Combinatorial Search Problems Research Articles

Intrinsic Prioritization in Answer Set Programming Based on an Adapted Notion of Tolerance

Answer set programming (ASP) is a declarative programming language suited to solve complex combinatorial search problems. Prioritized ASP is the subdiscipline of ASP which aims at prioritizing the models (answer sets) of ASP programs. Common approaches to prioritized ASP require an additional input, sometimes expressed as special literals within the ASP program, that guides the prioritization of the answer sets. In this paper, to the best of our knowledge, we propose the first approach that is able to prioritize answer sets solely based on the original ASP program. For this, we adapt the notion of tolerance from conditional reasoning according to System Z and establish tolerance partitions of ASP programs which can be used for prioritization.

Efficient Transit Network Design, Frequency Adjustment, and Fleet Calculation Using Genetic Algorithms

The main objective of this study was to implement a computational prototype in two stages: the first stage primarily focused on generating efficient routes based on an evolutionary algorithm. In other words, the complex computational problem was solved in the first stage. The second stage then shifted its focus towards determining the fleet size and frequencies using an allocation algorithm. This approach was designed to address the complex combinatorial search problem within a public transportation network. In the first stage, the prototype utilizes the metaheuristic known as Genetic Algorithms (GA). Within the GA operators, an innovative method called "aggregated crossover" is employed, with an additional mutation procedure that maintains feasible descendants. In the second stage, an allocation algorithm is used, taking into account the routes generated in the first stage. The results demonstrate that in the first stage, the GA metaheuristic consistently delivers highly efficient routes in each run, confirming that the combinatorial complexity of the problem is effectively resolved in this initial phase. These results were validated on Mandl's Swiss Road network, showing superior solutions compared to those presented in previous studies. Notably, the execution time for this process is only 35 minutes.

Certified Dominance and Symmetry Breaking for Combinatorial Optimisation

Symmetry and dominance breaking can be crucial for solving hard combinatorial search and optimisation problems, but the correctness of these techniques sometimes relies on subtle arguments. For this reason, it is desirable to produce efficient, machine-verifiable certificates that solutions have been computed correctly. Building on the cutting planes proof system, we develop a certification method for optimisation problems in which symmetry and dominance breaking is easily expressible. Our experimental evaluation demonstrates that we can efficiently verify fully general symmetry breaking in Boolean satisfiability (SAT) solving, thus providing, for the first time, a unified method to certify a range of advanced SAT techniques that also includes cardinality and parity (XOR) reasoning. In addition, we apply our method to maximum clique solving and constraint programming as a proof of concept that the approach applies to a wider range of combinatorial problems.

System Predictor: Grounding Size Estimator for Logic Programs under Answer Set Semantics

AbstractAnswer set programming is a declarative logic programming paradigm geared towards solving difficult combinatorial search problems. While different logic programs can encode the same problem, their performance may vary significantly. It is not always easy to identify which version of the program performs the best. We present the system predictor (and its algorithmic backend) for estimating the grounding size of programs, a metric that can influence a performance of a system processing a program. We evaluate the impact of predictor when used as a guide for rewritings produced by the answer set programming rewriting tools projector and lpopt. The results demonstrate potential to this approach.

Distributed randomized multiagent policy iteration in reinforcement learning

We propose a distributed randomized policy iteration algorithm for infinite horizon dynamic programming problems for which the control at each stage is m-dimensional. The traditional policy iteration algorithm involves performing a minimization over an m-dimensional constraint set and has a computational complexity that increases exponentially in m, resulting in an intractable combinatorial search problem. In each iteration, our algorithm performs a series of sequential minimizations followed by policy evaluation and policy improvement using the policy that attains the minimum cost over the sequential minimizations. Our algorithm is well-suited for parallel computation, has a complexity that increases linearly in m, and converges to an agent-by-agent optimal policy. We characterize sufficient conditions for which our algorithm generates a globally optimal policy that coincides with that obtained from standard policy iteration.

Memetic micro-genetic algorithms for cancer data classification

Fast and precise medical diagnosis of human cancer is crucial for treatment decisions. Gene selection consists of identifying a set of informative genes from microarray data to allow high predictive accuracy in human cancer classification. This task is a combinatorial search problem, and optimisation methods can be applied for its resolution. In this paper, two memetic micro-genetic algorithms (MμV1 and MμV2) with different hybridisation approaches are proposed for feature selection of cancer microarray data. Seven gene expression datasets are used for experimentation. The comparison with stochastic state-of-the-art optimisation techniques concludes that problem-dependent local search methods combined with micro-genetic algorithms improve feature selection of cancer microarray data.

Vector Symbolic Architectures as a Computing Framework for Emerging Hardware

This article reviews recent progress in the development of the computing framework vector symbolic architectures (VSA) (also known as hyperdimensional computing). This framework is well suited for implementation in stochastic, emerging hardware, and it naturally expresses the types of cognitive operations required for artificial intelligence (AI). We demonstrate in this article that the field-like algebraic structure of VSA offers simple but powerful operations on high-dimensional vectors that can support all data structures and manipulations relevant to modern computing. In addition, we illustrate the distinguishing feature of VSA, “computing in superposition,” which sets it apart from conventional computing. It also opens the door to efficient solutions to the difficult combinatorial search problems inherent in AI applications. We sketch ways of demonstrating that VSA are computationally universal. We see them acting as a framework for computing with distributed representations that can play a role of an abstraction layer for emerging computing hardware. This article serves as a reference for computer architects by illustrating the philosophy behind VSA, techniques of distributed computing with them, and their relevance to emerging computing hardware, such as neuromorphic computing.

Generalized Longest Path Problems

The longest simple path and snake-in-a-box are combinatorial search problems of considerable research interest. We create a common framework of longest constrained path in a graph that contains these two problems, as well as other interesting maximum path problems, as special cases. We analyze properties of this general framework, produce bounds on the path length that can be used as admissible heuristics for all problem types therein. For the special cases of longest simple path and snakes, these heuristics are shown to reduce the number of expansions when searching for a maximal path, which in some cases leads to reduced search time despite the significant overhead of computing these heuristics.

Certified Symmetry and Dominance Breaking for Combinatorial Optimisation

Symmetry and dominance breaking can be crucial for solving hard combinatorial search and optimisation problems, but the correctness of these techniques sometimes relies on subtle arguments. For this reason, it is desirable to produce efficient, machine-verifiable certificates that solutions have been computed correctly. Building on the cutting planes proof system, we develop a certification method for optimisation problems in which symmetry and dominance breaking are easily expressible. Our experimental evaluation demonstrates that we can efficiently verify fully general symmetry breaking in Boolean satisfiability (SAT) solving, thus providing, for the first time, a unified method to certify a range of advanced SAT techniques that also includes XOR and cardinality reasoning. In addition, we apply our method to maximum clique solving and constraint programming as a proof of concept that the approach applies to a wider range of combinatorial problems.

A Parallelizable Integer Linear Programming Approach for Tiling Finite Regions of the Plane with Polyominoes

The general problem of tiling finite regions of the plane with polyominoes is NP-complete, and so the associated computational geometry problem rapidly becomes intractable for large instances. Thus, the need to reduce algorithm complexity for tiling is important and continues as a fruitful area of research. Traditional approaches to tiling with polyominoes use backtracking, which is a refinement of the ‘brute-force’ solution procedure for exhaustively finding all solutions to a combinatorial search problem. In this work, we combine checkerboard colouring techniques with a recently introduced integer linear programming (ILP) technique for tiling with polyominoes. The colouring arguments often split large tiling problems into smaller subproblems, each represented as a separate ILP problem. Problems that are amenable to this approach are embarrassingly parallel, and our work provides proof of concept of a parallelizable algorithm. The main goal is to analyze when this approach yields a potential parallel speedup. The novel colouring technique shows excellent promise in yielding a parallel speedup for finding large tiling solutions with ILP, particularly when we seek a single (optimal) solution. We also classify the tiling problems that result from applying our colouring technique according to different criteria and compute representative examples using a combination of MATLAB and CPLEX, a commercial optimization package that can solve ILP problems. The collections of MATLAB programs PARIOMINOES (v3.0.0) and POLYOMINOES (v2.1.4) used to construct the ILP problems are freely available for download.

Strong Equivalence and Program Structure in Arguing Essential Equivalence between Logic Programs

AbstractAnswer set programming is a prominent declarative programming paradigm used in formulating combinatorial search problems and implementing different knowledge representation formalisms. Frequently, several related and yet substantially different answer set programs exist for a given problem. Sometimes these encodings may display significantly different performance. Uncovering precise formal links between these programs is often important and yet far from trivial. This paper presents formal results carefully relating a number of interesting program rewritings. It also provides the proof of correctness of system projector concerned with automatic program rewritings for the sake of efficiency.

Flexible and Explainable Solutions for Multi-Agent Path Finding Problems

The multi-agent path finding (MAPF) problem is a combinatorial search problem that aims at finding paths for multiple agents (e.g., robots) in an environment (e.g., an autonomous warehouse) such that no two agents collide with each other, and subject to some constraints on the lengths of paths. The real-world applications of MAPF require flexibility (e.g., solving variations of MAPF) as well as explainability. In this study, both of these challenges are addressed and some flexible and explainable solutions for MAPF and its variants are introduced.

Solution Enumeration by Optimality in Answer Set Programming

AbstractGiven a combinatorial search problem, it may be highly useful to enumerate its (all) solutions besides just finding one solution, or showing that none exists. The same can be stated about optimal solutions if an objective function is provided. This work goes beyond the bare enumeration of optimal solutions and addresses the computational task of solution enumeration by optimality (SEO). This task is studied in the context of answer set programming (ASP) where (optimal) solutions of a problem are captured with the answer sets of a logic program encoding the problem. Existing answer set solvers already support the enumeration of all (optimal) answer sets. However, in this work, we generalize the enumeration of optimal answer sets beyond strictly optimal ones, giving rise to the idea of answer set enumeration in the order of optimality (ASEO). This approach is applicable up to the best k answer sets or in an unlimited setting, which amounts to a process of sorting answer sets based on the objective function. As the main contribution of this work, we present the first general algorithms for the aforementioned tasks of answer set enumeration. Moreover, we illustrate the potential use cases of ASEO. First, we study how efficiently access to the next-best solutions can be achieved in a number of optimization problems that have been formalized and solved in ASP. Second, we show that ASEO provides us with an effective sampling technique for Bayesian networks.

Batch Repair with Heuristic Search

Recent work has raised the challenge of efficient automated troubleshooting in domains where repairing a set of components in a single repair action is cheaper than repairing each of them separately. This corresponds to cases where there is a non-negligible overhead to initiating a repair action and to testing the system after a repair action. The problem can be formalized as a combinatorial search problem, propose a new objective function to optimize, and investigate several search frameworks to solve it. The resulting search space is not monotone, but we are able to devise an admissible heuristic for it that enables solving it optimally in some cases with A*. Empirical evaluation on standard model-based diagnosis benchmark systems compare the A*-based approach with other search algorithms

All PSPACE-Complete Planning Problems Are Equal but Some Are More Equal than Others

Complexity analysis of planning is problematic. Even very simple planning languages are PSPACE-complete, yet cannot model many simple problems naturally. Many languages with much more powerful features are also PSPACE-complete. It is thus difficult to separate planning languages in a useful way and to get complexity figures that better reflect reality. This paper introduces new methods for complexity analysis of planning and similar combinatorial search problems, in order to achieve more precision and complexity separations than standard methods allow. Padding instances with the solution size yields a complexity measure that is immune to this factor and reveals other causes of hardness, that are otherwise hidden. Further combining this method with limited non-determinism improves the precision, making even finer separations possible. We demonstrate with examples how these methods can narrow the gap between theory and practice.

Counting Vertex-Disjoint Shortest Paths in Graphs

Finding a shortest path in a graph is at the core of many combinatorial search problems. A closely related problem refers to counting the number of shortest paths between two nodes. Such problems are solvable in polynomial time in the size of the graph. However, more realistic problem formulations could additionally specify constraints to satisfy. We study the problem of counting the shortest paths that are vertex disjoint and can satisfy additional constraints. Specifically, we look at the problems of counting vertex-disjoint shortest paths in edge-colored graphs, counting vertex-disjoint shortest paths with directional constraints, and counting vertex-disjoint shortest paths between multiple source-target pairs. We give a detailed theoretical analysis, and show formally that all of these three counting problems are NP-complete in general.

Глобальные ограничения при моделировании и решении задач в рамках парадигмы Constraint Programming.

At the moment, constraint programming technology is a powerful tool for solving combinatorial search and combinatorial optimization problems. To use this technology, any task must be formulated as a task of satisfying constraints. The role of the concept of global constraints in modeling and solving applied problems within the framework of the constraint programming paradigm can hardly be overestimated. The procedures that implement the algorithms of filtering global constraints are the elementary “building blocks” from which the model of a specific applied problem is built. Algorithms for filtering global constraints, as a rule, are supported by the corresponding developed theories that allow organizing high-performance computing. The choice of a particular software library is primarily determined by the extent to which the set and method of implementing global constraints corresponds tothe level of modern research in this area. The main focus of this article is focused on an overview of global constraints that are implemented within the most popular constraint programming libraries: Choco, GeCode, JaCoP, MiniZinc.

A Memetic Cellular Genetic Algorithm for Cancer Data Microarray Feature Selection

Gene selection aims at identifying a -small- subset of informative genes from the initial data to obtain high predictive accuracy for classification in human cancers. Gene selection can be considered as a combinatorial search problem and thus can be conveniently handled with optimization methods. This paper proposes a Memetic Cellular Genetic Algorithm (MCGA) to solve the Feature Selection problem of cancer microarray datasets. Benchmark gene expression datasets, i.e., colon, lymphoma, and leukaemia available in the literature were used for experimentation. MCGA is compared with other well-known metaheuristic' strategies. The results demonstrate that our proposal can provide efficient solutions to find a minimal subset of the genes.

Neuro-Symbolic Integration of Hopfield Neural Network for Optimal Maximum Random kSatisfiability (Maxrksat) Representation

Boolean satisfiability logical representation is a programming paradigm that has its foundations in mathematical logic. It has been classified as an NP-complete problem that difficult practical combinatorial optimization and search problems can be easily converted into it. Random Maximum kSatisfiability (MAX-RkSAT) composed of the most consistent mapping in a Boolean formula that generates a maximum number of random satisfied clauses. Many optimization and search problems can be easily expressed by mapping the problem into a Hopfield neural network (HNN) to minimize the optimal configuration of the corresponding Lyapunov energy function. In this paper, a hybrid computational model hs been proposed that incorporates the Random Maximum kSatisfiability (MAX-RkSAT) into the Hopfield neural network (HNN) for optimal Random Maximum kSatisfiability representation (HNN-MAX-RkSAT). Hopfield neural network learning will be integrated with the random maximum satisfiability to enhance the correct neural state of the network model representation. The computer simulation using C++ has been used to demonstrate the ability of MAX-RkSAT to be embedded optimally in Hopfield neural network to serve as Neuro-symbolic integration. The performance of the proposed hybrid HNN-MAXRkSAT model has been explored and compared with the existing model. The proposed HNN-MAXRkSAT demonstrates good agreement with the existing models measured in terms of Global minimum Ratio (Gm), Hamming Distance (HD), Mean Absolute Error (MAE) and network computation Time CPU time). The proposed framework explored that MAX-RkSAT can be optimally represented in HNN and subsequently provides an additional platform for neural-symbolic integration, representing the various types of satisfiability logic.

Resonator Networks, 1: An Efficient Solution for Factoring High-Dimensional, Distributed Representations of Data Structures.

The ability to encode and manipulate data structures with distributed neural representations could qualitatively enhance the capabilities of traditional neural networks by supporting rule-based symbolic reasoning, a central property of cognition. Here we show how this may be accomplished within the framework of Vector Symbolic Architectures (VSAs) (Plate, 1991; Gayler, 1998; Kanerva, 1996), whereby data structures are encoded by combining high-dimensional vectors with operations that together form an algebra on the space of distributed representations. In particular, we propose an efficient solution to a hard combinatorial search problem that arises when decoding elements of a VSA data structure: the factorization of products of multiple codevectors. Our proposed algorithm, called a resonator network, is a new type of recurrent neural network that interleaves VSA multiplication operations and pattern completion. We show in two examples-parsing of a tree-like data structure and parsing of a visual scene-how the factorization problem arises and how the resonator network can solve it. More broadly, resonator networks open the possibility of applying VSAs to myriad artificial intelligence problems in real-world domains. The companion article in this issue (Kent, Frady, Sommer, & Olshausen, 2020) presents a rigorous analysis and evaluation of the performance of resonator networks, showing it outperforms alternative approaches.