Mixed Dominating Set: A Parameterized Perspective
21 Jun 2017-pp 330-343
TL;DR: It is shown that unless the Set Cover Conjecture (SeCoCo) fails, mds does not admit an algorithm with running time \(\mathcal {O}((2-\epsilon )^{\mathsf{tw}(G))} n^{O}(1)})\) for any \(\epSilon >0\), where \(\mathsf(tw)(G)\) is the tree-width of G.
Abstract: In the mixed dominating set (mds) problem, we are given an n-vertex graph G and a positive integer k, and the objective is to decide whether there exists a set \(S \subseteq V(G) \cup E(G)\) of cardinality at most k such that every element \(x \in (V(G) \cup E(G)) \setminus S\) is either adjacent to or incident with an element of S. We show that mds can be solved in time \({7.465^k n^{\mathcal {O}(1)}} \) on general graphs, and in time \(2^{\mathcal {O}(\sqrt{k})} n^{\mathcal {O}(1)}\) on planar graphs. We complement this result by showing that mds does not admit an algorithm with running time \(2^{o(k)} n^{\mathcal {O}(1)}\) unless the Exponential Time Hypothesis (ETH) fails, and that it does not admit a polynomial kernel unless coNP \( \subseteq \mathsf{NP / poly}\). In addition, we provide an algorithm which, given a graph G together with a tree decomposition of width \(\mathsf{tw}\), solves mds in time \(6^{\mathsf{tw}} n^{\mathcal {O}(1)}\). We finally show that unless the Set Cover Conjecture (SeCoCo) fails, mds does not admit an algorithm with running time \(\mathcal {O}((2-\epsilon )^{\mathsf{tw}(G)} n^{\mathcal {O}(1)})\) for any \(\epsilon >0\), where \(\mathsf{tw}(G)\) is the tree-width of G.
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22 Nov 2006TL;DR: Koivisto et al. as discussed by the authors presented an O(2k n2 + n m) algorithm for the Steiner tree problem in graphs with n vertices, k terminals, and m edges with bounded integer weights.
Abstract: We present a fast algorithm for the subset convolution problem:given functions f and g defined on the lattice of subsets of ann-element set n, compute their subset convolution f*g, defined for S⊆ N by [ (f * g)(S) = [T ⊆ S] f(T) g(S/T),,]where addition and multiplication is carried out in an arbitrary ring. Via Mobius transform and inversion, our algorithm evaluates the subset convolution in O(n2 2n) additions and multiplications, substanti y improving upon the straightforward O(3n) algorithm. Specifically, if the input functions have aninteger range [-M,-M+1,...,M], their subset convolution over the ordinary sum--product ring can be computed in O(2n log M) time; the notation O suppresses polylogarithmic factors.Furthermore, using a standard embedding technique we can compute the subset convolution over the max--sum or min--sum semiring in O(2n M) time.To demonstrate the applicability of fast subset convolution, wepresent the first O(2k n2 + n m) algorithm for the Steiner tree problem in graphs with n vertices, k terminals, and m edges with bounded integer weights, improving upon the O(3kn + 2k n2 + n m) time bound of the classical Dreyfus-Wagner algorithm. We also discuss extensions to recent O(2n)-time algorithms for covering and partitioning problems (Bjorklund and Husfeldt, FOCS 2006; Koivisto, FOCS 2006).
280 citations
TL;DR: In this article, a fixed-parameter algorithm for the AMDS problem with bounded tree-width was proposed, achieving a fixedparameter complexity of O(3 2 2 2 + 2 2 -times 2 2 −2 2 2
Abstract: A mixed dominating set for a graph $G = (V,E)$ is a set $S\subseteq V \cup E$
such that every element $x \in (V \cup E) \backslash S$ is either adjacent or
incident to an element of $S$. The mixed domination number of a graph $G$,
denoted by $\gamma_m(G)$, is the minimum cardinality of mixed dominating sets
of $G$. Any mixed dominating set with the cardinality of $\gamma_m(G)$ is
called a minimum mixed dominating set. The mixed domination set (MDS) problem
is to find a minimum mixed dominating set for a graph $G$ and is known to be an
NP-complete problem. In this paper, we present a novel approach to find all of
the mixed dominating sets, called the AMDS problem, of a graph with bounded
tree-width $tw$. Our new technique of assigning power values to edges and
vertices, and combining with dynamic programming, leads to a fixed-parameter
algorithm of time $O(3^{tw^{2}}\times tw^2 \times |V|)$. This shows that MDS is
fixed-parameter tractable with respect to tree-width. In addition, we
theoretically improve the proposed algorithm to solve the MDS problem in
$O(6^{tw} \times |V|)$ time.
6 citations
TL;DR: A novel approach to find all of the mixed dominating sets of a graph with bounded tree-width, called the AMDS problem, is presented and it is shown that MDS is fixed-parameter tractable with respect to tree- width.
Abstract: A mixed dominating set for a graph $G = (V,E)$ is a set $S\subseteq V \cup E$ such that every element $x \in (V \cup E) \backslash S$ is either adjacent or incident to an element of $S$. The mixed domination number of a graph $G$, denoted by $\gamma_m(G)$, is the minimum cardinality of mixed dominating sets of $G$. Any mixed dominating set with the cardinality of $\gamma_m(G)$ is called a minimum mixed dominating set. The mixed domination set (MDS) problem is to find a minimum mixed dominating set for a graph $G$ and is known to be an NP-complete problem. In this paper, we present a novel approach to find all of the mixed dominating sets, called the AMDS problem, of a graph with bounded tree-width $tw$. Our new technique of assigning power values to edges and vertices, and combining with dynamic programming, leads to a fixed-parameter algorithm of time $O(3^{tw^{2}}\times tw^2 \times |V|)$. This shows that MDS is fixed-parameter tractable with respect to tree-width. In addition, we theoretically improve the proposed algorithm to solve the MDS problem in $O(6^{tw} \times |V|)$ time.
4 citations
Cites background from "Mixed Dominating Set: A Parameteriz..."
...t the MDS is fixed-parameter tractable with respect to tree-width. As defined later, the fundamental idea we use to solve the MDS problem is to assign power values to vertices. Recently, Jain et al. in Jain et al. (2017) enhanced the complexity(i) of Rajaati et al. (2016) to O (6w). Here they showed how to turn any set S V [Eto satisfy (i) the edges in Sform a matching, and (ii) the set of endpoints of edges in Sis ...
[...]
29 Jul 2019
TL;DR: In this article, a 2-approximation algorithm was proposed for the weighted version of the mixed dominating set problem, where all vertices and edges are assigned the same nonnegative weight.
Abstract: A mixed dominating set of a graph \(G = (V, E)\) is a mixed set D of vertices and edges, such that for every edge or vertex, if it is not in D, then it is adjacent or incident to at least one vertex or edge in D. The mixed domination problem is to find a mixed dominating set with a minimum cardinality. It has applications in system control and some other scenarios and it is NP-hard to compute an optimal solution. This paper studies approximation algorithms and hardness of the weighted mixed dominating set problem. The weighted version is a generalization of the unweighted version, where all vertices are assigned the same nonnegative weight \(w_v\) and all edges are assigned the same nonnegative weight \(w_e\), and the question is to find a mixed dominating set with a minimum total weight. Although the mixed dominating set problem has a simple 2-approximation algorithm, few approximation results for the weighted version are known. The main contributions of this paper include:
1.
for \(w_e\ge w_v\), a 2-approximation algorithm;
2.
for \(w_e\ge 2w_v\), inapproximability within ratio 1.3606 unless \(P=NP\) and within ratio 2 under UGC;
3.
for \(2w_v > w_e\ge w_v\), inapproximability within ratio 1.1803 unless \(P=NP\) and within ratio 1.5 under UGC;
4.
for \(w_e 0\).
2 citations
06 Aug 2019
TL;DR: A branch-and-search algorithm with running time bound of 4.172^k is given, which improves the previous bound of \(O^*(7.465^k)\), and the problem in planar graphs allows linear kernels by giving a kernel of 11k vertices.
Abstract: A mixed domination of a graph \(G = (V, E)\) is a mixed set D of vertices and edges such that for every edge or vertex, if it is not in D, then it is adjacent or incident to at least one vertex or edge in D. The Mixed Domination problem is to check whether there is a mixed domination of size at most k in a graph. Mixed domination is a mixture concept of vertex domination and edge domination, and the mixed domination problem has been studied from the view of approximation algorithms, parameterized algorithms, and so on. In this paper, we give a branch-and-search algorithm with running time bound of \(O^*(4.172^k)\), which improves the previous bound of \(O^*(7.465^k)\). For kernelization, it is known that the problem parameterized by k in general graphs is unlikely to have a polynomial kernel. We show the problem in planar graphs allows linear kernels by giving a kernel of 11k vertices.
1 citations
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TL;DR: The second edition of a quarterly column as discussed by the authors provides a continuing update to the list of problems (NP-complete and harder) presented by M. R. Garey and myself in our book "Computers and Intractability: A Guide to the Theory of NP-Completeness,” W. H. Freeman & Co., San Francisco, 1979.
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