考虑道路通行能力的应急避难点选址模型及算法

在k-中心点问题的基础上,考虑道路的通行能力限制,提出了k-避难点问题。在一般树图结构下,重点分析了1-避难点选址问题,并设计了有效的求解算法;在直线图结构下,首先改进了一般图1-避难点的求解算法,其次分析了2-避难点问题的特点,并给出了一个基于"二分思想"的求解算法,在此基础上,为一般的直线图k-避难点问题设计了求解算法,一般算法的时间复杂性为O(nlog^kn)。所提出的模型在理论上扩展了经典的k-中心点选址问题,所设计的求解算法能够为现实的应急管理规划提供良好的理论支持。     

Fast searching games on graphs

Given a graph, suppose that intruders hide on vertices or along edges of the graph. The fast searching problem is to find the minimum number of searchers required to capture all the intruders satisfying the constraint that every edge is traversed exactly once and searchers are not allowed to jump. In this paper, we prove lower bounds on the fast search number. We present a linear time algorithm to compute the fast search number of Halin graphs and their extensions. We present a quadratic time algorithm to compute the fast search number of cubic graphs.     

From theory to practice: maximizing revenues for on-line dial-a-ride

We consider the on-line dial-a-ride problem, where a server fulfills requests that arrive over time. Each request has a source, destination, and release time. We study a variation of this problem where each request also has a revenue that the server earns for fulfilling the request. The goal is to serve requests within a time limit while maximizing the total revenue. We first prove that no deterministic online algorithm can be competitive unless the input graph is complete and edge weights are unit. We therefore focus on these graphs and present a 2-competitive algorithm for this problem. We also consider two variations of this problem: (1) the input graph is complete bipartite and (2) there is a single node that is the source for every request, and present a 1-competitive algorithm for the former and an optimal algorithm for the latter. We also provide experimental results for the complete and complete bipartite graphs. Our simulations support our theoretical findings and demonstrate that our algorithms perform well under settings that reflect realistic dial-a-ride systems.     

A Multi-Exchange Neighborhood for Minimum Makespan Parallel Machine Scheduling Problems

We propose new local search algorithms for minimum makespan parallel machine scheduling problems, which perform multiple exchanges of jobs among machines. Inspired by the work of Thompson and Orlin (1989) on cyclic transfer neighborhood structures, we model multiple exchanges of jobs as special disjoint cycles and paths in a suitably defined improvement graph, by extending definitions and properties introduced in the context of vehicle routing problems (Thompson and Psaraftis, 1993) and of the capacitated minimum spanning tree problem (Ahuja et al., 2001). Several algorithms for searching the neighborhood are suggested.We report the results of a wide computational experimentation, on different families of benchmark instances, performed for the case of identical machines. This problem has been selected as a case study to perform a comparison among the alternative algorithms, and to discover families of instances for which the proposed neighborhood may be promising in practice. Based on the results of the experiments, we can suggest which among the many possible variants of the proposed approaches may be more promising for developing local search algorithms based on multi-exchange moves for related problems. Also, on some families of instances, which are very hard to solve exactly, the most promising multi-exchange algorithms were observed to dominate, in solution quality and in computational time, competitive benchmark heuristics.     

On the max-weight edge coloring problem

We study the following generalization of the classical edge coloring problem: Given a weighted graph, find a partition of its edges into matchings (colors), each one of weight equal to the maximum weight of its edges, so that the total weight of the partition is minimized. We explore the frontier between polynomial and NP-hard variants of the problem, with respect to the class of the underlying graph, as well as the approximability of NP-hard variants. In particular, we present polynomial algorithms for bounded degree trees and star of chains, as well as an approximation algorithm for bipartite graphs of maximum degree at most twelve which beats the best known approximation ratios.     

Improving robustness of next-hop routing

A weakness of next-hop routing is that following a link or router failure there may be no routes between some source-destination pairs, or packets may get stuck in a routing loop as the protocol operates to establish new routes. In this article, we address these weaknesses by describing mechanisms to choose alternate next hops. Our first contribution is to model the scenario as the following tree augmentation problem. Consider a mixed graph where some edges are directed and some undirected. The directed edges form a spanning tree pointing towards the common destination node. Each directed edge represents the unique next hop in the routing protocol. Our goal is to direct the undirected edges so that the resulting graph remains acyclic and the number of nodes with outdegree two or more is maximized. These nodes represent those with alternative next hops in their routing paths. We show that tree augmentation is NP-hard in general and present a simple \(\frac{1}{2}\)-approximation algorithm. We also study 3 special cases. We give exact polynomial-time algorithms for when the input spanning tree consists of exactly 2 directed paths or when the input graph has bounded treewidth. For planar graphs, we present a polynomial-time approximation scheme when the input tree is a breadth-first search tree. To the best of our knowledge, tree augmentation has not been previously studied.     

Multi-start iterated tabu search for the minimum weight vertex cover problem

The minimum weight vertex cover problem (MWVCP) is one of the most popular combinatorial optimization problems with various real-world applications. Given an undirected graph where each vertex is weighted, the MWVCP is to find a subset of the vertices which cover all edges of the graph and has a minimum total weight of these vertices. In this paper, we propose a multi-start iterated tabu search algorithm (MS-ITS) to tackle MWVCP. By incorporating an effective tabu search method, MS-ITS exhibits several distinguishing features, including a novel neighborhood construction procedure and a fast evaluation strategy. Extensive experiments on the set of public benchmark instances show that the proposed heuristic is very competitive with the state-of-the-art algorithms in the literature.     

Hybrid Evolutionary Algorithms for Graph Coloring

A recent and very promising approach for combinatorial optimization is to embed local search into the framework of evolutionary algorithms. In this paper, we present such hybrid algorithms for the graph coloring problem. These algorithms combine a new class of highly specialized crossover operators and a well-known tabu search algorithm. Experiments of such a hybrid algorithm are carried out on large DIMACS Challenge benchmark graphs. Results prove very competitive with and even better than those of state-of-the-art algorithms. Analysis of the behavior of the algorithm sheds light on ways to further improvement.     

Multiple facility location on a network with linear reliability order of edges

We study the problem of locating facilities on the nodes of a network to maximize the expected demand serviced. The edges of the input graph are subject to random failure due to a disruptive event. We consider a special type of failure correlation. The edge dependency model assumes that the failure of a more reliable edge implies the failure of all less reliable ones. Under this dependency model called Linear Reliability Order (LRO) we give two polynomial time exact algorithms. When two distinct LRO’s exist, we prove the total unimodularity of a linear programming formulation. In addition, we show that minimizing the sum of facility opening costs and expected cost of unserviced demand under two orderings reduces to a matching problem. We prove NP-hardness of the three orderings case and show that the problem with an arbitrary number of orderings generalizes the deterministic maximum coverage problem. When a demand point can be covered only if a facility exists within a distance limit, we show that the problem is NP-hard even for a single ordering.     

Heuristics for the network design problem with connectivity requirements

We consider the NP-complete problem of finding a spanning \(k\)-tree of minimum weight in a complete weighted graph. This problem has a number of applications in designing reliable backbone telecommunication networks. We propose effective algorithms based on a greedy strategy and several variable neighborhood search metaheuristics. We also develop an integer linear programming model for calculating a lower bound. Preliminary numerical experiments using random and real-word data sets are reported to show the effectiveness of our approach. In addition, we compare our approach with known metaheuristics.     

An adaptive multistart tabu search approach to solve the maximum clique problem

Given an undirected graph G=(V,E) with vertex set V={1,…,n} and edge set E?V×V. The maximum clique problem is to determine in G a clique (i.e., a complete subgraph) of maximum cardinality. This paper presents an effective algorithm for the maximum clique problem. The proposed multistart tabu search algorithm integrates a constrained neighborhood, a dynamic tabu tenure mechanism and a long term memory based restart strategy. Our proposed algorithm is evaluated on the whole set of 80 DIMACS challenge benchmarks and compared with five state-of-the-art algorithms. Computational results show that our proposed algorithm attains the largest known clique for 79 benchmarks.     

Graph coloring by multiagent fusion search

A multiagent fusion search is presented for the graph coloring problem. In this method, each of agents performs the fusion search, involving a local search working in a primary exploitation role and a recombination search in a navigation role, with extremely limited memory and interacts with others through a decentralized protocol, thus agents are able to explore in parallel as well as to achieve a collective performance. As the knowledge components implemented with available structural information and in formalized forms, the Quasi-Tabu local search and grouping-based recombination rules are especially useful in addressing neutrality and ruggedness of the problem landscape. The new method has been tested on some hard benchmark graphs, and has been shown competitive in comparison with several existing algorithms. In addition, the method provides new lower bound solutions when applied to two large graphs. Some search characteristics of the proposed method is also discussed.     

Efficient identifications of structural similarities for graphs

Measuring and detecting graph similarities is an important topic with numerous applications. Early algorithms often incur quadratic time or higher, making them unsuitable for graphs of very large scales. Motivated by the cooling process of an object in a thermodynamic system, we devise a new method for measuring graph similarities that can be carried out in linear time. Our algorithm, called Random Walker Termination (RWT), employs a large number of random walkers to capture the structure of a given graph using termination rates in a time sequence. To verify the effectiveness of the RWT algorithm, we use three major graph models, namely, the Erd?s-Rényi random graphs, the Watts-Strogatz small-world graphs, and the Barabási-Albert preferential-attachment graphs, to generate graphs of different sizes. We show that the RWT algorithm performs well for graphs generated by these models. Our experiment results agree with the actual similarities of generated graphs. Using self-similarity tests, we show that RWT is sufficiently stable to generate consistent results. We use the graph edge rerouting test and the cross model test to demonstrate that RWT can effectively identify structural similarities between graphs.     

Approximability of the subset sum reconfiguration problem

The subset sum problem is a well-known NP-complete problem in which we wish to find a packing (subset) of items (integers) into a knapsack with capacity so that the sum of the integers in the packing is at most the capacity of the knapsack and at least a given integer threshold. In this paper, we study the problem of reconfiguring one packing into another packing by moving only one item at a time, while at all times maintaining the feasibility of packings. First we show that this decision problem is strongly NP-hard, and is PSPACE-complete if we are given a conflict graph for the set of items in which each vertex corresponds to an item and each edge represents a pair of items that are not allowed to be packed together into the knapsack. We then study an optimization version of the problem: we wish to maximize the minimum sum among all packings in a reconfiguration. We show that this maximization problem admits a polynomial-time approximation scheme, while the problem is APX-hard if we are given a conflict graph.     

Graph Algorithms with Small Communication Costs

We study minimizing communication cost in parallel algorithm design, by minimizing the number of communication phases in coarse-grained parallel computers. There have been several recent papers dealing with parallel algorithms of small communication cost under different models. Most of these results are for computational geometry problems. For these problems it has been possible to decompose tasks into appropriate subproblems in a communication-efficient way. It appears to be somewhat more difficult to design parallel algorithms with small communication phases for graph theory problems. In this paper we focus on the design of deterministic algorithms with a small number of communication phases for the list ranking problem and the shortest path problem.     

Fast On-Line/Off-Line Algorithms for Optimal Reinforcement of a Network and its Connections with Principal Partition

The problem of computing the strength and performing optimal reinforcement for an edge-weighted graph G(V, E, w) is well-studied. In this paper, we present fast (sequential linear time and parallel logarithmic time) on-line algorithms for optimally reinforcing the graph when the reinforcement material is available continuously on-line. These are the first on-line algorithms for this problem. We invest O(|V|³|E|log|V|) time (equivalent to (|V|) invocations of the fastest known algorithms for optimal reinforcement) in preprocessing the graph before the start of our algorithms. It is shown that the output of our on-line algorithms is as good as that of the off-line algorithms. Thus our algorithms are better than the fastest off-line algorithms in situations when a sequence of more than (|V|) reinforcement problems need to be solved. The key idea is to make use of ideas underlying the theory of Principal Partition of a Graph. Our ideas are easily generalized to the general setting of polymatroid functions. We also present a new efficient algorithm for computation of the Principal Sequence of a graph.     

The permutation flow shop problem with blocking. A tabu search approach

This paper develops a fast tabu search algorithm to minimize makespan in a flow shop problem with blocking. Some properties of the problem associated with the blocks of jobs have been presented and discussed. These properties allow us to propose a specific neighbourhood of algorithms. Also, the multimoves are used that consist in performing several moves simultaneously in a single iteration and guide the search process to more promising areas of the solutions space, where good solutions can be found. It allow us to accelerate the convergence of the algorithm. Besides, a dynamic tabu list is proposed that assists additionally to avoid being trapped at a local optimum. The proposed algorithms are empirically evaluated and found to be relatively more effective in finding better solutions than attained by the leading approaches in a much shorter time. The presented ideas can be applied in many local search procedures.     

A Unifying Augmentation Algorithm for Two-Edge Connectivity and Biconnectivity

Given an undirected graph G and two vertex subsets H1 and H2, the bi-level augmentation problem is that of adding to G the smallest number of edges such that the resulting graph contains two internally vertex-disjoint paths between every pair of vertices in H1 and two edge-disjoint paths between every pair of vertices in H2. We present an algorithm to solve this problem in linear time. By properly setting H1 and H2, this augmentation algorithm subsumes existing optimal algorithms for several graph augmentation problems.     

Establishing symmetric connectivity in directional wireless sensor networks equipped with $$2\pi /3$$ antennas

In this paper, we study the antenna orientation problem concerning symmetric connectivity in directional wireless sensor networks. We are given a set of nodes each of which is equipped with one directional antenna with beam-width \(\theta = 2\pi /3\) and is initially assigned a transmission range 1 that yields a connected unit disk graph spanning all nodes. The objective of the problem is to compute an orientation of the antennas and to find a minimum transmission power range \(r=O(1)\) such that the induced symmetric communication graph is connected. We propose two algorithms that orient the antennas to yield symmetric connected communication graphs where the transmission power ranges are bounded by 6 and 5, which are currently the best results for this problem. We also study the performance of our algorithms through simulations.     

Some graph optimization problems with weights satisfying linear constraints

In this paper, we study several graph optimization problems in which the weights of vertices or edges are variables determined by several linear constraints, including maximum matching problem under linear constraints (max-MLC), minimum perfect matching problem under linear constraints (min-PMLC), shortest path problem under linear constraints (SPLC) and vertex cover problem under linear constraints (VCLC). The objective of these problems is to decide the weights that are feasible to the linear constraints, and find the optimal solutions of corresponding graph optimization problems among all feasible choices of weights. We find that these problems are NP-hard and are hard to be approximated in general. These findings suggest us to explore various special cases of them. In particular, we show that when the number of constraints is a fixed constant, all these problems are polynomially solvable. Moreover, if the total number of distinct weights is a fixed constant, then max-MLC, min-PMLC and SPLC are polynomially solvable, and VCLC has a 2-approximation algorithm. In addition, we propose approximation algorithms for various cases of max-MLC.