Ant colony optimisation with local search for the bandwidth minimisation problem on graphs

The bandwidth reduction problem is a well-known NP-complete graphlayout problem that consists of labeling the vertices of a graph with integer labels in such a way as to minimize the maximum absolute difference between the labels of adjacent vertices. The problem is isomorphic to the important problem of reordering the rows and columns of a symmetric matrix so that its non-zero entries are maximally close to the main diagonal — a problem which presents itself in a large number of domains in science and engineering. A considerable number of methods have been developed to reduce the bandwidth, among which graph-theoretic approaches are typically faster and more effective. In this paper, a hyper-heuristic approach based on genetic programming is presented for evolving graph-theoretic bandwidth reduction algorithms. The algorithms generated from our hyper-heuristic are extremely effective. We test the best of such evolved algorithms on a large set of standard benchmarks from the Harwell-Boeing sparse matrix collection against two state-of-the-art algorithms from the literature. Our algorithm outperforms both algorithms by a significant margin, clearly indicating the promise of the approach.

An efficient local search framework for the minimum weighted vertex cover problem

An important implementation issue in the design of hybrid evolutionary multiobjective optimization algorithms with local search (i.e., multiobjective memetic algorithms) is how to strike a balance between local search and global search. If local search is applied to all individuals at every generation, almost all computation time is spent by local search. As a result, global search ability of memetic algorithms is not well utilized. We can use three ideas for decreasing the computation load of local search. One idea is to apply local search to only a small number of individuals. This idea can be implemented by introducing a local search probability, which is used to choose only a small number of initial solutions for local search from the current population. Another idea is a periodical (i.e., intermittent) use of local search. This idea can be implemented by introducing a local search interval (e.g., every 10 generations), which is used to specify when local search is applied. The other idea is an early termination of local search. Local search for each initial solution is terminated after a small number of neighbors are examined. This idea can be implemented by introducing a local search length, which is the number of examined neighbors in a series of iterated local search from a single initial solution. In this paper, we discuss the use of these three ideas to strike a local-global search balance. Through computational experiments on a two-objective 500-item knapsack problem, we compare various settings of local search such as short local search from all individuals at every generation, long local search from only a few individuals at every generation, and periodical long local search from all individuals. Global search in this paper means genetic search by crossover and mutation in multiobjective memetic algorithms.

An $L(j,k)$-$labeling$ of a graph is a vertex labeling such that the difference between the labels of adjacent vertices is at least $j$ and that between vertices separated by a distance 2 is at least $k$. The minimum of the spans of all $L(j,k)$-labelings of $G$ is denoted by $\lambda_k^j(G)$. Recently, Haque and Jha [16] proved that if $G$ is a multiple direct product of complete graphs, then $\lambda_k^j(G)$ coincides with the trivial lower bound $(N-1)k$, where $N$ is the order of $G$ and $\frac{j}{k}$ is within a certain bound. In this paper, we suggest a new labeling method for such a graph $G$. With this method, we extend the range of $\frac{j}{k}$ such that $\lambda_k^j(G)= (N-1)k$ holds. Moreover, we obtain the upper bound of $\lambda_k^j(G)$ for the remaining cases in the range $\frac{j}{k}$.

Antibandwidth of complete [formula omitted]-ary trees

In this paper, we address the optimization problem arising in some practical applications in which we want to maximize the minimum difference between the labels of adjacent elements. For example, in the context of location models, the elements can represent sensitive facilities or chemicals and their labels locations, and the objective is to locate (label) them in a way that avoids placing some of them too close together (since it can be risky). This optimization problem is referred to as the antibandwidth maximization problem (AMP) and, modeled in terms of graphs, consists of labeling the vertices with different integers or labels such that the minimum difference between the labels of adjacent vertices is maximized. This optimization problem is the dual of the well-known bandwidth problem and it is also known as the separation problem or directly as the dual bandwidth problem. In this paper, we first review the previous methods for the AMP and then propose a heuristic algorithm based on the variable neighborhood search methodology to obtain high quality solutions. One of our neighborhoods implements ejection chains which have been successfully applied in the context of tabu search. Our extensive experimentation with 236 previously reported instances shows that the proposed procedure outperforms existing methods in terms of solution quality.

MEMPSODE: A global optimization software based on hybridization of population-based algorithms and local searches

The analysis of the biomechanics of the leg while cycling is based on assumptions that make the calculations easier or faster. Most of the models and optimisations based on the dynamic chains introduced by Gonzalez & Hull (Hull & Gonzalez, 1988: J. Biomechanics, 21(10), 839-849) and by Papadopoulos (Papadopoulos, 1987: Biomechanics in Sport, 6, 83-92) with 5 and 4 links respectively share many of these assumptions. For instance, a fixed or no distance from the seat to the centre of rotation of the hip, a constant length of the segments of the leg (thigh, shank and foot) or a constant angular speed of the crank. Previous work (Rios Soltero et al., 2014: Procedia Engineering, in press) has shown that these assumptions may have an influence on the inertial forces of the system. The present study shows that the generalised forces at the knee and ankle would also be different. As it was done before (Rios Soltero et al, 2014), the dynamic chain consisted of three leg segments (thigh, shank and foot), a crank and a grounding element. Also, three motion models were used to calculate the forces at each instant of the pedalling cycle in the sagittal plane at the knee and ankle (Figure 1.a). Here, the analysis was based on the force data at the pedal reported by Boyd et al. (1996, J. Biomechanics, 29(8), 1105-1110), and the motion capture of markers located at the leg and pedal from Rios Soltero et al. (2014). The crank speed was calculated from the pedal marker. To get the orientation angle of each segment, the first two models used a linear regression of their markers. One of them (Inters; intersections method) intersected each line of contiguous segments (thigh with shank and seat-tube, shank with foot and thigh, foot with crank and shank, and crank with seat-tube and foot) to create a dynamic chain at each instant. This allowed a change in length of the segments and the motion of the hip. The second method (Proj; projections method) only uses the orientation of the linear regression for each segment, while their lengths were constant as measured directly from the person. In this way, neither the hip nor the length of the segments were fixed. The third method (Kin; kinematics method) reproduced the assumptions of other studies: a hip fixed at the seat and unvarying length of the leg segments. The length of the leg segments was calculated according to anatomical tables (Winter, 2004: Biomechanics and motor control of human movement (3d ed.). John Wiley & Sons). To calculate the position of the thigh at each instant, an 8th-degree polynomial regression was calculated between the angle of the crank and that of the thigh. In all cases, the centre of mass and mass-moment of inertia of the thigh and shank were calculated using an analytical equation based on the geometry proposed by Durkin Teague et al. (2005: J. Biomechanical Engineering 127, 455-464) and Durkin and Dowling (2006: Annals of Biomedical Engineering 34, pp. 1483-1493). An elliptical frustum was used for the foot. The results of the analysis show a slight difference in the moments at the ankle and the knee for the first half of the cycle (Figure 1.b). The maximum absolute difference in the case of the knee is 15.4 N·m (between Inters and Kin) which is around 13% of the values' range. As for the ankle, it is 7.79 N·m (between Inters and Proj) with a 19% of the values' range. It is clear that there is no qualitative difference between the models. However, further analysis is needed to determine if there is a significant effect.

The electronic structure, the static and high frequency dielectric tensors, and the infrared (IR) spectrum of fayalite Fe2SiO4, the Fe‐rich end‐member of olivine solid solutions, are explored at an ab initio quantum mechanical level, by using an all‐electron Gaussian type basis set, the B3LYP hybrid DFT functional, and the CRYSTAL09 code. Mulliken population analysis and spin density maps illustrate the electronic structure, characterized by a nearly pure d6, high‐spin configuration of the transition metal atom. The full set of IR wavenumbers and intensities is computed. The availability of highly accurate synchrotron radiation data (Suto et al., Astron Astrophys 2002, 389, 568) permits a very accurate comparison between simulated and measured quantities, in primis wavenumbers (ν) and oscillator strengths (f). The mean absolute difference ${\overline{\vert\Delta\nu\vert}}$ is as small as 4 cm−1, and the maximum absolute difference |Δνmax| never exceeds 12 cm−1, whereas the summed absolute difference ΔF between fexp and fcalc is around 10%. Modes not detected in the experiment turn out to be (i) characterized by low computed intensity, or (ii) very close to a large intense peak. Computed and experimental IR reflectance curves are in striking agreement also. The nature of the vibrational modes is investigated by means of isotopic substitutions, which clarify the participation of the various atomic species to each mode. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2012

Local search for weighted sum coloring problem

This tutorial presents a novel search system—the Attractor-Based Search System (ABSS)—that can solve the Traveling Salesman Problem very efficiently with optimality guarantee. From the perspective of dynamical systems, a heuristic local search algorithm for an NP-complete combinatorial problem is a discrete dynamical system. In a local search system, an attractor drives the search trajectories into the vicinity of a globally optimal point in the solution space, and the convergence of local search trajectories makes the search system become a global and deterministic system. The attractor contains a small set of the most promising solutions to the problem. The attractor can reduce the problem size exponentially, and thus make the exhaustive search feasible. Therefore, this new search paradigm is called optimizing with attractor. The ABSS consists of two search phases: local search phase and exhaustive search phase. The local search process is used to quickly construct the attractor in the solution space, and the exhaustive search process is used to completely search the attractor to identify the optimal solution. Therefore, the exact optimal solution can be found quickly by combining local search and exhaustive search. This tutorial introduces the concept of an attractor in a local search system, and describes the process of optimizing with the attractor, using the Traveling Salesman Problem as the study platform.

The broadcast scheduling problem (BSP) in packet radio networks is a well-known NP-complete combinatorial optimization problem. The broadcast scheduling avoids packet collisions by allowing only one node transmission in each collision domain of a time division multiple access (TDMA) network. It also improves the transmission utilization by assigning one transmission time slot to one or more nodes; thus, each node transmits at least once in each time frame. An optimum transmission schedule could minimize the length of a time frame while minimizing the number of idle nodes. In this paper, we propose a new iterated local search (ILS) algorithm that consists of two special perturbation and local search operators to solve the BSPs. Computational experiments are applied to benchmark data sets and randomly generated problem instances. The experimental results show that our ILS approach is effective in solving the problems with only a few runtimes, even for very large networks with 2,500 nodes.

The Local Search algorithm (or Hill Climbing, or Iterative Improvement) is one of the simplest heuristics to solve the Satisfiability and Max-Satisfiability problems. Although it is not the best known Satisfiability algorithm even for the class of problems we study, the Local Search is a part of many satisfiability and max-satisfiability solvers, where it is used to find a good starting point for a more sophisticated heuristics, and to improve a candidate solution. In this paper we give an analysis of Local Search on random planted 3-CNF formulas. We show that a sharp transition of efficiency of Local Search occurs at density \(\varrho = \frac{7}{6} \ln n\). Specifically we show that if there is \(\kappa <\frac{7}{6}\) such that the clause-to-variable ratio is less than \(\kappa \ln n\) (n is the number of variables in a CNF) then Local Search whp does not find a satisfying assignment, and if there is \(\kappa >\frac{7}{6}\) such that the clause-to-variable ratio is greater than \(\kappa \ln n\) then the local search whp finds a satisfying assignment. As a byproduct we also show that for any constant \(\varrho \) there is \(\gamma \) such that Local Search applied to a random (not necessarily planted) 3-CNF with clause-to-variable ratio \(\varrho \) produces an assignment that satisfies at least \(\gamma n\) clauses less than the maximal number of satisfiable clauses.KeywordsLocal Search (LS)Satisfiable ClausesAlgorithm LSProper Local MinimumDistribution SatisfiesThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In Combinatorial Auctions, multiple goods (items) are available for auction simultaneously, and bidders bid for combinations of goods called bundles. In the single unit combinatorial auction problem, there is only single unit of each item present and the items are considered to be indivisible. This forms the supply side constraint resulting in a need by the seller to decide on which bundles to allocate for maximizing the revenue. Determining these winning bundles is an NP-complete problem and is known by the name of Winner Determination Problem (WDP) in literature. Combinatorial auction mechanism has a wide range of practical applications such as allocation of railroads, auction of adjacent pieces of real estate, auction of airport landing slots, distributed job shop scheduling etc. The optimal heuristic search algorithms like CASS and CABOB proposed for solving the WDP take a lot of CPU time for solving large problem instances. Though these algorithms give optimal solutions, they would be useful only in scenarios where time is not at all a constraint, and the bidding is a one time affair. This poses a limitation in conducting combinatorial auctions with multiple rounds of bidding and hence hinders the widespread use of combinatorial auctions on the patterns of traditional English auction and internet auctions. Thus faster solution of WDP holds the key to widespread use and popularity of combinatorial auctions. This paper attempts to address the above issue, and proposes a simple local search technique LSWDP (Local Search for Winner Determination Problem) that runs very fast and provides solutions quite close to optimal for a number of applications. It also outputs optimal solutions in many instances taking only a fraction of the CPU time taken by CASS. In addition LSWDP was found to outperform the anytime algorithm CASS given a time cutoff. This paper also presents an enhancement called LSCN (Local Search using Complimentary Neighborhood) that provides still better solutions in terms of close to optimal behavior and achieves a much better strike rate in hitting optimal solutions. LSCN runs slower than LSWDP but still takes only a fraction of the CPU time taken by CASS thus encouraging the use of local search for combinatorial auctions. Finally, an extension to multi-partition search has been proposed and analyzed.

In this paper, we investigate the efficiency of local search algorithms for some relatively unstructured networks. By local search, we mean that each vertex has a knowledge of its very close neighborhood, typically the identities, degrees and possibly the contents. Assuming there is no preprocessing step, we are interested in seeking a vertex of a given identity starting from any vertex. The time complexity of search is expressed as the number of vertices to explore before reaching the target or a neighbor of the target. For an arbitrary graph and a worst-case labelling of vertices, time complexity can be of the same order as the size of the graph itself. However, there is some hope that for large classes of graphs, the size of the exploration sequence would drastically decrease. Recent literature exhibits non trivial properties shared by the topologies of numerous real-world distributed networks: the combination of a low diameter, a specific degree distribution of the vertices and a sparse network are typically representative of a small-world topology. On the other hand, in a seminal paper, Kleinberg [Kle00] focuses on the ability of a user or a mobile agent to route along shorts paths within a class of random small-world graphs, the navigable small-worlds† using a greedy distributed algorithm (distances within a spanning subgraph are known or can be computed locally). Under some conditions, Kleinberg also proposes polynomial lower bounds valid for any distributed algorithm using the same knowledge of the network even if the diameter is polylogarithmic. However, Kleinberg’s model lacks an important property observed in real networks which are often scalefree. The degree distribution of the vertices tends to follow a power-law distribution, that is the number of vertices of degree δ is proportional to n ( 1 δ )k for an n-vertex graph and k a constant strictly greater than one. Moreover, it has been observed that for real-life networks, k tends to belong to the range [2,3]. For graphs built using Kleinberg’s model, the degree distribution is very different (close to a Poisson distribution). All local‡ distributed search algorithms presented in the literature work as follows: given the set of visited vertices, one should choose the next vertex to explore. The process stops as soon as the target is reached. In this paper, the underlying network is a random scale-free labeled graph whose node identities belong to the range [1,n]. Each vertex knows its own identity and its degree and we assume two models of local knowledge: some information about the neighbors can be known (strong model) or not (weak model). The first model is the more realistic one but our starting point is the second model. Note that Kleinberg’s model assumes more information than our strong model. Up to now, it is still an open problem to know if scale-free graphs are navigable in the polylogarithmic sense. In this paper, we answer negatively to this question for some random scale-free graphs built using a