approximation algorithm

Given a graph $G=(V,E)$,  a  function  $f:V\to \{0,1,2,3,4\}$ is a triple Roman  dominating function (TRDF)  of $G$, for each vertex $v\in V$,  (i) if $f (v ) = 0 $, then  $v$ must have either one neighbour in $V_4$, or either two neighbours in $V_2 \cup  V_3$ (one neighbour in $V_3$) or either three neighbours in $V_2 $, (ii) if $f (v ) = 1 $, then $v$ must have either one neighbour in  $V_3 \cup  V_4$  or either two neighbours in $V_2 $, and if $f (v ) = 2 $, then $v$ must have one neighbour in $V_2 \cup  V_3\cup  V_4$. The triple Roman  domination number of $G$ is the  minimum weight of an TRDF  $f$  of $G$, where the weight of $f$ is $\sum_{v\in V}f(v)$.  The triple  Roman  domination problem is to compute the  triple Roman  domination number of a given graph.  In this paper, we study the triple  Roman  domination problem. We show that   the problem is NP-complete for  the  star convex bipartite  and the   comb convex bipartite graphs and is APX-complete for graphs of degree at~most~4. We propose a linear-time algorithm for computing  the triple Roman  domination number of proper interval graphs.  We also   give an $( 2 H(\Delta(G)+1) -1  )$-approximation algorithm  for solving the problem  for any graph $G$,  where   $  \Delta(G)$ is the maximum degree of $G$ and $H(d)$ denotes the first $d$ terms of the harmonic  series. In addition, we prove  that  for any $\varepsilon>0$  there is no  $(1/4-\varepsilon)\ln|V|$-approximation  polynomial-time   algorithm for solving  the problem on bipartite and split  graphs, unless NP $\subseteq$ DTIME $(|V|^{O(\log\log|V |)})$.

In this study, we examine biorthogonal wavelets that are tailored to a specific discrete pseudo-differential equation ‎‎‎of the form $T_{\sigma}u = f$, where $T_{\sigma}$ is an invertible discrete pseudo-differential operator defined on ‎‎‎the lattice $\mathbb{Z}^{n}$ for every $f\in\ell^{2}(\mathbb{Z}^{n})‎$‎. Our focus is on computing Galerkin approximations ‎‎‎of the solution to this problem using an adaptive algorithm.‎

The travelling salesman problem is one of the well-known NP-hard problems, and there are various versions of the problem with respect to its different specifications of the constraints and assumptions. Especially, the symmetric travelling salesman problem has been considered in numerous routing models. The critical node detection problem has received increasing attention throughout the routing models. The critical node has the most important role in the routing problems, and if it is out of service then the optimal solution will be hit by a large undesirable cost. The critical node is defined as the node whose deletion from the network results in the largest decrease in the optimal cost. It is proved the critical node of the network is the critical node for the optimal tour, too. Thus, the critical node is considered to obtain a good approximate solution in a reasonable iteration. The 2-opt heuristic is applied by the critical node in the symmetric traveling salesman problem and the iterations are reduced significantly. Then, the pseudo-critical node is defined and detected in the approximate solution, whose removal results in the largest decrease of the approximate cost. So, the 2-opt heuristic is applied by the pseudo-critical node and the optimal or a nearby optimal solution is obtained.

We consider a family of problems that combine network design and facility location. Such problems arise in many practical applications in different fields such as telecommunications, transportation networks, logistic, and energy supply networks. In facility location problems, we want to decide which facilities to open and how to assign clients to the open facilities so as to minimize the sum of the facility opening costs and client connection costs. These problems typically do not involve decisions concerning the routing of the clients’ demands to the open facilities; once we decided on the set of open facilities, each client is served by the closest open facility. In network design problems, on the other hand, we generally want to design and dimension a minimum-cost routing network providing sufficient capacities to route all clients’ demands to their destinations. These problems involve deciding on the routing of each client’s demand. But, in contrast to facility location problems, demands’ destinations are predetermined. In many modern day applications, however, all these decisions are interdependent and affect each other. Hence, they should be taken simultaneously. The aim of this work is to survey models, algorithmic approaches and methodologies concerning such combined network design facility location problems.

This paper considered the cost constrained vehicle scheduling problem under the constraint that the total number of vehicles is known in advance. Each depot has a different time processing cost. The goal of this problem is to find a feasible minimum cost schedule for vehicles. A mathematical formulation of the problem is developed and the complexity of the problem when there are more than two depots is investigated. It is proved that in this case, the problem is NP-complete. Also, it is showed that there is not any constant ratio approximation algorithm for the problem, i.e., it is in the complexity class APX.

By definition, web-services composition works on developing merely optimum coordination among a number of available web-services to provide a new composed web-service intended to satisfy some users requirements for which a single web service is not (good) enough. In this article, the formulation of the automatic web-services composition is proposed as several set-cover problems and an approximation algorithm has been exploited to solve them. In proposed method, the web-service composition has been carried out within two main phases, the top-down expansion of the composition tree, and the production of composed service by bottom-up traversal of composition tree. In the first phase, the production of a composition tree (similar to the production of tree in problemsolving by searching) is proposed by starting from the output or post-conditions of the requested service towards its input or pre-conditions. Each node or state of the tree is a set of inputs and/or outputs or conditions, and services as tree edges illustrate the transition from one node to another. In the second phase, finding the path from the leaves of the produced composition tree to the root is considered equal to reaching the output of requested service, and this path specifies the involved services and the composition plan. The requested service input set determines the available leaves of the composition tree. To achieve each non-leaf node of the tree, a set-cover problem is produced and solved using a greedy approximation algorithm. If the production and solving of the set-cover problems continues hierarchically until it reaches the root node, the composition plan and cost of the required composition service will be specified. The main focus of this research is the joint sequential and parallel composition with the aim of producing near-optimal and QoS-aware composed services.

In this paper, we study the directed multicut and directed multimultiway cut problems. The input to the directed multi-multiway cut problem is a weighted directed graph G = (V, E) and k sets S1, S2, ..., Sk of vertices. The goal is to find a subset of edges of minimum total weight whose removal will disconnect all the connections between the vertices in each set Si , for 1 ≤ i ≤ k. A special case of this problem is the directed multicut problem whose input consists of a weighted directed graph G = (V, E) and a set of ordered pairs of vertices (s1, t1), ...,(sk, tk). The goal is to find a subset of edges of minimum total weight whose removal will make for any i, 1 ≤ i ≤ k, there is no directed path from si to ti . In this paper, we present two approximation algorithms for these problems. The so called region growing paradigm is modified and used for these two cut problems on directed graphs. using this paradigm, we give an approximation algorithm for each problem such that both algorithms have the approximation factor of O(k) the same as the previous works done on these problems. However, the previous works need to solve k linear programming, whereas our algorithms require only one linear programming. Therefore, our algorithms improve the running time of the previous algorithms.