A greedy algorithm for the fault-tolerant connected dominating set in a general graph

Using a connected dominating set (CDS) to serve as the virtual backbone of a wireless network is an effective way to save energy and alleviate broadcasting storm. Since nodes may fail due to an accidental damage or energy depletion, it is desirable that the virtual backbone is fault tolerant. A node set \(C\) is an \(m\) -fold connected dominating set ( \(m\) -fold CDS) of graph \(G\) if every node in \(V(G)\setminus C\) has at least \(m\) neighbors in \(C\) and the subgraph of \(G\) induced by \(C\) is connected. In this paper, we will present a greedy algorithm to compute an \(m\) -fold CDS in a general graph, which has size at most \(2+\ln (\Delta +m-2)\) times that of a minimum \(m\) -fold CDS, where \(\Delta \) is the maximum degree of the graph. This result improves on the previous best known performance ratio of \(2H(\Delta +m-1)\) for this problem, where \(H(\cdot )\) is the Harmonic number.     

A simple greedy approximation algorithm for the minimum connected $$$$-Center problem

In this paper, we consider the connected \(k\)-Center (\(CkC\)) problem, which can be seen as the classic \(k\)-Center problem with the constraint of internal connectedness, i.e., two nodes in a cluster are required to be connected by an internal path in the same cluster. \(CkC\) was first introduced by Ge et al. (ACM Trans Knowl Discov Data 2:7, 2008), in which they showed the \(NP\)-completeness for this problem and claimed a polynomial time approximation algorithm for it. However, the running time of their algorithm might not be polynomial, as one key step of their algorithm involves the computation of an \(NP\)-hard problem. We first present a simple polynomial time greedy-based \(2\)-approximation algorithm for the relaxation of \(CkC\)—the \(CkC^*\). Further, we give a \(6\)-approximation algorithm for \(CkC\).     

A heuristic approximation algorithm of minimum dominating set based on rough set theory

The minimum dominating set of graph has been widely used in many fields, but its solution is NP-hard. The complexity and approximation accuracy of existing algorithms need to be improved. In this paper, we introduce rough set theory to solve the dominating set of undirected graph. First, the adjacency matrix of undirected graph is used to establish an induced decision table, and the minimum dominating set of undirected graph is equivalent to the minimum attribute reduction of its induced decision table. Second, based on rough set theory, the significance of attributes (i.e., vertices) based on the approximate quality is defined in induced decision table, and a heuristic approximation algorithm of minimum dominating set is designed by using the significance of attributes (i.e., vertices) as heuristic information. This algorithm uses forward and backward search mechanism, which not only ensures to find a minimal dominating set, but also improves the approximation accuracy of minimum dominating set. In addition, a cumulative strategy is used to calculate the positive region of induced decision table, which effectively reduces the computational complexity. Finally, the experimental results on public datasets show that our algorithm has obvious advantages in running time and approximation accuracy of the minimum dominating set.

     

Algorithms for the minimum weight k-fold (connected) dominating set problem

In this paper, we study the problem of computing a minimum weight k-fold dominating set (MWkDS) or a minimum weight k-fold connected dominating set (MWkCDS) in a unit ball graph (UBG). Using slab decomposition and dynamic programming, we give two exact algorithms for the computation of MWkDS and MWkCDS which can be executed in polynomial time if the thickness of the graph is bounded above.     

A PTAS for the minimum weighted dominating set problem with smooth weights on unit disk graphs

In the minimum weighted dominating set problem (MWDS), we are given a unit disk graph with non-negative weight on each vertex. The MWDS seeks a subset of the vertices of the graph with minimum total weight such that each vertex of the graph is either in the subset or adjacent to some nodes in the subset. A?weight function is called smooth, if the ratio of the weights of any two adjacent nodes is upper bounded by a constant. MWDS is known to be NP-hard. In this paper, we give the first polynomial time approximation scheme (PTAS) for MWDS with smooth weights on unit disk graphs, which achieves a (1+ε)-approximation for MWDS, for any ε>0.     

A complexity dichotomy and a new boundary class for the dominating set problem

We study the computational complexity of the dominating set problem for hereditary graph classes, i.e., classes of simple unlabeled graphs closed under deletion of vertices. Every hereditary class can be defined by a set of its forbidden induced subgraphs. There are numerous open cases for the complexity of the problem even for hereditary classes with small forbidden structures. We completely determine the complexity of the problem for classes defined by forbidding a five-vertex path and any set of fragments with at most five vertices. Additionally, we also prove polynomial-time solvability of the problem for some two classes of a similar type. The notion of a boundary class is a helpful tool for analyzing the computational complexity of graph problems in the family of hereditary classes. Three boundary classes were known for the dominating set problem prior to this paper. We present a new boundary class for it.     

On computing a minimum secure dominating set in block graphs

In a graph \(G=(V,E)\), a set \(D \subseteq V\) is said to be a dominating set of G if for every vertex \(u\in V{\setminus }D\), there exists a vertex \(v\in D\) such that \(uv\in E\). A secure dominating set of the graph G is a dominating set D of G such that for every \(u\in V{\setminus }D\), there exists a vertex \(v\in D\) such that \(uv\in E\) and \((D{\setminus }\{v\})\cup \{u\}\) is a dominating set of G. Given a graph G and a positive integer k, the secure domination problem is to decide whether G has a secure dominating set of cardinality at most k. The secure domination problem has been shown to be NP-complete for chordal graphs via split graphs and for bipartite graphs. In Liu et al. (in: Proceedings of 27th workshop on combinatorial mathematics and computation theory, 2010), it is asked to find a polynomial time algorithm for computing a minimum secure dominating set in a block graph. In this paper, we answer this by presenting a linear time algorithm to compute a minimum secure dominating set in block graphs. We then strengthen the known NP-completeness of the secure domination problem by showing that the secure domination problem is NP-complete for undirected path graphs and chordal bipartite graphs.     

A branch-and-bound algorithm for the minimum cut linear arrangement problem

Given an edge-weighted graph G of order n, the minimum cut linear arrangement problem (MCLAP) asks to find a one-to-one map from the vertices of G to integers from 1 to n such that the largest of the cut values c ₁,…,c _n?1 is minimized, where c _i , i∈{1,…,n?1}, is the total weight of the edges connecting vertices mapped to integers 1 through i with vertices mapped to integers i+1 through n. In this paper, we present a branch-and-bound algorithm for solving this problem. A salient feature of the algorithm is that it employs a dominance test which allows reducing the redundancy in the enumeration process drastically. The test is based on the use of a tabu search procedure developed to solve the MCLAP. We report computational results for both the unweighted and weighted graphs. In particular, we focus on calculating the cutwidth of some well-known graphs from the literature.     

New analysis and computational study for the planar connected dominating set problem

The connected dominating set (CDS) problem is a well studied NP-hard problem with many important applications. Dorn et al. (Algorithmica 58:790–810 2010) introduce a branch-decomposition based algorithm design technique for NP-hard problems in planar graphs and give an algorithm (DPBF algorithm) which solves the planar CDS problem in \(O(2^{9.822\sqrt{n}}n+n^3)\) time and \(O(2^{8.11\sqrt{n}}n+n^3)\) time, with a conventional method and fast matrix multiplication in the dynamic programming step of the algorithm, respectively. We show that DPBF algorithm solves the planar CDS problem in \(O(2^{9.8\sqrt{n}}n+n^3)\) time with a conventional method and in \(O(2^{8.08\sqrt{n}}n+n^3)\) time with a fast matrix multiplication. For a graph \(G\), let \({\hbox {bw}}(G)\) be the branchwidth of \(G\) and \(\gamma _c(G)\) be the connected dominating number of \(G\). We prove \({\hbox {bw}}(G)\le 2\sqrt{10\gamma _c(G)}+32\). From this result, the planar CDS problem admits an \(O(2^{23.54\sqrt{\gamma _c(G)}}\gamma _c(G)+n^3)\) time fixed-parameter algorithm. We report computational study results on the practical performance of DPBF algorithm, which show that the size of instances can be solved by the algorithm mainly depends on the branchwidth of the instances, coinciding with the theoretical analysis. For graphs with small or moderate branchwidth, the CDS problem instances with size up to a few thousands edges can be solved in a practical time and memory space.     

A primal-dual algorithm for the minimum power partial cover problem

Journal of Combinatorial Optimization - In this paper, we study the minimum power partial cover problem (MinPPC). Suppose X is a set of points and $${\mathcal {S}}$$ is a set of sensors on the...     

A simple approximation algorithm for minimum weight partial connected set cover

Let \(LTQ_n\) be the n-dimensional locally twisted cube. Hsieh and Tu (Theor Comput Sci 410(8–10):926–932, 2009) proposed an algorithm to construct n edge-disjoint spanning trees rooted at a particular vertex 0 in \(LTQ_n\). Later on, Lin et al. (Inf Process Lett 110(10):414–419, 2010) proved that Hsieh and Tu’s spanning trees are indeed independent spanning trees (ISTs for short), i.e., all spanning trees are rooted at the same vertex r and for any other vertex \(v(\ne r)\), the paths from v to r in any two trees are internally vertex-disjoint. Shortly afterwards, Liu et al. (Theor Comput Sci 412(22):2237–2252, 2011) pointed out that \(LTQ_n\) fails to be vertex-transitive for \(n\geqslant 4\) and proposed an algorithm for constructing n ISTs rooted at an arbitrary vertex in \(LTQ_n\). Although this algorithm can simultaneously construct n ISTs, it is hard to be parallelized for the construction of each spanning tree. In this paper, from a modification of Hsieh and Tu’s algorithm, we present a fully parallelized scheme to construct n ISTs rooted at an arbitrary vertex in \(LTQ_n\) in \({\mathcal O}(n)\) time using \(2^n\) vertices of \(LTQ_n\) as processors.     

An iterated greedy algorithm for the flowshop scheduling problem with blocking

This paper proposes an iterated greedy algorithm for solving the blocking flowshop scheduling problem for makespan minimization. Moreover, it presents an improved NEH-based heuristic, which is used as the initial solution procedure for the iterated greedy algorithm. The effectiveness of both procedures was tested on some of Taillard’s benchmark instances that are considered to be blocking flowshop instances. The experimental evaluation showed the efficiency of the proposed algorithm, in spite of its simple structure, in comparison with a state-of-the-art algorithm. In addition, new best solutions for Taillard’s instances are reported for this problem, which can be used as a basis of comparison in future studies.     

Approximation algorithms for minimum weight partial connected set cover problem

In the Minimum Weight Partial Connected Set Cover problem, we are given a finite ground set \(U\), an integer \(q\le |U|\), a collection \(\mathcal {E}\) of subsets of \(U\), and a connected graph \(G_{\mathcal {E}}\) on vertex set \(\mathcal {E}\), the goal is to find a minimum weight subcollection of \(\mathcal {E}\) which covers at least \(q\) elements of \(U\) and induces a connected subgraph in \(G_{\mathcal {E}}\). In this paper, we derive a “partial cover property” for the greedy solution of the Minimum Weight Set Cover problem, based on which we present (a) for the weighted version under the assumption that any pair of sets in \(\mathcal {E}\) with nonempty intersection are adjacent in \(G_{\mathcal {E}}\) (the Minimum Weight Partial Connected Vertex Cover problem falls into this range), an approximation algorithm with performance ratio \(\rho (1+H(\gamma ))+o(1)\), and (b) for the cardinality version under the assumption that any pair of sets in \(\mathcal {E}\) with nonempty intersection are at most \(d\)-hops away from each other (the Minimum Partial Connected \(k\)-Hop Dominating Set problem falls into this range), an approximation algorithm with performance ratio \(2(1+dH(\gamma ))+o(1)\), where \(\gamma =\max \{|X|:X\in \mathcal {E}\}\), \(H(\cdot )\) is the Harmonic number, and \(\rho \) is the performance ratio for the Minimum Quota Node-Weighted Steiner Tree problem.     

An effective iterated greedy algorithm for the mixed no-idle permutation flowshop scheduling problem

In the no-idle flowshop, machines cannot be idle after finishing one job and before starting the next one. Therefore, start times of jobs must be delayed to guarantee this constraint. In practice machines show this behavior as it might be technically unfeasible or uneconomical to stop a machine in between jobs. This has important ramifications in the modern industry including fiber glass processing, foundries, production of integrated circuits and the steel making industry, among others. However, to assume that all machines in the shop have this no-idle constraint is not realistic. To the best of our knowledge, this is the first paper to study the mixed no-idle extension where only some machines have the no-idle constraint. We present a mixed integer programming model for this new problem and the equations to calculate the makespan. We also propose a set of formulas to accelerate the calculation of insertions that is used both in heuristics as well as in the local search procedures. An effective iterated greedy (IG) algorithm is proposed. We use an NEH-based heuristic to construct a high quality initial solution. A local search using the proposed accelerations is employed to emphasize intensification and exploration in the IG. A new destruction and construction procedure is also shown. To evaluate the proposed algorithm, we present several adaptations of other well-known and recent metaheuristics for the problem and conduct a comprehensive set of computational and statistical experiments with a total of 1750 instances. The results show that the proposed IG algorithm outperforms existing methods in the no-idle and in the mixed no-idle scenarios by a significant margin.     

Parameterized dominating set problem in chordal graphs: complexity and lower bound

In this paper, we study the parameterized dominating set problem in chordal graphs. The goal of the problem is to determine whether a given chordal graph G=(V,E) contains a dominating set of size k or not, where k is an integer parameter. We show that the problem is W[1]-hard and it cannot be solved in time unless 3SAT can be solved in subexponential time. In addition, we show that the upper bound of this problem can be improved to when the underlying graph G is an interval graph.     

A practical greedy approximation for the directed Steiner tree problem

The directed Steiner tree (DST) NP-hard problem asks, considering a directed weighted graph with n nodes and m arcs, a node r called root and a set of k nodes X called terminals, for a minimum cost directed tree rooted at r spanning X. The best known polynomial approximation ratio for DST is a \(O(k^\varepsilon )\)-approximation greedy algorithm. However, a much faster k-approximation, returning the shortest paths from r to X, is generally used in practice. We give two new algorithms : a fast k-approximation called Greedy\(_\text {FLAC}\) running in \(O(m \log (n)k + \min (m, nk)nk^2)\) and a \(O(\sqrt{k})\)-approximation called Greedy\(_\text {FLAC}^\triangleright \) running in \(O(nm + n^2 \log (n)k +n^2 k^3)\). We provide computational results to show that, Greedy\(_\text {FLAC}\) rivals in practice with the running time of the fast k-approximation and returns solution with smaller cost in practice.     

A faster strongly polynomial time algorithm to solve the minimum cost tension problem

This paper presents a strongly polynomial time algorithm for the minimum cost tension problem, which runs in \(O(\max \{m^3n, m^2 \log n(m+n \log n)\})\) time, where n and m denote the number of nodes and number of arcs, respectively. Our algorithm improves upon the previous strongly polynomial time of \(O(n^4 m^3 \log n)\) due to Hadjiat and Maurras (Discret Math 165(166):377–394, 1997).