A study of heuristic combinations for hyper-heuristic systems for the uncapacitated examination timetabling problem

Research in the domain of examination timetabling is moving towards developing methods that generalise well over a range of problems. This is achieved by implementing hyper-heuristic systems to find the best heuristic or heuristic combination to allocate examinations when constructing a timetable for a problem. Heuristic combinations usually take the form of a list of low-level heuristics that are applied sequentially. This study proposes an alternative representation for heuristic combinations, namely, a hierarchical combination of heuristics. Furthermore, the heuristics in each combination are applied simultaneously rather than sequentially. The study also introduces a new low-level heuristic, namely, highest cost. A set of heuristic combinations of this format have been tested on the 13 Carter benchmarks. The quality of the examination timetables induced using these combinations are comparable to, and in some cases better than, those produced by hyper-heuristic systems combining and applying heuristic combinations sequentially.     

Linear combinations of heuristics for examination timetabling

Although they are simple techniques from the early days of timetabling research, graph colouring heuristics are still attracting significant research interest in the timetabling research community. These heuristics involve simple ordering strategies to first select and colour those vertices that are most likely to cause trouble if deferred until later. Most of this work used a single heuristic to measure the difficulty of a vertex. Relatively less attention has been paid to select an appropriate colour for the selected vertex. Some recent work has demonstrated the superiority of combining a number of different heuristics for vertex and colour selection. In this paper, we explore this direction and introduce a new strategy of using linear combinations of heuristics for weighted graphs which model the timetabling problems under consideration. The weights of the heuristic combinations define specific roles that each simple heuristic contributes to the process of ordering vertices. We include specific explanations for the design of our strategy and present the experimental results on a set of benchmark real world examination timetabling problem instances. New best results for several instances have been obtained using this method when compared with other constructive methods applied to this benchmark dataset.     

Register Allocation in Structured Programs

In this article we look at the register allocation problem. In the literature this problem is frequently reduced to the general graph coloring problem and the solutions to the problem are obtained from graph coloring heuristics. Hence, no algorithm with a good performance guarantee is known. Here we show that when attention is restricted tostructured programswhich we define to be programs whose control-flow graphs are series-parallel, there is an efficient algorithm that produces a solution which is within a factor of 2 of the optimal solution. We note that even with the previous restriction the resulting coloring problem is NP-complete.We also consider how to delete a minimum number of edges from arbitrary control-flow graphs to make them series-parallel and to apply our algorithm. We show that this problem is Max SNP hard. However, we define the notion of anapproximate articulation pointand we give efficient algorithms to find approximate articulation points. We present a heuristic for the edge deletion problem based on this notion which seems to work well when the given graph is close to series-parallel.     

A graph-based hyper-heuristic for educational timetabling problems

This paper presents an investigation of a simple generic hyper-heuristic approach upon a set of widely used constructive heuristics (graph coloring heuristics) in timetabling. Within the hyper-heuristic framework, a tabu search approach is employed to search for permutations of graph heuristics which are used for constructing timetables in exam and course timetabling problems. This underpins a multi-stage hyper-heuristic where the tabu search employs permutations upon a different number of graph heuristics in two stages. We study this graph-based hyper-heuristic approach within the context of exploring fundamental issues concerning the search space of the hyper-heuristic (the heuristic space) and the solution space. Such issues have not been addressed in other hyper-heuristic research. These approaches are tested on both exam and course benchmark timetabling problems and are compared with the fine-tuned bespoke state-of-the-art approaches. The results are within the range of the best results reported in the literature. The approach described here represents a significantly more generally applicable approach than the current state of the art in the literature. Future work will extend this hyper-heuristic framework by employing methodologies which are applicable on a wider range of timetabling and scheduling problems.     

Adaptive automated construction of hybrid heuristics for exam timetabling and graph colouring problems

In this paper, we present a random iterative graph based hyper-heuristic to produce a collection of heuristic sequences to construct solutions of different quality. These heuristic sequences can be seen as dynamic hybridisations of different graph colouring heuristics that construct solutions step by step. Based on these sequences, we statistically analyse the way in which graph colouring heuristics are automatically hybridised. This, to our knowledge, represents a new direction in hyper-heuristic research. It is observed that spending the search effort on hybridising Largest Weighted Degree with Saturation Degree at the early stage of solution construction tends to generate high quality solutions. Based on these observations, an iterative hybrid approach is developed to adaptively hybridise these two graph colouring heuristics at different stages of solution construction. The overall aim here is to automate the heuristic design process, which draws upon an emerging research theme on developing computer methods to design and adapt heuristics automatically. Experimental results on benchmark exam timetabling and graph colouring problems demonstrate the effectiveness and generality of this adaptive hybrid approach compared with previous methods on automatically generating and adapting heuristics. Indeed, we also show that the approach is competitive with the state of the art human produced methods.     

Four classes of perfectly orderable graphs

A graph is called “perfectly orderable” if its vertices can be ordered in such a way that, for each induced subgraph F, a certain “greedy” coloring heuristic delivers an optimal coloring of F. No polynomial-time algorithm to recognize these graphs is known. We present four classes of perfectly orderable graphs: Welsh–Powell perfect graphs, Matula perfect graphs, graphs of Dilworth number at most three, and unions of two threshold graphs. Graphs in each of the first three classes are recognizable in a polynomial time. In every graph that belongs to one of the first two classes, we can find a largest clique and an optimal coloring in a linear time.     

On the resolution of certain discrete univariate max–min problems

Graph coloring is a classical NP-hard combinatorial optimization problem with many practical applications. A broad range of heuristic methods exist for tackling the graph coloring problem: from fast greedy algorithms to more time-consuming metaheuristics. Although the latter produce better results in terms of minimizing the number of colors, the former are widely employed due to their simplicity. These heuristic methods are centralized since they operate by using complete knowledge of the graph. However, in real-world environmets where each component only interacts with a limited number of other components, the only option is to apply decentralized methods. This paper explores a novel and simple algorithm for decentralized graph coloring that uses a fixed number of colors and iteratively reduces the edge conflicts in the graph. We experimentally demonstrate that, for most of the tested instances, the new algorithm outperforms a recent and very competitive algorithm for decentralized graph coloring in terms of coloring quality. In our experiments, the fixed number of colors used by the new algorithm is controlled in a centralized manner.     

Circular-arc graph coloring: On chords and circuits in the meeting graph

In compilers register allocation in loops is usually performed by coloring a corresponding circular-arc graph. Generally, the problem of finding the chromatic number of circular-arc graphs is known to be NP-complete. Thus, approximation algorithms should be considered. In this paper we propose heuristics based on decomposition of a so called meeting graph into a set of circuits. We explain the importance of the meeting graph for our solutions and prove properties of our decomposition of the graph into circuits. We derive inequalities relating the number of circuits in the decomposition to the size of the maximum stable set of chords, and present experimental results. Finally, we discuss the quality of our heuristics for circular-arc graph coloring.     

Solving Graph Coloring Problems with the Douglas-Rachford Algorithm

We present the Douglas-Rachford algorithm as a successful heuristic for solving graph coloring problems. Given a set of colors, these types of problems consist in assigning a color to each node of a graph, in such a way that every pair of adjacent nodes are assigned with different colors. We formulate the graph coloring problem as an appropriate feasibility problem that can be effectively solved by the Douglas-Rachford algorithm, despite the nonconvexity arising from the combinatorial nature of the problem. Different modifications of the graph coloring problem and applications are also presented. The good performance of the method is shown in various computational experiments.     

A GRASP for Coloring Sparse Graphs

We first present a literature review of heuristics and metaheuristics developed for the problem of coloring graphs. We then present a Greedy Randomized Adaptive Search Procedure (GRASP) for coloring sparse graphs. The procedure is tested on graphs of known chromatic number, as well as random graphs with edge probability 0.1 having from 50 to 500 vertices. Empirical results indicate that the proposed GRASP implementation compares favorably to classical heuristics and implementations of simulated annealing and tabu search. GRASP is also found to be competitive with a genetic algorithm that is considered one of the best currently available for graph coloring.     

A class of heuristics for the constrained forest problem

The constrained forest problem seeks a minimum-weight spanning forest in an undirected edge-weighted graph such that each tree spans at least a specified number of vertices. We present a structured class of greedy heuristics for this NP-hard problem, and identify the best heuristic.     

Combining Swaps and Node Weights in an Adaptive Greedy Approach for the Maximum Clique Problem

In this work, the NP-hard maximum clique problem on graphs is considered. Starting from basic greedy heuristics, modifications and improvements are proposed and combined in a two-phase heuristic procedure. In the first phase an improved greedy procedure is applied starting from each node of the graph; on the basis of the results of this phase a reduced subset of nodes is selected and an adaptive greedy algorithm is repeatedly started to build cliques around such nodes. In each restart the selection of nodes is biased by the maximal clique generated in the previous execution. Computational results are reported on the DIMACS benchmarks suite. Remarkably, the two-phase procedure successfully solves the difficult Brockington-Culberson instances, and is generally competitive with state-of-the-art much more complex heuristics.     

最大度为5的平面图是11-线性可染的

A linear coloring of a graph G is a proper vertex coloring such that the graph induced by the vertices of any two color classes is the union of vertex-disjoint paths.The linear chromatic number lc(G) o...     

Generalized hyper-heuristics for solving 2D Regular and Irregular Packing Problems

The idea behind hyper-heuristics is to discover some combination of straightforward heuristics to solve a wide range of problems. To be worthwhile, such a combination should outperform the single heuristics. This article presents a GA-based method that produces general hyper-heuristics that solve two-dimensional regular (rectangular) and irregular (convex polygonal) bin-packing problems. A hyper-heuristic is used to define a high-level heuristic that controls low-level heuristics. The hyper-heuristic should decide when and where to apply each single low-level heuristic, depending on the given problem state. In this investigation two kinds of heuristics were considered: for selecting the figures (pieces) and objects (bins), and for placing the figures into the objects. Some of the heuristics were taken from the literature, others were adapted, and some other variations developed by us. We chose the most representative heuristics of their type, considering their individual performance in various studies and also in an initial experimentation on a collection of benchmark problems. The GA included in the proposed model uses a variable-length representation, which evolves combinations of condition-action rules producing hyper-heuristics after going through a learning process which includes training and testing phases. Such hyper-heuristics, when tested with a large set of benchmark problems, produce outstanding results for most of the cases. The testbed is composed of problems used in other similar studies in the literature. Some additional instances for the testbed were randomly generated.     

平面图线性色数的一个上界

如果图G的一个正常染色满足染任意两种颜色的顶点集合导出的子图是一些点不交的路的并,则称这个正常染色为图G的线性染色．图G的线性色数用lc（G）表示,是指G的所有线性染色中所用的最少颜色的个数．证明了：若G是一个最大度△（G）≠5,6的平面图,则lc（G）≤2△（G）．     

Tabu search for the BWC problem

Given a graph G and positive integers B and W, the BWC problem asks about the existence of a coloring of G, with B black and W white vertices, such that there is no edge between a black and a white vertex. We suggest a heuristic, based on tabu search, which yields quite good results for this problem. We compare the performance of our algorithm to that of several other known heuristics, and also to what one might expect based on some theoretical results we obtained for the checked graphs.     

A semidefinite programming-based heuristic for graph coloring

The Lovász θ-function is a well-known polynomial lower bound on the chromatic number. Any near-optimal solution of its semidefinite programming formulation carries valuable information on how to color the graph. A self-contained presentation of the role of this formulation in obtaining heuristics for the graph coloring problem is presented. These heuristics could be useful for coloring medium sized graphs as numerical results on DIMACS benchmark graphs indicate.     

An adaptive memory algorithm for the k-coloring problem

Let G=(V,E) be a graph with vertex set V and edge set E. The k-coloring problem is to assign a color (a number chosen in {1,…,k}) to each vertex of G so that no edge has both endpoints with the same color. The adaptive memory algorithm is a hybrid evolutionary heuristic that uses a central memory. At each iteration, the information contained in the central memory is used for producing an offspring solution which is then possibly improved using a local search algorithm. The so obtained solution is finally used to update the central memory. We describe in this paper an adaptive memory algorithm for the k-coloring problem. Computational experiments give evidence that this new algorithm is competitive with, and simpler and more flexible than, the best known graph coloring algorithms.     

A single-resource revenue management problem with random resource consumptions

We study a single-resource multi-class revenue management problem where the resource consumption for each class is random and only revealed at departure. The model is motivated by cargo revenue management problems in the airline and other shipping industries. We study how random resource consumption distribution affects the optimal expected profit and identify a preference acceptance order on classes. For a special case where the resource consumption for each class follows the same distribution, we fully characterize the optimal control policy. We then propose two easily computable heuristics: (i) a class-independent heuristic through parameter scaling, and (ii) a decomposition heuristic that decomposes the dynamic programming formulation into a collection of one-dimensional problems. We conduct extensive numerical experiments to investigate the performance of the two heuristics and compared them with several widely studied heuristic policies. Our results show that both heuristics work very well, with class-independent heuristic slightly better between the two. In particular, they consistently outperform heuristics that ignore demand and/or resource consumption uncertainty. Our results demonstrate the importance of considering random resource consumption as another problem dimension in revenue management applications.     

Tutorial on Surrogate Constraint Approaches for Optimization in Graphs

Surrogate constraint methods have been embedded in a variety of mathematical programming applications over the past thirty years, yet their potential uses and underlying principles remain incompletely understood by a large segment of the optimization community. In a number of significant domains of combinatorial optimization, researchers have produced solution strategies without recognizing that they can be derived as special instances of surrogate constraint methods. Once the connection to surrogate constraint ideas is exposed, additional ways to exploit this framework become visible, frequently offering opportunities for improvement.We provide a tutorial on surrogate constraint approaches for optimization in graphs, illustrating the key ideas by reference to independent set and graph coloring problems, including constructions for weighted independent sets which have applications to associated covering and weighted maximum clique problems. In these settings, the surrogate constraints can be generated relative to well-known packing and covering formulations that are convenient for exposing key notions. The surrogate constraint approaches yield widely used heuristics for identifying independent sets as simple special cases, and also afford previously unidentified heuristics that have greater power in these settings. Our tutorial also shows how the use of surrogate constraints can be placed within the context of vocabulary building strategies for independent set and coloring problems, providing a framework for applying surrogate constraints that can be used in other applications.At a higher level, we show how to make use of surrogate constraint information, together with specialized algorithms for solving associated sub-problems, to obtain stronger objective function bounds and improved choice rules for heuristic or exact methods. The theorems that support these developments yield further strategies for exploiting surrogate constraint relaxations, both in graph optimization and integer programming generally.