Breaking Instance-Independent Symmetries In Exact Graph Coloring

Code optimization and high level synthesis can be posed as constraint satisfaction and optimization problems, such as graph coloring used in register allocation. Graph coloring is also used to model more traditional CSPs relevant to AI, such as planning, time-tabling and scheduling. Provably optimal solutions may be desirable for commercial and defense applications. Additionally, for applications such as register allocation and code optimization, naturally-occurring instances of graph coloring are often small and can be solved optimally. A recent wave of improvements in algorithms for Boolean satisfiability (SAT) and 0-1 Integer Linear Programming (ILP) suggests generic problem-reduction methods, rather than problem-specific heuristics, because (1) heuristics may be upset by new constraints, (2) heuristics tend to ignore structure, and (3) many relevant problems are provably inapproximable. Problem reductions often lead to highly symmetric SAT instances, and symmetries are known to slow down SAT solvers. In this work, we compare several avenues for symmetry breaking, in particular when certain kinds of symmetry are present in all generated instances. Our focus on reducing CSPs to SAT allows us to leverage recent dramatic improvement in SAT solvers and automatically benefit from future progress. We can use a variety of black-box SAT solvers without modifying their source code because our symmetry-breaking techniques are static, i.e., we detect symmetries and add symmetry breaking predicates (SBPs) during pre-processing. An important result of our work is that among the types of instance-independent SBPs we studied and their combinations, the simplest and least complete constructions are the most effective. Our experiments also clearly indicate that instance-independent symmetries should mostly be processed together with instance-specific symmetries rather than at the specification level, contrary to what has been suggested in the literature

Analysis of combinatorial search spaces for a class of NP-hard problems, An

2011 Spring.Includes bibliographical references.Given a finite but very large set of states X and a real-valued objective function ƒ defined on X, combinatorial optimization refers to the problem of finding elements of X that maximize (or minimize) ƒ. Many combinatorial search algorithms employ some perturbation operator to hill-climb in the search space. Such perturbative local search algorithms are state of the art for many classes of NP-hard combinatorial optimization problems such as maximum k-satisfiability, scheduling, and problems of graph theory. In this thesis we analyze combinatorial search spaces by expanding the objective function into a (sparse) series of basis functions. While most analyses of the distribution of function values in the search space must rely on empirical sampling, the basis function expansion allows us to directly study the distribution of function values across regions of states for combinatorial problems without the need for sampling. We concentrate on objective functions that can be expressed as bounded pseudo-Boolean functions which are NP-hard to solve in general. We use the basis expansion to construct a polynomial-time algorithm for exactly computing constant-degree moments of the objective function ƒ over arbitrarily large regions of the search space. On functions with restricted codomains, these moments are related to the true distribution by a system of linear equations. Given low moments supplied by our algorithm, we construct bounds of the true distribution of ƒ over regions of the space using a linear programming approach. A straightforward relaxation allows us to efficiently approximate the distribution and hence quickly estimate the count of states in a given region that have certain values under the objective function. The analysis is also useful for characterizing properties of specific combinatorial problems. For instance, by connecting search space analysis to the theory of inapproximability, we prove that the bound specified by Grover's maximum principle for the Max-Ek-Lin-2 problem is sharp. Moreover, we use the framework to prove certain configurations are forbidden in regions of the Max-3-Sat search space, supplying the first theoretical confirmation of empirical results by others. Finally, we show that theoretical results can be used to drive the design of algorithms in a principled manner by using the search space analysis developed in this thesis in algorithmic applications. First, information obtained from our moment retrieving algorithm can be used to direct a hill-climbing search across plateaus in the Max-k-Sat search space. Second, the analysis can be used to control the mutation rate on a (1+1) evolutionary algorithm on bounded pseudo-Boolean functions so that the offspring of each search point is maximized in expectation. For these applications, knowledge of the search space structure supplied by the analysis translates to significant gains in the performance of search

People Efficiently Explore the Solution Space of the Computationally Intractable Traveling Salesman Problem to Find Near-Optimal Tours

Humans need to solve computationally intractable problems such as visual search, categorization, and simultaneous learning and acting, yet an increasing body of evidence suggests that their solutions to instantiations of these problems are near optimal. Computational complexity advances an explanation to this apparent paradox: (1) only a small portion of instances of such problems are actually hard, and (2) successful heuristics exploit structural properties of the typical instance to selectively improve parts that are likely to be sub-optimal. We hypothesize that these two ideas largely account for the good performance of humans on computationally hard problems. We tested part of this hypothesis by studying the solutions of 28 participants to 28 instances of the Euclidean Traveling Salesman Problem (TSP). Participants were provided feedback on the cost of their solutions and were allowed unlimited solution attempts (trials). We found a significant improvement between the first and last trials and that solutions are significantly different from random tours that follow the convex hull and do not have self-crossings. More importantly, we found that participants modified their current better solutions in such a way that edges belonging to the optimal solution (“good” edges) were significantly more likely to stay than other edges (“bad” edges), a hallmark of structural exploitation. We found, however, that more trials harmed the participants' ability to tell good from bad edges, suggesting that after too many trials the participants “ran out of ideas.” In sum, we provide the first demonstration of significant performance improvement on the TSP under repetition and feedback and evidence that human problem-solving may exploit the structure of hard problems paralleling behavior of state-of-the-art heuristics

Lightweight String Reasoning in Model Finding

International audienceModels play a key role in assuring software quality in the model-driven approach. Precise models usually require the definition of well-formedness rules to specify constraints that cannot be expressed graphically. The Object Constraint Language (OCL) is a de-facto standard to define such rules. Techniques that check the satisfiability of such models and find corresponding instances of them are important in various activities, such as model-based testing and validation. Several tools for these activities have been developed, but to our knowledge, none of them supports OCL string operations on scale that is sufficient for, e.g., model-based testing. As, in contrast, many industrial models do contain such operations, there is evidently a gap. We present a lightweight solver that is specifically tailored to generate large solutions for tractable string constraints in model finding, and that is suitable for directly express the main operations of the OCL datatype String. It is based on constraint logic programming (CLP) and constraint handling rules (CHR), and can be seamlessly combined with other constraint solvers in CLP. We have integrated our solver into the EMFtoCSP model finder, and we show that our implementation efficiently solves several common string constraints on a large instances

Fitness function distributions over generalized search neighborhoods in the q-ary hypercube

Evolutionary Computation, 21(4): 561-590, 2013The frequency distribution of a fitness function over regions of its domain is an important quantity for understanding the behavior of algorithms that employ randomized sampling to search the function. In general, exactly characterizing this distribution is at least as hard as the search problem, since the solutions typically live in the tails of the distribution. However, in some cases it is possible to efficiently retrieve a collection of quantities (called moments) that describe the distribution. In this paper, we consider functions of bounded epistasis that are defined over length-n strings from a finite alphabet of cardinality q. Many problems in combinatorial optimization can be specified as search problems over functions of this type. Employing Fourier analysis of functions over finite groups, we derive an efficient method for computing the exact moments of the frequency distribution of fitness functions over Hamming regions of the q-ary hypercube. We then use this approach to derive equations that describe the expected fitness of the offspring of any point undergoing uniform mutation. The results we present provide insight into the statistical structure of the fitness function for a number of combinatorial problems. For the graph coloring problem, we apply our results to efficiently compute the average number of constraint violations that lie within a certain number of steps of any coloring. We derive an expression for the mutation rate that maximizes the expected fitness of an offspring at each fitness level. We also apply the results to the slightly more complex frequency assignment problem, a relevant application in the domain of the telecommunications industry. As with the graph coloring problem, we provide formulas for the average value of the fitness function in Hamming regions around a solution and the expectation-optimal mutation rate.Spanish Ministry of Science and Innovation and FEDER under contract TIN2008-06491-C04-01 (the M∗ project). Andalusian Government under contract P07-TIC-03044 (DIRICOM project). Air Force Office of Scientific Re- search, Air Force Materiel Command, USAF, under grant number FA9550-08-1-0422

Exact computation of the fitness-distance correlation for pseudoboolean functions with one global optimum

Abstract. Landscape theory provides a formal framework in which combinatorial optimization problems can be theoretically characterized as a sum of a special kind of landscapes called elementary landscapes. The decomposition of the objective function of a problem into its elementary components can be exploited to compute summary statistics. We present closed-form expressions for the fitness-distance correlation (FDC) based on the elementary landscape decomposition of the problems defined over binary strings in which the objective function has one global optimum. We present some theoretical results that raise some doubts on using FDC as a measure of problem difficulty

Multi-objectivising Combinatorial Optimisation Problems by means of Elementary Landscape Decompositions

In the last decade, many works in combinatorial optimisation have shown that, due to the advances in multi-objective optimisation, the algorithms from this field could be used for solving single-objective problems as well. In this sense, a number of papers have proposed multi-objectivising single-objective problems in order to use multi-objective algorithms in their optimisation. In this paper, we follow up this idea by presenting a methodology for multi-objectivising combinatorial optimisation prob- lems based on elementary landscape decompositions of their objective function. Under this framework, each of the elementary landscapes obtained from the decomposition is considered as an independent objective function to optimise. In order to illustrate this general methodology, we consider four problems from different domains: the quadratic assignment problem and the linear ordering problem (permutation domain), the 0-1 unconstrained quadratic optimisation problem (binary domain), and the frequency assignment problem (integer domain). We implemented two widely known multi-objective algorithms, NSGA-II and SPEA2, and compared their perfor- mance with that of a single-objective GA. The experiments conducted on a large benchmark of instances of the four problems show that the multi-objective algorithms clearly outperform the single-objective approaches. Furthermore, a discussion on the results suggests that the multi-objective space generated by this decomposition enhances the exploration ability, thus permitting NSGA-II and SPEA2 to obtain better results in the majority of the tested instances.TIN2016-78365R IT-609-1

On Formal Methods for Large-Scale Product Configuration

<p>In product development companies mass customization is widely used to achieve better customer satisfaction while keeping costs down. To efficiently implement mass customization, product platforms are often used. A product platform allows building a wide range of products from a set of predefined components. The process of matching these components to customers' needs is called product configuration. Not all components can be combined with each other due to restrictions of various kinds, for example, geometrical, marketing and legal reasons. Product design engineers develop configuration constraints to describe such restrictions. The number of constraints and the complexity of the relations between them are immense for complex product like a vehicle. Thus, it is both error-prone and time consuming to analyze, author and verify the constraints manually. Software tools based on formal methods can help engineers to avoid making errors when working with configuration constraints, thus design a correct product faster.</p> <p>This thesis introduces a number of formal methods to help engineers maintain, verify and analyze product configuration constraints. These methods provide automatic verification of constraints and computational support for analyzing and refactoring constraints. The methods also allow verifying the correctness of one specific type of constraints, item usage rules, for sets of mutually-exclusive required items, and automatic verification of equivalence of different formulations of the constraints. The thesis also introduces three methods for efficient enumeration of valid partial configurations, with benchmarking of the methods on an industrial dataset.</p> <p>Handling large-scale industrial product configuration problems demands high efficiency from the software methods. This thesis investigates a number of search-based and knowledge-compilation-based methods for working with large product configuration instances, including Boolean satisfiability solvers, binary decision diagrams and decomposable negation normal form. This thesis also proposes a novel method based on supervisory control theory for efficient reasoning about product configuration data. The methods were implemented in a tool, to investigate the applicability of the methods for handling large product configuration problems. It was found that search-based Boolean satisfiability solvers with incremental capabilities are well suited for industrial configuration problems.</p> <p>The methods proposed in this thesis exhibit good performance on practical configuration problems, and have a potential to be implemented in industry to support product design engineers in creating and maintaining configuration constraints, and speed up the development of product platforms and new products.</p

The role of Walsh structure and ordinal linkage in the optimisation of pseudo-Boolean functions under monotonicity invariance.

Optimisation heuristics rely on implicit or explicit assumptions about the structure of the black-box fitness function they optimise. A review of the literature shows that understanding of structure and linkage is helpful to the design and analysis of heuristics. The aim of this thesis is to investigate the role that problem structure plays in heuristic optimisation. Many heuristics use ordinal operators; which are those that are invariant under monotonic transformations of the fitness function. In this thesis we develop a classification of pseudo-Boolean functions based on rank-invariance. This approach classifies functions which are monotonic transformations of one another as equivalent, and so partitions an infinite set of functions into a finite set of classes. Reasoning about heuristics composed of ordinal operators is, by construction, invariant over these classes. We perform a complete analysis of 2-bit and 3-bit pseudo-Boolean functions. We use Walsh analysis to define concepts of necessary, unnecessary, and conditionally necessary interactions, and of Walsh families. This helps to make precise some existing ideas in the literature such as benign interactions. Many algorithms are invariant under the classes we define, which allows us to examine the difficulty of pseudo-Boolean functions in terms of function classes. We analyse a range of ordinal selection operators for an EDA. Using a concept of directed ordinal linkage, we define precedence networks and precedence profiles to represent key algorithmic steps and their interdependency in terms of problem structure. The precedence profiles provide a measure of problem difficulty. This corresponds to problem difficulty and algorithmic steps for optimisation. This work develops insight into the relationship between function structure and problem difficulty for optimisation, which may be used to direct the development of novel algorithms. Concepts of structure are also used to construct easy and hard problems for a hill-climber