The Center Location Improvement Problem Under the Hamming Distance

In this paper, we consider the center location improvement problems under the sum-type and bottleneck-type Hamming distance. For the sum-type problem, we show that achieving an algorithm with a worst-case ratio of O(log |V|) is NP-hard, and for the bottleneck-type problem, we present a strongly polynomial algorithm.     

Critical edges/nodes for the minimum spanning tree problem: complexity and approximation

In this paper, we study the complexity and the approximation of the k most vital edges (nodes) and min edge (node) blocker versions for the minimum spanning tree problem (MST). We show that the k most vital edges MST problem is NP-hard even for complete graphs with weights 0 or 1 and 3-approximable for graphs with weights 0 or 1. We also prove that the k most vital nodes MST problem is not approximable within a factor n ^1?? , for any ?>0, unless NP=ZPP, even for complete graphs of order n with weights 0 or 1. Furthermore, we show that the min edge blocker MST problem is NP-hard even for complete graphs with weights 0 or 1 and that the min node blocker MST problem is NP-hard to approximate within a factor 1.36 even for graphs with weights 0 or 1.     

Just-in-time scheduling with controllable processing times on parallel machines

We study scheduling problems with controllable processing times on parallel machines. Our objectives are to maximize the weighted number of jobs that are completed exactly at their due date and to minimize the total resource allocation cost. We consider four different models for treating the two criteria. We prove that three of these problems are NP\mathcal{NP} -hard even on a single machine, but somewhat surprisingly, the problem of maximizing an integrated objective function can be solved in polynomial time even for the general case of a fixed number of unrelated parallel machines. For the three NP\mathcal{NP} -hard versions of the problem, with a fixed number of machines and a discrete resource type, we provide a pseudo-polynomial time optimization algorithm, which is converted to a fully polynomial time approximation scheme.     

Probabilistic graph-coloring in bipartite and split graphs

We revisit in this paper the stochastic model for minimum graph-coloring introduced in (Murat and Paschos in Discrete Appl. Math. 154:564–586, 2006), and study the underlying combinatorial optimization problem (called probabilistic coloring) in bipartite and split graphs. We show that the obvious 2-coloring of any connected bipartite graph achieves standard-approximation ratio 2, that when vertex-probabilities are constant probabilistic coloring is polynomial and, finally, we propose a polynomial algorithm achieving standard-approximation ratio 8/7. We also handle the case of split graphs. We show that probabilistic coloring is NP-hard, even under identical vertex-probabilities, that it is approximable by a polynomial time standard-approximation schema but existence of a fully a polynomial time standard-approximation schema is impossible, even for identical vertex-probabilities, unless P=NP. We finally study differential-approximation of probabilistic coloring in both bipartite and split graphs. Part of this research has been performed while the second author was with the LAMSADE on a research position funded by the CNRS.     

Perfect Circular Arc Coloring

The circular arc coloring problem is to find a minimum coloring of a set of arcs of a circle so that no two overlapping arcs share a color. This NP-hard problem arises in a rich variety of applications and has been studied extensively. In this paper we present an O(n² m) combinatorial algorithm for optimally coloring any set of arcs that corresponds to a perfect graph, and propose a new approach to the general circular arc coloring problem.Partially supported by Project 02139 of Education Ministry of China.Supported in part by the Research Grants Council of Hong Kong (Project No. HKU7054/03P) and a seed funding for basic research of HKU.     

A “maximum node clustering” problem

In this note we introduce a graph problem, called Maximum Node Clustering (MNC). We prove that the problem (which is easily shown to be strongly NP-complete) can be approximated in polynomial time within a ratio arbitrarily close to 2. For the special case where the graph is a tree, the problem is NP-complete in the ordinary sense; for this case we present a pseudopolynomial algorithm based on dynamic programming, and a related Fully Polynomial Time Approximation Scheme (FPTAS). Also, the tree case is shown to be exactly solvable in time, where n is the number of nodes.     

Optimal algorithms for uncovering synteny problem

The syntenic distance between two genomes is the minimum number of fusions, fissions, and translocations that can transform one genome to the other, ignoring the gene order within chromosomes. As the problem is NP-hard in general, some particular classes of synteny instances, such as linear synteny, exact synteny and nested synteny, are examined in the literature. In this paper, we propose a new special class of synteny instances, called uncovering synteny. We first present a polynomial time algorithm to solve the connected case of uncovering synteny optimally. By performing only intra-component moves, we then solve the unconnected case of uncovering synteny. We will further calculate the diameters of connected and unconnected uncovering synteny, respectively.     

Fixed-parameter tractability of anonymizing data by suppressing entries

A popular model for protecting privacy when person-specific data is released is k -anonymity. A dataset is k-anonymous if each record is identical to at least (k−1) other records in the dataset. The basic k-anonymization problem, which minimizes the number of dataset entries that must be suppressed to achieve k-anonymity, is NP-hard and hence not solvable both quickly and optimally in general. We apply parameterized complexity analysis to explore algorithmic options for restricted versions of this problem that occur in practice. We present the first fixed-parameter algorithms for this problem and identify key techniques that can be applied to this and other k-anonymization problems.     

The complexity of influence maximization problem in the deterministic linear threshold model

The influence maximization is an important problem in the field of social network. Informally it is to select few people to be activated in a social network such that their aggregated influence can make as many as possible people active. Kempe et al. gave a $(1-{1 \over e})$ -approximation algorithm for this problem in the linear threshold model and the independent cascade model. In addition, Chen et al. proved that the exact computation of the influence given a seed set is #P-hard in the linear threshold model. Both of the two models are based on randomized propagation, however such information might be obtained by surveys and data mining techniques. This will make great difference on the complexity of the problem. In this note, we study the complexity of the influence maximization problem in deterministic linear threshold model. We show that in the deterministic linear threshold model, there is no n ^1??? -factor polynomial time approximation for the problem unless P=NP. We also show that the exact computation of the influence given a seed set can be solved in polynomial time.     

Polynomially solvable special cases of the quadratic bottleneck assignment problem

Quadratic bottleneck assignment problems (QBAP) are obtained by replacing the addition of cost terms in the objective function of a quadratic (sum) assignment problem by taking their maximum. Since the QBAP is an NP\mathcal{NP}-hard problem, polynimially solvable special cases of the QBAP are of interest. In this paper we specify conditions on the cost matrices of QBAP leading to special cases which can be solved to optimality in polynomial time. In particular, the following three cases are discussed: (i) any permutation is optimal (constant QBAP), (ii) a certain specified permutation is optimal (constant permutation QBAP) and (iii) the solution can be found algorithmically by a polynomial algorithm. Moreover, the max-cone of bottleneck Monge matrices is investigated, its generating matrices are identified and it is used as a tool in proving polynomiality results.     

Capacity inverse minimum cost flow problem

Given a directed graph G=(N,A) with arc capacities u _ij and a minimum cost flow problem defined on G, the capacity inverse minimum cost flow problem is to find a new capacity vector [^(u)]\hat{u} for the arc set A such that a given feasible flow [^(x)]\hat{x} is optimal with respect to the modified capacities. Among all capacity vectors [^(u)]\hat{u} satisfying this condition, we would like to find one with minimum ||[^(u)]-u||\|\hat{u}-u\| value. We consider two distance measures for ||[^(u)]-u||\|\hat{u}-u\| , rectilinear (L ₁) and Chebyshev (L _∞) distances. By reduction from the feedback arc set problem we show that the capacity inverse minimum cost flow problem is NP\mathcal{NP} -hard in the rectilinear case. On the other hand, it is polynomially solvable by a greedy algorithm for the Chebyshev norm. In the latter case we propose a heuristic for the bicriteria problem, where we minimize among all optimal solutions the number of affected arcs. We also present computational results for this heuristic.     

Partitioning 2-edge-colored complete multipartite graphs into monochromatic cycles, paths and trees

In this paper we consider the problem of partitioning complete multipartite graphs with edges colored by 2 colors into the minimum number of vertex disjoint monochromatic cycles, paths and trees, respectively. For general graphs we simply address the decision version of these three problems the 2-PGMC, 2-PGMP and 2-PGMT problems, respectively. We show that both 2-PGMC and 2-PGMP problems are NP-complete for complete multipartite graphs and the 2-PGMT problem is NP-complete for bipartite graphs. This also implies that all these three problems are NP-complete for general graphs, which solves a question proposed by the authors in a previous paper. Nevertheless, we show that the 2-PGMT problem can be solved in polynomial time for complete multipartite graphs. Research supported by NSFC.     

The nearest neighbor Spearman footrule distance for bucket, interval, and partial orders

Comparing and ranking information is an important topic in social and information sciences, and in particular on the web. Its objective is to measure the difference of the preferences of voters on a set of candidates and to compute a consensus ranking. Commonly, each voter provides a total order of all candidates. Recently, this approach was generalized to bucket orders, which allow ties. In this work we further generalize and consider total, bucket, interval and partial orders. The disagreement between two orders is measured by the nearest neighbor Spearman footrule distance, which has not been studied so far. For two bucket orders and for a total and an interval order the nearest neighbor Spearman footrule distance is shown to be computable in linear time, whereas for a total and a partial order the computation is NP-hard, 4-approximable and fixed-parameter tractable. Moreover, in contrast to the well-known efficient solution of the rank aggregation problem for total orders, we prove the NP-completeness for bucket orders and establish a 4-approximation.     

Maximizing Profits of Routing in WDM Networks

Let G = (V, E) be a ring (or chain) network representing an optical wavelength division multiplexing (WDM) network with k channels, where each edge e_j has an integer capacity c_j. A request s_i,t_i is a pair of two nodes in G. Given m requests s_i,t_i, i = 1, 2, ..., m, each with a profit value p_i, we would like to design/route a k-colorable set of paths for some (may not be all) of the m requests such that each edge e_j in G is used at most c_j times and the total profit of the set of designed paths is maximized. Here two paths cannot have the same color (channel) if they share some common edge(s).This problem arises in optical communication networks. In this paper, we present a polynomial-time algorithm to solve the problem when G is a chain. When G is a ring, however, the optimization problem is NP-hard (Wan and Liu, 1998), we present a 2-approximation algorithm based on our solution to the chain network. Similarly, some results in a bidirected chain and a bidirected ring are obtained.     

Revised GRASP with path-relinking for the linear ordering problem

The linear ordering problem (LOP) is an NP\mathcal{NP}-hard combinatorial optimization problem with a wide range of applications in economics, archaeology, the social sciences, scheduling, and biology. It has, however, drawn little attention compared to other closely related problems such as the quadratic assignment problem and the traveling salesman problem. Due to its computational complexity, it is essential in practice to develop solution approaches to rapidly search for solution of high-quality. In this paper we propose a new algorithm based on a greedy randomized adaptive search procedure (GRASP) to efficiently solve the LOP. The algorithm is integrated with a Path-Relinking (PR) procedure and a new local search scheme. We tested our implementation on the set of 49 real-world instances of input-output tables (LOLIB instances) proposed in Reinelt (Linear ordering library (LOLIB) 2002). In addition, we tested a set of 30 large randomly-generated instances proposed in Mitchell (Computational experience with an interior point cutting plane algorithm, Tech. rep., Mathematical Sciences, Rensellaer Polytechnic Institute, Troy, NY 12180-3590, USA 1997). Most of the LOLIB instances were solved to optimality within 0.87 seconds on average. The average gap for the randomly-generated instances was 0.0173% with an average running time of 21.98 seconds. The results indicate the efficiency and high-quality of the proposed heuristic procedure.     

Complexity of determining the most vital elements for the p-median and p-center location problems

We consider the k most vital edges (nodes) and min edge (node) blocker versions of the p-median and p-center location problems. Given a weighted connected graph with distances on edges and weights on nodes, the k most vital edges (nodes) p-median (respectively p-center) problem consists of finding a subset of k edges (nodes) whose removal from the graph leads to an optimal solution for the p-median (respectively p-center) problem with the largest total weighted distance (respectively maximum weighted distance). The complementary problem, min edge (node) blocker p-median (respectively p-center), consists of removing a subset of edges (nodes) of minimum cardinality such that an optimal solution for the p-median (respectively p-center) problem has a total weighted distance (respectively a maximum weighted distance) at least as large as a specified threshold. We show that k most vital edges p-median and k most vital edges p-center are NP-hard to approximate within a factor $\frac{7}{5}-\epsilon$ and $\frac{4}{3}-\epsilon$ respectively, for any ?>0, while k most vital nodes p-median and k most vital nodes p-center are NP-hard to approximate within a factor $\frac{3}{2}-\epsilon$ , for any ?>0. We also show that the complementary versions of these four problems are NP-hard to approximate within a factor 1.36.     

Partitioning a weighted partial order

The problem of partitioning a partially ordered set into a minimum number of chains is a well-known problem. In this paper we study a generalization of this problem, where we not only assume that the chains have bounded size, but also that a weight w _i is given for each element i in the partial order such that w _i ≤w _j if i ≺ j. The problem is then to partition the partial order into a minimum-weight set of chains of bounded size, where the weight of a chain equals the weight of the heaviest element in the chain. We prove that this problem is -hard, and we propose and analyze lower bounds for this problem. Based on these lower bounds, we exhibit a 2-approximation algorithm, and show that it is tight. We report computational results for a number of real-world and randomly generated problem instances.     

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.     

Complexity analysis for maximum flow problems with arc reversals

We provide a comprehensive study on network flow problems with arc reversal capabilities. The problem is to identify the arcs to be reversed in order to achieve a maximum flow from source(s) to sink(s). The problem finds its applications in emergency transportation management, where the lanes of a road network could be reversed to enable flow in the opposite direction. We study several network flow problems with the arc reversal capability and discuss their complexity. More specifically, we discuss the polynomial time algorithms for the maximum dynamic flow problem with arc reversal capability having a single source and a single sink, and for the maximum (static) flow problem. The presented algorithms are based on graph transformations and reductions to polynomially solvable flow problems. In addition, we show that the quickest transshipment problem with arc reversal capability and the problem of minimizing the total cost resulting from arc switching costs are NP\mathcal{NP} -hard.