On connected domination in unit ball graphs

Given a simple undirected graph, the minimum connected dominating set problem is to find a minimum cardinality subset of vertices D inducing a connected subgraph such that each vertex outside D has at least one neighbor in D. Approximations of minimum connected dominating sets are often used to represent a virtual routing backbone in wireless networks. This paper first proposes a constant-ratio approximation algorithm for the minimum connected dominating set problem in unit ball graphs and then introduces and studies the edge-weighted bottleneck connected dominating set problem, which seeks a minimum edge weight in the graph such that the corresponding bottleneck subgraph has a connected dominating set of size k. In wireless network applications this problem can be used to determine an optimal transmission range for a network with a predefined size of the virtual backbone. We show that the problem is hard to approximate within a factor better than 2 in graphs whose edge weights satisfy the triangle inequality and provide a 3-approximation algorithm for such graphs. We also show that for fixed k the problem is polynomially solvable in unit disk and unit ball graphs.     

On connected dominating sets of restricted diameter

A connected dominating set (CDS) is commonly used to model a virtual backbone of a wireless network. To bound the distance that information must travel through the network, we explicitly restrict the diameter of a CDS to be no more than s leading to the concept of a dominating s-club. We prove that for any fixed positive integer s it is NP-complete to determine if a graph has a dominating s  -club, even when the graph has diameter s+1$s+1$. As a special case it is NP-complete to determine if a graph of diameter two has a dominating clique. We then propose a compact integer programming formulation for the related minimization problem, enhance the approach with variable fixing rules and valid inequalities, and present computational results.     

A linear algorithm for the domination number of a series-parallel graph

A vertex u in an undirected graph G = (V, E) is said to dominate all its adjacent vertices and itself. A subset D of V is a dominating set in G if every vertex in G is dominated by a vertex in D, and is a minimum dominating set in G if no other dominating set in G has fewer vertices than D. The domination number of G is the cardinality of a minimum dominating set in G.The problem of determining, for a given positive integer k and an undirected graph G, whether G has a dominating set D in G satisfying ¦D¦ ≤ k, is a well-known NP-complete problem. Cockayne have presented a linear time algorithm for finding a minimum dominating set in a tree. In this paper, we will present a linear time algorithm for finding a minimum dominating set in a series-parallel graph.     

An exact algorithm for minimum CDS with shortest path constraint in wireless networks

In this paper, we study a minimum connected dominating set problem (CDS) in wireless networks, which selects a minimum CDS with property that all intermediate nodes inside every pairwise shortest path should be included. Such a minimum CDS (we name this problem as SPCDS) is an important tache of some other algorithms for constructing a minimum CDS. We prove that finding such a minimum SPCDS can be achieved in polynomial time and design an exact algorithm with time complexity O(δ ² n), where δ is the maximum node degree in communication graph.     

Minimum 2-tuple dominating set of permutation graphs

For a fixed positive integer k, a k-tuple dominating set of a graph G=(V,E) is a subset D?V such that every vertex in V is dominated by at least k vertex in D. The k-tuple domination number γ _×k (G) is the minimum size of a k-tuple dominating set of G. The special case when k=1 is the usual domination. The case when k=2 was called double domination or 2-tuple domination. A 2-tuple dominating set D ₂ is said to be minimal if there does not exist any D′?D ₂ such that D′ is a 2-tuple dominating set of G. A 2-tuple dominating set D ₂, denoted by γ _×2(G), is said to be minimum, if it is minimal as well as it gives 2-tuple domination number. In this paper, we present an efficient algorithm to find a minimum 2-tuple dominating set on permutation graphs with n vertices which runs in O(n ²) time.     

Dominance of capacities by k-additive belief functions

In this paper we deal with the set of k-additive belief functions dominating a given capacity. We follow the line introduced by Chateauneuf and Jaffray for dominating probabilities and continued by Grabisch for general k-additive measures. First, we show that the conditions for the general k-additive case lead to a very wide class of functions and this makes that the properties obtained for probabilities are no longer valid. On the other hand, we show that these conditions cannot be improved. We solve this situation by imposing additional constraints on the dominating functions. Then, we consider the more restrictive case of k-additive belief functions. In this case, a similar result with stronger conditions is proved. Although better, this result is not completely satisfactory and, as before, the conditions cannot be strengthened. However, when the initial capacity is a belief function, we find a subfamily of the set of dominating k-additive belief functions from which it is possible to derive any other dominant k-additive belief function, and such that the conditions are even more restrictive, obtaining the natural extension of the result for probabilities. Finally, we apply these results in the fields of Social Welfare Theory and Decision Under Risk.     

The k-Dominating Graph

Given a graph G, the k-dominating graph of G, D _k (G), is defined to be the graph whose vertices correspond to the dominating sets of G that have cardinality at most k. Two vertices in D _k (G) are adjacent if and only if the corresponding dominating sets of G differ by either adding or deleting a single vertex. The graph D _k (G) aids in studying the reconfiguration problem for dominating sets. In particular, one dominating set can be reconfigured to another by a sequence of single vertex additions and deletions, such that the intermediate set of vertices at each step is a dominating set if and only if they are in the same connected component of D _k (G). In this paper we give conditions that ensure D _k (G) is connected.     

Fall colouring of bipartite graphs and cartesian products of graphs

The question of whether a graph can be partitioned into k independent dominating sets, which is the same as having a fallk-colouring, is considered. For k=3, it is shown that a graph G can be partitioned into three independent dominating sets if and only if the cartesian product G□K₂ can be partitioned into three independent dominating sets. The graph K₂ can be replaced by any graph H such that there is a mapping f:Q_n→H, where f is a type-II graph homomorphism.The cartesian product of two trees is considered, as well as the complexity of partitioning a bipartite graph into three independent dominating sets, which is shown to be NP-complete. For other values of k, iterated cartesian products are considered, leading to a result that shows for what values of k the hypercubes can be partitioned into k independent dominating sets.     

Algorithms for graphs with small octopus

A d-octopus of a graph G=(V,E) is a subgraph T=(W,F) of G such that W is a dominating set of G, and T is the union of d (not necessarily disjoint) shortest paths of G that have one endpoint in common. First, we study the complexity of finding and approximating a d-octopus of a graph. Then we show that for some NP-complete graph problems that are hard to approximate in general there are efficient approximation algorithms with worst case performance ratio c·d for some small constant c>0 (depending on the problem) assuming that the input graph G is given together with a d-octopus of G. For example, there is a linear time algorithm to approximate the bandwidth of a graph within a factor of 8d. Furthermore, the minimum number of subsets in a partition of the vertex set of a graph into clusters of diameter at most k can be approximated in linear time within a factor of 3d (for k=2) and 2d (for k⩾3). Finally, we show that there are O(n^7d+2) time algorithms to compute a minimum cardinality dominating set, respectively, total dominating set for graphs having a d-octopus.     

Optimizing Bernoulli Routing Policies for Balancing Loads on Call Centers and Minimizing Transmission Costs

We address the problem of assigning probabilities at discrete time instants for routing toll-free calls to a given set of call centers to minimize a weighted sum of transmission costs and load variability at the call centers during the next time interval.We model the problem as a tripartite graph and decompose the finding of an optimal probability assignment in the graph into the following problems: (i) estimating the true arrival rates at the nodes for the last time period; (ii) computing routing probabilities assuming that the estimates are correct. We use a simple approach for arrival rate estimation and solve the routing probability assignment by formulating it as a convex quadratic program and using the affine scaling algorithm to obtain an optimal solution.We further address a practical variant of the problem that involves changing routing probabilities associated with k nodes in the graph, where k is a prespecified number, to minimize the objective function. This involves deciding which k nodes to select for changing probabilities and determining the optimal value of the probabilities. We solve this problem using a heuristic that ranks all subsets of k nodes using gradient information around a given probability assignment.The routing model and the heuristic are evaluated for speed of computation of optimal probabilities and load balancing performance using a Monte Carlo simulation. Empirical results for load balancing are presented for a tripartite graph with 99 nodes and 17 call center gates.     

Constant memory routing in quasi-planar and quasi-polyhedral graphs

We address the problem of online route discovery for a class of graphs that can be embedded either in two or in three-dimensional space. In two dimensions we propose the class of quasi-planar graphs and in three dimensions the class of quasi-polyhedral graphs. In the former case such graphs are geometrically embedded in R² and have an underlying backbone that is planar with convex faces; however within each face arbitrary edges (with arbitrary crossings) are allowed. In the latter case, these graphs are geometrically embedded in R³ and consist of a backbone of convex polyhedra and arbitrary edges within each polyhedron. In both cases we provide a routing algorithm that guarantees delivery. Our algorithms need only “remember” the source and destination nodes and one (respectively, two) reference nodes used to store information about the underlying face (respectively, polyhedron) currently being traversed. The existence of the backbone is used only in proofs of correctness of the routing algorithm; the particular choice is irrelevant and does not affect the behaviour of the algorithm.     

Routing-efficient CDS construction in Disk-Containment Graphs

In wireless networks, Connected Dominating Sets (CDSs) are widely used as virtual backbones for communications. On one hand, reducing the backbone size will reduce the maintenance overhead. So how to minimize the CDS size is a critical issue. On the other hand, when evaluating the performance of a wireless network, the hop distance between two communication nodes, which reflect the energy consumption and response delay, is of great importance. Hence how to minimize the routing cost is also a key problem for constructing the network virtual backbone. In this paper, we study the problem of constructing applicable CDS in wireless networks in terms of size and routing cost. We formulate a wireless network as a Disk-Containment Graph (DCG), which is a generalization of the Unit-Disk Graph (UDG), and we develop an efficient algorithm to construct CDS in such kind of graphs. The algorithm contains two parts and is flexible to balance the performance between the two metrics. We also analyze the algorithm theoretically. It is shown that our algorithm has provable performance in minimizing the CDS size and reducing the communication distance for routing.     

Dominating sets and domatic number of circular arc graphs

A dominating set of a graph G = (N,E) is a subset S of nodes such that every node is either in S or adjacent to a node which is in S. The domatic number of G is the size of a maximum cardinality partition of N into dominating sets. The problems of finding a minimum cardinality dominating set and the domatic number are both NP-complete even for special classes of graphs. In the present paper we give an O(n∣E∣) time algorithm that finds a minimum cardinality dominating set when G is a circular arc graph (intersection graph of arcs on a circle). The domatic number problem is solved in O(n² log n) time when G is a proper circular arc graph, and it is shown NP-complete for general circular arc graphs.     

Rainbow domination on trees

This paper studies a variation of domination in graphs called rainbow domination. For a positive integer k, a k-rainbow dominating function of a graph G is a function f from V(G) to the set of all subsets of {1,2,…,k} such that for any vertex v with f(v)=0? we have ∪_u∈NG(v)f(u)={1,2,…,k}. The 1-rainbow domination is the same as the ordinary domination. The k-rainbow domination problem is to determine the k-rainbow domination number of a graph G, that is the minimum value of ∑_v∈V(G)|f(v)| where f runs over all k-rainbow dominating functions of G. In this paper, we prove that the k-rainbow domination problem is NP-complete even when restricted to chordal graphs or bipartite graphs. We then give a linear-time algorithm for the k-rainbow domination problem on trees. For a given tree T, we also determine the smallest k such that .     

The design of an efficient load balancing algorithm employing block design

In order to maintain load balancing in a distributed system, we should obtain workload information from all the nodes on network. This processing requiresO(v ²) communication overhead, wherev is the number of nodes. In this paper, we present a new synchronous dynamic distributed load balancing algorithm on a (v, k+1, 1)-configured network applying a symmetric balanced incomplete block design, wherev=k ²+k+1. Our algorithm needs only $O(v\sqrt v )$ communication overhead and each node receives workload information from all the nodes without redundancy. Therefore, load balancing is maintained since every link has the same amount of traffic for transferring workload information.     

Scheduling transmissions in a network

Consider k stations wishing to transmit messages over a network of channels to a common receiver. The capacity of a channel is the maximum amount of data which can be transmitted in a time unit. In addition to the transmission stations, the network contains switching nodes. Given that the jth station has σ_j messages (j = 1,…, k) to transmit, it is desired to find a schedule with minimum completion time T. The amount of data sent over a channel may vary in time. A schedule is stationary if the amount of data sent in a time unit is constant throughout the schedule. It is first shown that for every schedule there exists a stationary schedule with the same completion time. Thus, the search for an optimum schedule is confined to stationary schedules. The problem of finding an optimum stationary schedule is formulated as a multisource single-sink network flow problem, in which the net amount of outgoing flow from each source (transmission station) within one time unit is $\text{σ}{}_{\mathrm{j}}\text{T}$. An O(k|E||V|²) time algorithm to find the minimum T similar to Dinic's flow algorithm is suggested. Using Sleator and Tarjan's techniques an O(k²|E||V|log|V|) algorithm is derived. The running time of both algorithms is independent of the σ_j's and the capacities.     

Enumerating K best paths in length order in DAGs

We address the problem of finding the K best paths connecting a given pair of nodes in a directed acyclic graph (DAG) with arbitrary lengths. One of the main results in this paper is the proof that a tree representing the kth shortest path is obtained by an arc exchange in one of the previous (k − 1) trees (each of which contains a previous best path). An O(m + K(n + log K)) time and O(K + m) space algorithm is designed to explicitly determine the K shortest paths in a DAG with n nodes and m arcs. The algorithm runs in O(m + Kn) time using O(K + m) space in DAGs with integer length arcs. Empirical results confirming the superior performance of the algorithm to others found in the literature for randomly generated graphs are reported.     

Minimum monopoly in regular and tree graphs

In this paper we consider a graph optimization problem called minimum monopoly problem, in which it is required to find a minimum cardinality set S⊆V, such that, for each u∈V, |N[u]∩S|?|N[u]|/2 in a given graph G=(V,E). We show that this optimization problem does not have a polynomial-time approximation scheme for k-regular graphs (k?5), unless P=NP. We show this by establishing two L-reductions (an approximation preserving reduction) from minimum dominating set problem for k-regular graphs to minimum monopoly problem for 2k-regular graphs and to minimum monopoly problem for (2k-1)-regular graphs, where k?3. We also show that, for tree graphs, a minimum monopoly set can be computed in linear time.     

k-Abelian pattern matching

Two words are called k-abelian equivalent, if they share the same multiplicities for all factors of length at most k. We present an optimal linear time algorithm for identifying all occurrences of factors in a text that are k-abelian equivalent to some pattern P. Moreover, an optimal algorithm for finding the largest k for which two words are k-abelian equivalent is given. Solutions for online versions of the k-abelian pattern matching problem are also proposed.     

On approximation of max-vertex-cover

We consider the max-vertex-cover (MVC) problem, i.e., find k vertices from an undirected and edge-weighted graph G=(V,E), where |V|=n⩾k, such that the total edge weight covered by the k vertices is maximized. There is a 3/4-approximation algorithm for MVC, based on a linear programming relaxation. We show that the guaranteed ratio can be improved by a simple greedy algorithm for k>(3/4)n, and a simple randomized algorithm for k>(1/2)n. Furthermore, we study a semidefinite programming (SDP) relaxation based approximation algorithms for MVC. We show that, for a range of k, our SDP-based algorithm achieves the best performance guarantee among the four types of algorithms mentioned in this paper.