Approximation hardness of min-max tree covers

We prove the first inapproximability bounds to study approximation hardness for a min-max k-tree cover problem and its variants. The problem is to find a set of k trees to cover vertices of a given graph with metric edge weights, so as to minimize the maximum total edge weight of any of the k trees. Our technique can also be applied to improve inapproximability bounds for min-max problems that use other covering objectives, such as stars, paths, and tours.     

A primal-dual method for approximating tree cover with two weights

The tree cover (TC) problem is to compute a minimum weight connected edge set, given a connected and edge-weighted graph G, such that its vertex set forms a vertex cover for G. Unlike related problems of vertex cover or edge dominating set, weighted TC is not yet known to be approximable in polynomial time as well as the unweighted version is. Moreover, the best approximation algorithm known so far for weighted TC is far from practical in its efficiency. In this paper we consider a restricted version of weighted TC, as a first step towards better approximation of general TC, where only two edge weights differing by at least a factor of 2 are available. It will be shown that a factor 2 approximation can be attained efficiently (in the complexity of max flow) in this case by a primal-dual method. Even under the limited weights as such, the primal-dual arguments used will be seen to be quite involved, having a nontrivial style of dual assignments as an essential part, unlike the case of uniform weights.     

Constructing a perfect matching is in random NC

We show that the problem of constructing a perfect matching in a graph is in the complexity class Random NC; i.e., the problem is solvable in polylog time by a randomized parallel algorithm using a polynomial-bounded number of processors. We also show that several related problems lie in Random NC. These include:

1. Constructing a perfect matching of maximum weight in a graph whose edge weights are given in unary notation;
2. Constructing a maximum-cardinality matching;
3. Constructing a matching covering a set of vertices of maximum weight in a graph whose vertex weights are given in binary;
4. Constructing a maximums-t flow in a directed graph whose edge weights are given in unary.

     

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.     

Interactive decision making for uncertain minimum spanning tree problems with total importance based on a risk-management approach

This paper deals with a minimum spanning tree problem where each edge cost includes uncertainty and importance measure. In risk management to avoid adverse impacts derived from uncertainty, a d-confidence interval for the total cost derived from robustness is introduced. Then, by maximizing the considerable region as well as minimizing the cost-importance ratio, a biobjective minimum spanning tree problem is proposed. Furthermore, in order to satisfy the objects of the decision maker and to solve the proposed model in mathematical programming, fuzzy goals for the objects are introduced as satisfaction functions, and an exact solution algorithm is developed using interactive decision making and deterministic equivalent transformations. Numerical examples are provided to compare our proposed model with some previous models.     

An iterative rounding 2-approximation algorithm for the k-partial vertex cover problem

We study a generalization of the vertex cover problem. For a given graph with weights on the vertices and an integer k, we aim to find a subset of the vertices with minimum total weight, so that at least k edges in the graph are covered. The problem is called the k-partial vertex cover problem. There are some 2-approximation algorithms for the problem. In the paper we do not improve on the approximation ratios of the previous algorithms, but we derive an iterative rounding algorithm. We present our technique in two algorithms. The first is an iterative rounding algorithm and gives a （2 ＋ Q/OPT ）-approximation for the k-partial vertex cover problem where Q is the largest finite weight in the problem definition and OPT is the optimal value for the instance. The second algorithm uses the first as a subroutine and achieves an approximation ratio of 2.     

Minimum edge ranking spanning trees of split graphs

Given a graph G, the minimum edge ranking spanning tree problem (MERST) is to find a spanning tree of G whose edge ranking is minimum. However, this problem is known to be NP-hard for general graphs. In this paper, we show that the problem MERST has a polynomial time algorithm for split graphs, which have useful applications in practice. The result is also significant in the sense that this is a first non-trivial graph class for which the problem MERST is found to be polynomially solvable. We also show that the problem MERST for threshold graphs can be solved in linear time, where threshold graphs are known to be split.     

2-Approximation algorithm for finding a clique with minimum weight of vertices and edges

The problem of finding a minimum clique (with respect to the total weight of its vertices and edges) of fixed size in a complete undirected weighted graph is considered along with some of its important subclasses. Approximability issues are analyzed. The inapproximability of the problem is proved for the general case. A 2-approximation efficient algorithm with time complexity O(n ²) is suggested for the cases when vertex weights are nonnegative and edge weights either satisfy the triangle inequality or are squared pairwise distances for some point configuration of Euclidean space.     

Approximation algorithms for the maximum 2-peripatetic salesman problem

Under study is the problem of finding two edge-disjoint Hamiltonian cycles (salesman routes) of maximal total weight in a complete undirected graph. For the case of edge weights from the interval [1, q], a polynomial algorithm is constructed with the guaranteed accuracy estimate \(\frac{{3q + 2}}{{4q + 1}}\). For the case of weights 1 and 2 and two different weight functions corresponding to the two routes, a polynomial algorithm with the accuracy estimate \(\frac{{11\rho - 8}}{{18\rho - 15}}\) is presented, where ρ is the accuracy estimate of an algorithm for solving a similar minimum optimization problem.     

The most vital nodes with respect to independent set and vertex cover

Given an undirected graph with weights on its vertices, the k most vital nodes independent set (k most vital nodes vertex cover) problem consists of determining a set of k vertices whose removal results in the greatest decrease in the maximum weight of independent sets (minimum weight of vertex covers, respectively). We also consider the complementary problems, minimum node blocker independent set (minimum node blocker vertex cover) that consists of removing a subset of vertices of minimum size such that the maximum weight of independent sets (minimum weight of vertex covers, respectively) in the remaining graph is at most a specified value. We show that these problems are NP-hard on bipartite graphs but polynomial-time solvable on unweighted bipartite graphs. Furthermore, these problems are polynomial also on cographs and graphs of bounded treewidth. Results on the non-existence of ptas are presented, too.     

Fuzzy facility location-allocation problem under the Hurwicz criterion

Facility location-allocation (FLA) problem has been widely studied by operational researchers due to its many practical applications. Many researchers have studied the FLA problem in a deterministic environment. However, the models they proposed cannot accommodate satisfactorily various customer demands in the real world. Thus, we consider the FLA problem with uncertainties. In this paper, a new model named α-cost model under the Hurwicz criterion is presented with fuzzy demands. In order to solve this model, the simplex algorithm, fuzzy simulations and a genetic algorithm are integrated to produce a hybrid intelligent algorithm. Finally, some numerical examples are presented to illustrate the effectiveness of the proposed algorithm.     

Asymptotically optimal approach to the approximate solution of several problems of covering a graph by nonadjacent cycles

We consider the m-Cycle Cover Problem of covering a complete undirected graph by m vertex-nonadjacent cycles of extremal total edge weight. The so-called TSP approach to the construction of an approximation algorithm for this problem with the use of a solution of the traveling salesman problem (TSP) is presented. Modifications of the algorithm for the Euclidean Max m-Cycle Cover Problem with deterministic instances (edge weights) in a multidimensional Euclidean space and the Random Min m-Cycle Cover Problem with random instances UNI(0,1) are analyzed. It is shown that both algorithms have time complexity O(n ³) and are asymptotically optimal for the number of covering cycles m = o(n) and \(m \leqslant \frac{{n^{1/3} }}{{\ln n}}\), respectively.     

Partial inverse maximum spanning tree in which weight can only be decreased under $$l_p$$-norm

The maximum or minimum spanning tree problem is a classical combinatorial optimization problem. In this paper, we consider the partial inverse maximum spanning tree problem in which the weight function can only be decreased. Given a graph, an acyclic edge set, and an edge weight function, the goal of this problem is to decrease weights as little as possible such that there exists with respect to function containing the given edge set. If the given edge set has at least two edges, we show that this problem is APX-Hard. If the given edge set contains only one edge, we present a polynomial time algorithm.     

The Proportional Colouring Problem: Optimizing Buffers in Wireless Mesh Networks

In this paper, we consider a new edge colouring problem motivated by wireless mesh networks optimization: the proportional edge colouring problem. Given a graph G with positive weights associated to its edges, we want to find a proper edge colouring which assigns to each edge at least a proportion (given by its weight) of all the colours. If such colouring exists, we want to find one using the minimum number of colours. We proved that deciding if a weighted graph admits a proportional edge colouring is polynomial while determining its proportional edge chromatic number is NP-hard. We also give a lower and an upper bound that can be polynomially computed. We finally characterize some graphs and weighted graphs for which we can determine the proportional edge chromatic number.     

Revisiting dynamic programming for finding optimal subtrees in trees

In this paper we revisit an existing dynamic programming algorithm for finding optimal subtrees in edge weighted trees. This algorithm was sketched by Maffioli in a technical report in 1991. First, we adapt this algorithm for the application to trees that can have both node and edge weights. Second, we extend the algorithm such that it does not only deliver the values of optimal trees, but also the trees themselves. Finally, we use our extended algorithm for developing heuristics for the k-cardinality tree problem in undirected graphs G with node and edge weights. This NP-hard problem consists of finding in the given graph a tree with exactly k edges such that the sum of the node and the edge weights is minimal. In order to show the usefulness of our heuristics we conduct an extensive computational analysis that concerns most of the existing problem instances. Our results show that with growing problem size the proposed heuristics reach the performance of state-of-the-art metaheuristics. Therefore, this study can be seen as a cautious note on the scaling of metaheuristics.     

On the approximability of the minimum strictly fundamental cycle basis problem

We consider the problem of finding a strictly fundamental cycle basis of minimum weight in the cycle space associated with an undirected connected graph G, where a nonnegative weight is assigned to each edge of G and the total weight of a basis is defined as the sum of the weights of all the cycles in the basis. Several heuristics have been proposed to tackle this NP-hard problem, which has some interesting applications. In this paper we show that this problem is APX-hard, even when restricted to unweighted graphs, and hence does not admit a polynomial-time approximation scheme, unless P=NP. Using a recent result on the approximability of lower-stretch spanning trees (Elkin et al. (2005) [7]), we obtain that the problem is approximable within O(log²nloglogn) for arbitrary graphs. We obtain tighter approximability bounds for dense graphs. In particular, the problem restricted to complete graphs admits a polynomial-time approximation scheme.     

一类最小支撑树的逆问题及其求解方法

本文针对传统的基于边的最小支撑树逆问题，提出了一类基于点边更新策略的最小支撑树逆问题．更新一个点是指减少与此点相关联的某些边的权值．根据是否含有更新点的费用，考虑了两类模型，它们均可转化为森林上的最小(费用)点覆盖的求解问题，算法的复杂性都是O(mn)，其中m=|E|n=|V|。     

Finding the K best policies in a finite-horizon Markov decision process

Directed hypergraphs represent a general modelling and algorithmic tool, which have been successfully used in many different research areas such as artificial intelligence, database systems, fuzzy systems, propositional logic and transportation networks. However, modelling Markov decision processes using directed hypergraphs has not yet been considered.In this paper we consider finite-horizon Markov decision processes (MDPs) with finite state and action space and present an algorithm for finding the K best deterministic Markov policies. That is, we are interested in ranking the first K deterministic Markov policies in non-decreasing order using an additive criterion of optimality. The algorithm uses a directed hypergraph to model the finite-horizon MDP. It is shown that the problem of finding the optimal policy can be formulated as a minimum weight hyperpath problem and be solved in linear time, with respect to the input data representing the MDP, using different additive optimality criteria.     

Matching interdiction

We introduce two interdiction problems involving matchings, one dealing with edge removals and the other dealing with vertex removals. Given is an undirected graph G with positive weights on its edges. In the edge interdiction problem, every edge of G has a positive cost and the task is to remove a subset of the edges constrained to a given budget, such that the weight of a maximum matching in the resulting graph is minimized. The vertex interdiction problem is analogous to the edge interdiction problem, with the difference that vertices instead of edges are removed. Hardness results are presented for both problems under various restrictions on the weights, interdiction costs and graph classes. Furthermore, we study the approximability of the edge and vertex interdiction problem on different graph classes. Several approximation-hardness results are presented as well as two constant-factor approximations, one of them based on iterative rounding. A pseudo-polynomial algorithm for solving the edge interdiction problem on graphs with bounded treewidth is proposed which can easily be adapted to the vertex interdiction problem. The algorithm presents a general framework to apply dynamic programming for solving a large class of problems in graphs with bounded treewidth. Additionally, we present a method to transform pseudo-polynomial algorithms for the edge interdiction problem into fully polynomial approximation schemes, using a scaling and rounding technique.     

The interval Branch-and-Prune algorithm for the discretizable molecular distance geometry problem with inexact distances

The Distance Geometry Problem in three dimensions consists in finding an embedding in ${\mathbb{R}^3}$ of a given nonnegatively weighted simple undirected graph such that edge weights are equal to the corresponding Euclidean distances in the embedding. This is a continuous search problem that can be discretized under some assumptions on the minimum degree of the vertices. In this paper we discuss the case where we consider the full-atom representation of the protein backbone and some of the edge weights are subject to uncertainty within a given nonnegative interval. We show that a discretization is still possible and propose the iBP algorithm to solve the problem. The approach is validated by some computational experiments on a set of artificially generated instances.