A methodology to find the elementary landscape decomposition of combinatorial optimization problems

A small number of combinatorial optimization problems have search spaces that correspond to elementary landscapes, where the objective function f is an eigenfunction of the Laplacian that describes the neighborhood structure of the search space. Many problems are not elementary; however, the objective function of a combinatorial optimization problem can always be expressed as a superposition of multiple elementary landscapes if the underlying neighborhood used is symmetric. This paper presents theoretical results that provide the foundation for algebraic methods that can be used to decompose the objective function of an arbitrary combinatorial optimization problem into a sum of subfunctions, where each subfunction is an elementary landscape. Many steps of this process can be automated, and indeed a software tool could be developed that assists the researcher in finding a landscape decomposition. This methodology is then used to show that the subset sum problem is a superposition of two elementary landscapes, and to show that the quadratic assignment problem is a superposition of three elementary landscapes.     

Algebraic theory of recombination spaces

A new mathematical representation is proposed for the configuration space structure induced by recombination, which we call "P-structure." It consists of a mapping of pairs of objects to the power set of all objects in the search space. The mapping assigns to each pair of parental "genotypes" the set of all recombinant genotypes obtainable from the parental ones. It is shown that this construction allows a Fourier decomposition of fitness landscapes into a superposition of "elementary landscapes." This decomposition is analogous to the Fourier decomposition of fitness landscapes on mutation spaces. The elementary landscapes are obtained as eigenfunctions of a Laplacian operator defined for P-structures. For binary string recombination, the elementary landscapes are exactly the p-spin functions (Walsh functions), that is, the same as the elementary landscapes of the string point mutation spaces (i.e., the hypercube). This supports the notion of a strong homomorphism between string mutation and recombination spaces. However, the effective nearest neighbor correlations on these elementary landscapes differ between mutation and recombination and among different recombination operators. On average, the nearest neighbor correlation is higher for one-point recombination than for uniform recombination. For one-point recombination, the correlations are higher for elementary landscapes with fewer interacting sites as well as for sites that have closer linkage, confirming the qualitative predictions of the Schema Theorem. We conclude that the algebraic approach to fitness landscape analysis can be extended to recombination spaces and provides an effective way to analyze the relative hardness of a landscape for a given recombination operator.     

Embedded landscapes

In this paper we introduce embedded landscapes as an extension of NK landscapes and MAXSAT problems. This extension is valid for problems where the representation can be expressed as a simple sum of subfunctions over subsets of the representation domain. This encompasses many additive constraint problems and problems expressed as the interaction of subcomponents, where the critical features of the subcomponents are represented by subsets of bits in the domain. We show that embedded landscapes of fixed maximum epistasis K are exponentially sparse in epistatic space with respect to all possible functions. We show we can compute many important statistical features of these functions in polynomial time including all the epistatic interactions and the statistical moments of hyperplanes about the function mean and hyperplane mean. We also show that embedded landscapes of even small fixed K can be NP-complete. We can conclude that knowing the epistasis and many of the hyperplane statistics is not enough to solve the exponentially difficult part of these general problems and that the difficulty of the problem lies not in the epistasis itself but in the interaction of the epistatic parts.     

Exact approaches for static data segment allocation problem in an information network

In a large distributed database, data are geographically distributed across several separate servers (or data centers). This helps in distributing load in the access network. It also helps to serve data locally where it is required. There are various approaches based on the granularity of data for efficient data distribution in a communication network. The file allocation problem (FAP) locates files to servers, the segment allocation problem (SAP) locates database segments, and the mirror location problem (MLP) locates replicas of the entire database. The placement of such data to multiple servers can be modeled as an optimization problem. The major decisions influencing optimization involves the location of servers, allocation of content and assignment of users. In this paper, we study the segment allocation problem (SAP), which is also known as the partial mirroring problem. This approach is more tractable than the file allocation problem in realistic cases and also eliminates the overhead of (constant) update costs that is incurred in the mirror location problem. Our contribution is two-fold: Firstly, earlier works on SAP assume pre-defined segments. We build a data partitioning method using well-known facility location models. We quantify the performance of the partitioning method. We show that the method partitions the database within a reasonable limit of error. Secondly, we introduce a new model for the segment allocation problem in which the segments are completely connected to each other by high-bandwidth links and contains a cost benefit for inter-segment traffic flows. We formulate this problem as an MILP and build exact solution approaches to solve large scale problems. We demonstrate some structural properties of the problem that make it solvable, using a Benders decomposition algorithm. Computational results validate the superiority of the decomposition approach.     

A hybrid approach based on stochastic competitive Hopfield neural network and efficient genetic algorithm for frequency assignment problem

This paper presents a hybrid efficient genetic algorithm (EGA) for the stochastic competitive Hopfield (SCH) neural network, which is named SCH–EGA. This approach aims to tackle the frequency assignment problem (FAP). The objective of the FAP in satellite communication system is to minimize the co-channel interference between satellite communication systems by rearranging the frequency assignment so that they can accommodate increasing demands. Our hybrid algorithm involves a stochastic competitive Hopfield neural network (SCHNN) which manages the problem constraints, when a genetic algorithm searches for high quality solutions with the minimum possible cost. Our hybrid algorithm, reflecting a special type of algorithm hybrid thought, owns good adaptability which cannot only deal with the FAP, but also cope with other problems including the clustering, classification, and the maximum clique problem, etc. In this paper, we first propose five optimal strategies to build an efficient genetic algorithm. Then we explore three hybridizations between SCHNN and EGA to discover the best hybrid algorithm. We believe that the comparison can also be helpful for hybridizations between neural networks and other evolutionary algorithms such as the particle swarm optimization algorithm, the artificial bee colony algorithm, etc. In the experiments, our hybrid algorithm obtains better or comparable performance than other algorithms on 5 benchmark problems and 12 large problems randomly generated. Finally, we show that our hybrid algorithm can obtain good results with a small size population.     

Landscapes Synthesis Achieved through Erosion and Deposition Process Simulation

This paper describes an original approach to terrain evolution in landscapes synthesis. In order to create some realistic landforms, we simulate geologically contrasted terrains and apply to them deterministic erosion processes. This allows us to relate the erosion on any point of the landsurface to local geological parameters. Any height field may be chosen as an initial topographic surface. Small perturbations may be introduced to avoid unpleasant regularities. A 3D model defines the geological parameters of each point according to its elevation. Our method is iterative: at each step, rock removal and possible alluvial deposition are computed at each point of the landsurface. The available erosion laws simulate mechanical erosion, chemical dissolution and alluvial deposition. At the end of each iteration, a new landsurface and the corresponding river network are created. Landsurfaces can be visualized at the final stage by two rendering algorithms including natural textures mapping. The stream network and the ridges may also be visualized.     

Non-myopic vehicle and route selection in dynamic DARP with travel time and workload objectives

In dial-a-ride problems, a fleet of n vehicles is routed to transport people between pick-up and delivery locations. We consider an elementary version of the problem where trip requests arrive in time and require an immediate vehicle assignment (which triggers an appropriate route update of the selected vehicle). In this context, a relatively general objective can be stated as a weighted sum of the system's effort and the customers' inconvenience. However, optimizing almost any objective in this immensely complex stochastic system is prohibitively difficult. Thus the earlier work has largely resorted to heuristic cost functions that arise, e.g., from the corresponding static systems. By using the framework of Markov decision processes and the classical M/M/1 queue as a highly abstract model for a single vehicle, we explain why certain intuitive cost functions indeed give satisfactory results in the dynamic system, and also give an explicit interpretation of different components appearing in a general cost function. The resulting family of heuristic control policies is demonstrated to offer a desired type of performance thus justifying the assumed analogy between a multi-queue and dial-a-ride systems.     

利用遗传算法求解文件分配问题

文件分配问题是计算机网络和分布式系统中一个非常重要的问题．本文提出一种用遗传算法求解文件分配问题的新方法，该方法能较好地解决工程中的文件优化分配问题，还可以应用到其它资源需要分配的领域.     

Fitness landscape of the cellular automata majority problem: View from the “Olympus”

In this paper we study cellular automata (CAs) that perform the computational Majority task. This task is a good example of what the phenomenon of emergence in complex systems is. We take an interest in the reasons that make this particular fitness landscape a difficult one. The first goal is to study the landscape as such, and thus it is ideally independent from the actual heuristics used to search the space. However, a second goal is to understand the features a good search technique for this particular problem space should possess. We statistically quantify in various ways the degree of difficulty of searching this landscape. Due to neutrality, investigations based on sampling techniques on the whole landscape are difficult to conduct. So, we go exploring the landscape from the top. Although it has been proved that no CA can perform the task perfectly, several efficient CAs for this task have been found. Exploiting similarities between these CAs and symmetries in the landscape, we define the Olympus landscape which is regarded as the “heavenly home” of the best local optima known (blok). Then we measure several properties of this subspace. Although it is easier to find relevant CAs in this subspace than in the overall landscape, there are structural reasons that prevent a searcher from finding overfitted CAs in the Olympus. Finally, we study dynamics and performance of genetic algorithms on the Olympus in order to confirm our analysis and to find efficient CAs for the Majority problem with low computational cost.     

A Model for Minimizing Active Processor Time

We introduce the following elementary scheduling problem. We are given a collection of n jobs, where each job J _i has an integer length ? _i as well as a set T _i of time intervals in which it can be feasibly scheduled. Given a parameter B, the processor can schedule up to B jobs at a timeslot t so long as it is “active” at t. The goal is to schedule all the jobs in the fewest number of active timeslots. The machine consumes a fixed amount of energy per active timeslot, regardless of the number of jobs scheduled in that slot (as long as the number of jobs is non-zero). In other words, subject to ? _i units of each job i being scheduled in its feasible region and at each slot at most B jobs being scheduled, we are interested in minimizing the total time during which the machine is active. We present a linear time algorithm for the case where jobs are unit length and each T _i is a single interval, assuming that jobs are given in sorted order. For general T _i , we show that the problem is NP-complete even for B=3. However when B=2, we show that it can be efficiently solved. In addition, we consider a version of the problem where jobs have arbitrary lengths and can be preempted at any point in time. For general B, the problem can be solved by linear programming. For B=2, the problem amounts to finding a triangle-free 2-matching on a special graph. We extend the algorithm of Babenko et al. (Proceedings of the 16th Annual International Conference on Computing and Combinatorics, pp. 120–129, 2010) to handle our variant, and also to handle non-unit length jobs. This yields an \(O(\sqrt{L} m)\) time algorithm to solve the preemptive scheduling problem for B=2, where L=∑ _i ? _i . We also show that for B=2 and unit length jobs, the optimal non-preemptive schedule has active time at most 4/3 times that of the optimal preemptive schedule; this bound extends to several versions of the problem when jobs have arbitrary length.     

Algorithms for computing the maximum weight region decomposable into elementary shapes

Motivated by the image segmentation problem, we consider the following geometric optimization problem: Given a weighted n × n pixel grid, find the maximum weight region whose shape is decomposable into a set of disjoint elementary shapes. We give efficient algorithms for several interesting shapes. This is in strong contrast to finding the maximum weight region that is the union of elementary shapes for the corresponding cases—a problem that we prove to be NP-hard. We implemented one of the algorithms and demonstrate its applicability for image segmentation.     

COMPARATIVE ANALYSIS OF A HYBRID DE ALGORITHM WITH THE VNS ALGORITHM AND ITS VARIATION SVNS TO SOLVE A REAL-WORLD FREQUENCY ASSIGNMENT PROBLEM

The Frequency Assignment is a very important task in the planning of the GSM networks, and it still continues to be a critical task for current (and future) mobile communication operators. In this work, we compare a hybrid Differential Evolution algorithm with the Variable Neighbourhood Search algorithm and also its variant Skewed Variable Neighbourhood Search to solve a real-world Frequency Assignment problem (FAP) in GSM Networks. The results that are shown use accurate interference information. That information was also adopted by other researchers and it represents a real GSM network, granting, therefore, an really important applicability. Furthermore, we have analyzed and compared our approach with other algorithms proposed so far to this problem. Hence, our approach using the SVNS algorithm has proven to be efficient in solving this problem, and permitted us to obtain good results. In fact, with this work we have contributed to the FAP problem with an additional comparison between approaches using metaheuristics based on trajectory (VNS and SVNS) and others based on population (DE).     

Verifying parallel programs with dynamic communication structures

We address the verification problem of networks of communicating pushdown systems modeling communicating parallel programs with procedure calls. Processes in such networks can read the control state of the other processes according to a given communication structure (specifying the observability rights between processes). The reachability problem of such models is undecidable in general. First, we define a class of networks that effectively preserves recognizability (hence, its reachability problem is decidable). Then, we consider networks where the communication structure can change dynamically during the execution according to a phase graph. The reachability problem for these dynamic networks being undecidable in general, we define a subclass for which it becomes decidable. Then, we consider reachability when the switches in the communication structures are bounded. We show that this problem is undecidable even for one switch. We define a natural class of models for which this problem is decidable. This class can be used in the definition of an efficient semi-decision procedure for the analysis of the general model of dynamic networks. Our techniques allowed to find bugs in two versions of a Windows NT Bluetooth driver.     

Optimization algorithms for large-scale real-world instances of the frequency assignment problem

Nowadays, mobile communications are experiencing a strong growth, being more and more indispensable. One of the key issues in the design of mobile networks is the frequency assignment problem (FAP). This problem is crucial at present and will remain important in the foreseeable future. Real-world instances of FAP typically involve very large networks, which can be handled only by heuristic methods. In the present work, we are interested in optimizing frequency assignments for problems described in a mathematical formalism that incorporates actual interference information, measured directly on the field, as is done in current GSM networks. To achieve this goal, a range of metaheuristics have been designed, adapted, and rigourously compared on two actual GSM networks modeled according to the latter formalism. To generate quickly and reliably high-quality solutions, all metaheuristics combine their global search capabilities with a local-search method specially tailored for this domain. The experiments and statistical tests show that in general, all metaheuristics are able to improve upon results published in previous studies, but two of the metaheuristics emerge as the best performers: a population-based algorithm (Scatter Search) and a trajectory based (1+1) Evolutionary Algorithm. Finally, the analysis of the frequency plans obtained offers insight about how the interference cost is reduced in the optimal plans.     

A Combined column generation and heuristics for railway short-term rolling stock planning with regular inspection constraints

The aim of railway rolling stock planning problem is to find an optimal allocation of train-sets for a given set of trips in the train timetable in order to minimize the total cost. We propose a column generation and Lagrangian relaxation heuristics for short-term rolling stock planning problems with regular inspection constraints. The problem is formulated as a subtour traveling salesman problem to find a set of elementary shortest cycles that cover all trips in the timetable. In the proposed method, a tight lower bound is obtained from the continuous relaxation of Dantzig–Wolfe reformulation by column generation. The pricing problem can be formulated as an elementary shortest cycle problem with resource constraints. A labeling algorithm is applied to solve the pricing problem. In order to reduce the computational effort, we apply a general state space augmenting algorithm to solve the pricing problems. Computational results show that the proposed column generation and Lagrangian relaxation heuristics can find good lower and upper bounds for 300 trips within reasonable computing time.     

A heuristic solution procedure for the dynamic lot sizing problem with remanufacturing and product recovery

Here we discuss the lot sizing problem of product returns and remanufacturing. Let us consider a forecast of demands and product returns over a finite planning horizon — the problem is to determine an optimal production plan. This consists of either manufacturing new products or remanufacturing returned units, and in this way meets both demands at minimum costs. The costs of course are the fixed set-up expenses associated with manufacturing and/or remanufacturing lots and also the inventory holding costs of stocks kept on hand.In addition to showing that a general instance of this problem is NP-Hard, we develop an alternative mixed-integer model formulation for this problem and contrast it to the formulation commonly used in the literature. We show that when integrality constraints are relaxed, our formulation obtains better bounds. Our formulation incorporates the fact that every optimal solution can be decomposed into a series of well-structured blocks with distinct patterns in the way in which set-ups for manufacturing and remanufacturing occur. We then construct a dynamic programming based heuristic that exploits the block structure of the optimal solution. We also propose some improvement schemes as well. Finally, our numerical testing shows that the heuristic performs very well as intended.     

Incompleteness and incomparability in preference aggregation: Complexity results

We consider how to combine the preferences of multiple agents despite the presence of incompleteness and incomparability in their preference relations over possible candidates. The set of preference relations of an agent may be incomplete because, for example, there is an ongoing preference elicitation process. There may also be incomparability as this is useful, for example, in multi-criteria scenarios. We focus here on the problem of computing the possible and necessary winners, that is, those candidates which can be, or always are, the most preferred for the agents. Possible and necessary winners are useful in many scenarios including preference elicitation. First, we show that testing possible and necessary winners is in general a computationally intractable problem for STV with unweighted votes and the cup rule with weighted votes, as is providing a good approximation of such sets. Then, we identify general properties of the preference aggregation function, such as independence to irrelevant alternatives, which are sufficient for such sets to be computed in polynomial time. Finally, we show how possible and necessary winners can be used to focus preference elicitation. We show that it is computationally intractable for the cup rule with weighted votes in the worst-case to decide when to terminate elicitation. However, we identify a property of the preference aggregation function that allows us to decide when to terminate elicitation in polynomial time, by focusing on possible and necessary winners.     

On Minimizing the Cost of Location Management in Mobile Environments

A mobile host typically has a home agent that maintains a registry of its current location. This registry is normally updated every time a host changes its current network. The update cost could be reduced using a two-tier update process in which a registry is updated using special agents, called proxy agents. We study the problem of selecting proxy agents to minimize the cost of search associated with this two-tier update approach. We show that the problem can be formulated as p-center or p-median finding problems. We focus on the p-center formulation. Due to the intractability of the problem, we introduce a distributed strategy to solve the general problem and show that it yields an approximate solution for arbitrary networks. We present an implementation of the distributed strategy that produces an optimal solution for ring networks. We prove that the optimal solution for rings is fault tolerant and resilient to topology changes.