method to sh…
We introduce a hierarchy of problems between the \textsc{Dominating Set} problem and the \textsc{Power Dominating Set} (PDS) problem called the $\ell$-round power dominating set ($\ell$-round PDS, for short) problem. For $\ell=1$, this is the \textsc{Dominating Set} problem, and for $\ell\geq n-1$, this is the PDS problem; here $n$ denotes the number of nodes in the input graph. In PDS the goal is to find a minimum size set of nodes $S$ that power dominates all the nodes, where a node $v$ is power dominated if (1) $v$ is in $S$ or it has a neighbor in $S$, or (2) $v$ has a neighbor $u$ such th…
The concept of generalized domination unifies well-known variants of domination-like and independence problems, such as Dominating Set, Independent Set, Perfect Code, etc. A generalized domination (also called $[σ,ρ]$-Dominating Set}) problem consists in finding a subset of vertices in a graph such that every vertex is satisfied with respect to two given sets of constraints $σ$ and $ρ$. Very few problems are known not to be FPT when parameterized by tree-width, as usually this restriction allows one to write efficient algorithms to solve the considered problems. The main result of this article…
The NP-hard Metric Dimension problem is to decide for a given graph G and a positive integer k whether there is a vertex subset of size at most k that separates all vertex pairs in G. Herein, a vertex v separates a pair {u,w} if the distance (length of a shortest path) between v and u is different from the distance of v and w. We give a polynomial-time computable reduction from the Bipartite Dominating Set problem to Metric Dimension on maximum degree three graphs such that there is a one-to-one correspondence between the solution sets of both problems. There are two main consequences of this:…
We study the parameterized complexity of approximating the $k$-Dominating Set (DomSet) problem where an integer $k$ and a graph $G$ on $n$ vertices are given as input, and the goal is to find a dominating set of size at most $F(k) \cdot k$ whenever the graph $G$ has a dominating set of size $k$. When such an algorithm runs in time $T(k) \cdot poly(n)$ (i.e., FPT-time) for some computable function $T$, it is said to be an $F(k)$-FPT-approximation algorithm for $k$-DomSet. We prove the following for every computable functions $T, F$ and every constant $\varepsilon > 0$: $\bullet$ Assuming $W[1]\…
For a simple graph $G=(V,E)$ without any isolated vertex, a cosecure dominating set $D$ of $G$ satisfies the following two properties (i) $S$ is a dominating set of $G$, (ii) for every vertex $v \in S$ there exists a vertex $u \in V \setminus S$ such that $uv \in E$ and $(S \setminus \{v\}) \cup \{u\}$ is a dominating set of $G$. The minimum cardinality of a cosecure dominating set of $G$ is called cosecure domination number of $G$ and is denoted by $γ_{cs}(G)$. The Minimum Cosecure Domination problem is to find a cosecure dominating set of a graph $G$ of cardinality $γ_{cs}(G)$. The decision …
In this article, we study a generalized version of the maximum independent set and minimum dominating set problems, namely, the maximum $d$-distance independent set problem and the minimum $d$-distance dominating set problem on unit disk graphs for a positive integer $d>0$. We first show that the maximum $d$-distance independent set problem and the minimum $d$-distance dominating set problem belongs to NP-hard class. Next, we propose a simple polynomial-time constant-factor approximation algorithms and PTAS for both the problems.
The minimum dominating set problem has wide applications in network science and related fields. It consists of assembling a node set of global minimum size such that any node of the network is either in this set or is adjacent to at least one node of this set. Although this is a difficult optimization problem in general, we show it can be exactly solved by a generalized leaf-removal process if the network contains no core. If the network has an extensive core, we estimate the size of minimum dominating sets by a mean-field theory and implement a belief-propagation algorithm to obtain near-opti…
This paper studies the computational complexity of the Edge Packing problem and the Vertex Packing problem. The edge packing problem (denoted by $\bar{EDS}$) and the vertex packing problem (denoted by $\bar{DS} $) are linear programming duals of the edge dominating set problem and the dominating set problem respectively. It is shown that these two problems are equivalent to the set packing problem with respect to hardness of approximation and parametric complexity. It follows that $\bar{EDS}$ and $\bar{DS}$ cannot be approximated asymptotically within a factor of $O(N^{1/2-ε})$ for any $ε>0$ u…
We first devise a branching algorithm that computes a minimum independent dominating set on any graph with running time O*(2^0.424n) and polynomial space. This improves the O*(2^0.441n) result by (S. Gaspers and M. Liedloff, A branch-and-reduce algorithm for finding a minimum independent dominating set in graphs, Proc. WG'06). We then show that, for every r>3, it is possible to compute an r-((r-1)/r)log_2(r)-approximate solution for min independent dominating set within time O*(2^(nlog_2(r)/r)).
We study the parameterized complexity of dominating sets in geometric intersection graphs. In one dimension, we investigate intersection graphs induced by translates of a fixed pattern Q that consists of a finite number of intervals and a finite number of isolated points. We prove that Dominating Set on such intersection graphs is polynomially solvable whenever Q contains at least one interval, and whenever Q contains no intervals and for any two point pairs in Q the distance ratio is rational. The remaining case where Q contains no intervals but does contain an irrational distance ratio is sh…
In the Minimum Installation Path problem, we are given a graph $G$ with edge weights $w(.)$ and two vertices $s,t$ of $G$. We want to assign a non-negative power $p(v)$ to each vertex $v$ of $G$ so that the edges $uv$ such that $p(u)+p(v)$ is at least $w(uv)$ contain some $s$-$t$-path, and minimize the sum of assigned powers. In the Minimum Barrier Shrinkage problem, we are given a family of disks in the plane and two points $x$ and $y$ lying outside the disks. The task is to shrink the disks, each one possibly by a different amount, so that we can draw an $x$-$y$ curve that is disjoint from t…
We study the hardness of approximation of clause minimum and literal minimum representations of pure Horn functions in $n$ Boolean variables. We show that unless P=NP, it is not possible to approximate in polynomial time the minimum number of clauses and the minimum number of literals of pure Horn CNF representations to within a factor of $2^{\log^{1-o(1)} n}$. This is the case even when the inputs are restricted to pure Horn 3-CNFs with $O(n^{1+\varepsilon})$ clauses, for some small positive constant $\varepsilon$. Furthermore, we show that even allowing sub-exponential time computation, it i…
In this paper we give a randomized reduction for the Rank Syndrome Decoding problem and Rank Minimum Distance problem for rank codes. Our results are based on an embedding from linear codes equipped with Hamming distance unto linear codes over an extension field equipped with the rank metric. We prove that if both previous problems for rank metric are in ZPP = RP$\cap$coRP, then we would have NP=ZPP. We also give complexity results for the respective approximation problems in rank metric.
In this paper we present further studies of recurrent configurations of Chip-firing games on Eulerian directed graphs (simple digraphs), a class on the way from undirected graphs to general directed graphs. A computational problem that arises naturally from this model is to find the minimum number of chips of a recurrent configuration, which we call the minimum recurrent configuration (MINREC) problem. We point out a close relationship between MINREC and the minimum feedback arc set (MINFAS) problem on Eulerian directed graphs, and prove that both problems are NP-hard.
It is shown that in star-free graphs the maximum independent set problem, the minimum dominating set problem and the minimum independent dominating set problem are approximable up to constant factor by any maximal independent set.
Let $G=(V, E)$ be a simple and undirected graph. For some integer $k\geq 1$, a set $D\subseteq V$ is said to be a k-dominating set in $G$ if every vertex $v$ of $G$ outside $D$ has at least $k$ neighbors in $D$. Furthermore, for some real number $α$ with $0<α\leq1$, a set $D\subseteq V$ is called an $α$-dominating set in $G$ if every vertex $v$ of $G$ outside $D$ has at least $α\times d_v$ neighbors in $D$, where $d_v$ is the degree of $v$ in $G$. The cardinality of a minimum $k$-dominating set and a minimum $α$-dominating set in $G$ is said to be the $k$-domination number and the $α$-dominati…
The shortest common superstring (SCS) problem has been studied at great length because of its connections to the de novo assembly problem in computational genomics. The base problem is APX-complete, but several generalizations of the problem have also been studied. In particular, previous results include that SCS with Negative strings (SCSN) is in Log-APX (though there is no known hardness result) and SCS with Wildcards (SCSW) is Poly-APX-hard. Here, we prove two new hardness results: (1) SCSN is Log-APX-hard (and therefore Log-APX-complete) by a reduction from Minimum Set Cover and (2) SCS wi…
Let $G = (V,E)$ be a simple, undirected and connected graph. A connected dominating set $S \subseteq V$ is a secure connected dominating set of $G$, if for each $ u \in V\setminus S$, there exists $v\in S$ such that $(u,v) \in E$ and the set $(S \setminus \{ v \}) \cup \{ u \} $ is a connected dominating set of $G$. The minimum size of a secure connected dominating set of $G$ denoted by $ γ_{sc} (G)$, is called the secure connected domination number of $G$. Given a graph $ G$ and a positive integer $ k,$ the Secure Connected Domination (SCDM) problem is to check whether $ G $ has a secure conn…
Unit disk graphs are the set of graphs which represent the intersection of disk graphs and interval graphs. These graphs are of great importance due to their structural similarity with wireless communication networks. Firefighter problem on unit disk graph is interesting as it models the virus spreading in an wireless network and asks for a solution to stop it. In this paper, we consider the MIN-BUDGET firefighter problem where the goal is to determine the minimum number of firefighters required and the nodes to place them at each time instant to save a given set of vertices of a given graph a…
We show that the problem k-Dominating Set and its several variants including k-Connected Dominating Set, k-Independent Dominating Set, and k-Dominating Clique, when parameterized by the solution size k, are W[1]-hard in either multiple-interval graphs or their complements or both. On the other hand, we show that these problems belong to W[1] when restricted to multiple-interval graphs and their complements. This answers an open question of Fellows et al. In sharp contrast, we show that d-Distance k-Dominating Set for d >= 2 is W[2]-complete in multiple-interval graphs and their complements. We…
We give logarithmic lower bounds for the approximability of the Minimum Dominating Set problem in connected (alpha,beta)-Power Law Graphs. We give also a best up to now upper approximation bound on the problem for the case of the parameters beta>2. We develop also a new functional method for proving lower approximation bounds and display a sharp phase transition between approximability and inapproximability of the underlying problem. This method could also be of independent interest.
