Hard Combinatorial Problems Research Articles

Warmstarting of Model-Based Algorithm Configuration

The performance of many hard combinatorial problem solvers depends strongly on their parameter settings, and since manual parameter tuning is both tedious and suboptimal the AI community has recently developed several algorithm configuration (AC) methods to automatically address this problem. While all existing AC methods start the configuration process of an algorithm A from scratch for each new type of benchmark instances, here we propose to exploit information about A's performance on previous benchmarks in order to warmstart its configuration on new types of benchmarks. We introduce two complementary ways in which we can exploit this information to warmstart AC methods based on a predictive model. Experiments for optimizing a flexible modern SAT solver on twelve different instance sets show that our methods often yield substantial speedups over existing AC methods (up to 165-fold) and can also find substantially better configurations given the same compute budget.

Robotic Optimal Assembly Sequence Using Improved Cuckoo Search Algorithm

The demand for manufacturing newer products are increasing day-by-day, keeping this demand in mind many modern manufacturing processes have been evolved to meet the demand and supply the product in time. Even though many modern methods have been evolved, still there is lack in time to meet the consumer’s requirements. This is due to assembly, which takes 20% of cost in manufacturing. To do effective assembly, optimal sequence is required; achieving the optimal assembly sequence is a difficulty process because it is one of them Non Probabilistic (NP) hard combinatorial problems. Achieving an effective optimal assembly sequence involves more than one objective function to develop the fitness equation (number of directional changes, gripper changes, time of assembly etc.), which converts the problem into discrete optimization problem. At the starting stages of assembly planning, researchers implemented mathematical models to achieve the feasible solution. These methods performs very poorly when comes to large part assemblies. Meanwhile, Artificial Intelligence (AI) techniques are evolved to solve the Assembly Sequence Planning (ASP) Problems. Performances of these methods are quite impressive in solving ASP problems, but most of these algorithms fall in local optimal during execution. More over these methods consumes more time for getting optimal solution especially for the more part assemblies. Keeping the above difficulties in mind, in this paper an Improved Cuckoo Search (ICS) algorithm is implemented to obtain the optimal solution. The proposed algorithm is compared by considering two assemblies (wall rack assembly and eccentric milling machine) with the algorithms like Genetic Algorithm (GA), Ant Colony Optimization (ACO), Grey Wolf Optimization (GWO), Particle Swarm Optimization (PSO) algorithm and Hybrid Ant Wolf Algorithm (HAWA). The results of the different algorithms are compared in terms of number of iterations and fitness values with the proposed algorithm. The results show that the proposed algorithm performs better than the compared algorithms.

A Heuristic Algorithm for a Low Autocorrelation Binary Sequence Problem With Odd Length and High Merit Factor

A low autocorrelation binary sequence (LABS) problem is a hard combinatorial problem and its solutions are important in many practical applications. Till now, the largest best-known skew-symmetric sequence with merit factor greater than 9 had a length of 189. In this paper, a new heuristic algorithm is presented for the LABS problem. The proposed algorithm stores promising solutions and this mechanism enables the algorithm to perform local searches on these solutions in a systematic way. Our algorithm was tested on skew-symmetric sequences and the obtained results are compared with results of the state-of-the-art algorithms. The proposed algorithm was able to find some new best-known skew-symmetric solutions with merit factor greater than 9 in sequence lengths over 200. The obtained results improve the suggestion from 1985 (Beenker et al. ) and 1987 (Bernasconi) greatly, where the merit factor is approximately equal to 6 for long skew-symmetric sequences with length up to 199. Now, the largest best-known skew-symmetric sequence with merit factor greater than 9 has the length 225. Additionally, now all merit factors are greater than 8.5 on the interval from 159 up to 225 for odd lengths.

Efficient benchmarking of algorithm configurators via model-based surrogates

The optimization of algorithm (hyper-)parameters is crucial for achieving peak performance across a wide range of domains, ranging from deep neural networks to solvers for hard combinatorial problems. However, the proper evaluation of new algorithm configuration (AC) procedures (or configurators) is hindered by two key hurdles. First, AC scenarios are hard to set up, including the target algorithm to be optimized and the problem instances to be solved. Second, and even more significantly, they are computationally expensive: a single configurator run involves many costly runs of the target algorithm. Here, we propose a benchmarking approach that uses surrogate scenarios, which are computationally cheap while remaining close to the original AC scenarios. These surrogate scenarios approximate the response surface corresponding to true target algorithm performance using a regression model. In our experiments, we construct and evaluate surrogate scenarios for hyperparameter optimization as well as for AC problems that involve performance optimization of solvers for hard combinatorial problems. We generalize previous work by building surrogates for AC scenarios with multiple problem instances, stochastic target algorithms and censored running time observations. We show that our surrogate scenarios capture overall important characteristics of the original AC scenarios from which they were derived, while being much easier to use and orders of magnitude cheaper to evaluate.

Greedy Sampling of Graph Signals

Sampling is a fundamental topic in graph signal processing, having found applications in estimation, clustering, and video compression. In contrast to traditional signal processing, the irregularity of the signal domain makes selecting a sampling set non-trivial and hard to analyze. Indeed, though conditions for graph signal interpolation from noiseless samples exist, they do not lead to a unique sampling set. The presence of noise makes choosing among these sampling sets a hard combinatorial problem. Although greedy sampling schemes are commonly used in practice, they have no performance guarantee. This work takes a twofold approach to address this issue. First, universal performance bounds are derived for the Bayesian estimation of graph signals from noisy samples. In contrast to currently available bounds, they are not restricted to specific sampling schemes and hold for any sampling sets. Second, this paper provides near-optimal guarantees for greedy sampling by introducing the concept of approximate submodularity and updating the classical greedy bound. It then provides explicit bounds on the approximate supermodularity of the interpolation mean-square error showing that it can be optimized with worst-case guarantees using greedy search even though it is not supermodular. Simulations illustrate the derived bound for different graph models and show an application of graph signal sampling to reduce the complexity of kernel principal component analysis.

Co-Bipartite Neighborhood Edge Elimination Orderings

In SODA 2001, Raghavan and Spinrad introduced robust algorithms as a way to solve hard combinatorial graph problems in polynomial time even when the input graph falls slightly outside a graph class for which a polynomial-time algorithm exists. As a leading example, the Maximum Clique problem on unit disk graphs (intersection graphs of unit disks in the plane) was shown to have a robust, polynomial-time algorithm by proving that such graphs admit a co-bipartite neighborhood edge elimination ordering (CNEEO). This begs the question whether other graph classes also admit a CNEEO.In this paper, we answer this question positively, and identify many graph classes that admit a CNEEO, including several graph classes for which no polynomial-time recognition algorithm exists (unless P=NP). As a consequence, we obtain robust, polynomial-time algorithms for Maximum Clique on all identified graph classes.We also prove some negative results, and identify graph classes that do not admit a CNEEO. This implies an almost-perfect dichotomy for subclasses of perfect graphs.

Exploring parallel multi-GPU local search strategies in a metaheuristic framework

Optimization tasks are often complex, CPU-time consuming and usually deal with finding the best (or good enough) solution among alternatives for a given problem. Parallel metaheuristics have been used in many real-world and scientific applications to efficiently solve these kind of problems. Local Search (LS) is an essential component for some metaheuristics and, very often, represents the dominant computational effort accomplished by an algorithm. Several metaheuristic approaches try to adapt traditional LS models to parallel platforms without considering the intrinsic features of the available architectures. In this work, we present a novel local search strategy, so-called Distributed Variable Neighborhood Descent (DVND), specially designed for CPU and multi-GPU environment. Furthermore, a new neighborhood search strategy, so-called Multi Improvement, is introduced, taking advantage of GPU massive parallelism in order to boost up LS procedures. A hard combinatorial problem is considered as case of study, the Minimum Latency Problem (MLP). For tackling this problem, a hybrid metaheuristic algorithm is considered, which combines good quality initial solutions, generated by a Greedy Randomized Adaptive Search Procedures, with a flexible and powerful refinement procedure, inside the scope of an Iterated Local Search. The DVND was compared to the classic local search procedures, producing results that outperformed the best known sequential algorithm found in the literature. The speedups ranged from 7.3 to 13.7, for the larger MLP instances with 500 to 1000 clients. Results demonstrate the effectiveness of the proposed techniques in terms of solution quality, performance and scalability.

A novel chemical reaction optimization for the distributed permutation flowshop scheduling problem with makespan criterion

The Permutation Flowshop Scheduling Problem (PFSP) is among the most investigated scheduling problems in the fields of Operational Research (OR) and management science. During the last six decades, it has gained much attention and interest thanks to its applicability in a variety of domains such as industrial engineering and economics. Recently, the PFSP with multi-factory environment has been proposed in shop scheduling sphere. Since the problem is known to be NP-hard, exact algorithms can be extremely costly, computationally speaking. Chemical Reaction Optimization (CRO) is lastly proposed by Lam and Li (2010) to optimize hard combinatorial problems. Due to its ability to escape from local optima, CRO has demonstrated excellent performance in solving a variety of scheduling problems, such as flexible job-shop scheduling, grid scheduling, network scheduling etc. In such a paper, we address the Distributed Permutation Flowshop Scheduling Problem (DPFSP) with an artificial chemical reaction metaheuristic which objective is to minimize the maximum completion time. In the proposed CRO, the effective NEH heuristic is adapted to generate the initial population of molecules. Furthermore, a well-designed One-Point (OP) crossover and an effective greedy strategy are embedded in the CRO algorithm in order to ameliorate the solution quality. Moreover, the influence of the parameter setting on the CRO algorithm is being investigated on the base of the Taguchi method. To validate the performance of the proposed algorithm, intensive experiments are carried out on 720 large instances which are extended from the well known Taillard benchmark. The results prove the efficiency of the proposed algorithm in comparison with some powerful algorithms. It is also seen that more than 200 best-known solutions are improved.

Reactive Dialectic Search Portfolios for MaxSAT

Metaheuristics have been developed to provide general purpose approaches for solving hard combinatorial problems. While these frameworks often serve as the starting point for the development of problem-specific search procedures, they very rarely work efficiently in their default state. We combine the ideas of reactive search, which adjusts key parameters during search, and algorithm configuration, which fine-tunes algorithm parameters for a given set of problem instances, for the automatic compilation of a portfolio of highly reactive dialectic search heuristics for MaxSAT. Even though the dialectic search metaheuristic knows nothing more about MaxSAT than how to evaluate the cost of a truth assignment, our automatically generated solver defines a new state of the art for random weighted partial MaxSAT instances. Moreover, when combined with an industrial MaxSAT solver, the self-assembled reactive portfolio was able to win four out of nine gold medals at the recent 2016 MaxSAT Evaluation on random, crafted, and industrial partial and weighted-partial MaxSAT instances.

The Symbolic Interior Point Method

Numerical optimization is arguably the most prominent computational framework in machine learning and AI. It can be seen as an assembly language for hard combinatorial problems ranging from classification and regression in learning, to computing optimal policies and equilibria in decision theory, to entropy minimization in information sciences. Unfortunately, specifying such problems in complex domains involving relations, objects and other logical dependencies is cumbersome at best, requiring considerable expert knowledge, and solvers require models to be painstakingly reduced to standard forms. To overcome this, we introduce a rich modeling framework for optimization problems that allows convenient codification of symbolic structure. Rather than reducing this symbolic structure to a sparse or dense matrix, we represent and exploit it directly using algebraic decision diagrams (ADDs). Combining efficient ADD-based matrix-vector algebra with a matrix-free interior-point method, we develop an engine that can fully leverage the structure of symbolic representations to solve convex linear and quadratic optimization problems. We demonstrate the flexibility of the resulting symbolic-numeric optimizer on decision making and compressed sensing tasks with millions of non-zero entries.

Optimal shaping and positioning of energy-efficient buildings

Due to the number of variables and the complexity of objective functions, optimal design of an energy-efficient building is hard combinatorial problem of multi-objective optimisation. Therefore, it is necessary to describe structure and its position in surroundings precisely but by as few variables as possible. This paper presents methodology for finding adequate methodology for defining geometry and orientation of a given building, as well as its elements of importance for energy-efficiency analysis.

Towards a Distributed Implementation of Chemical Reaction Optimization for the Multi-factory Permutation Flowshop Scheduling Problem

The Distributed Permutation Flowshop Scheduling Problem (DPFSP) is one of the most computationally complex problems. It has gained a wide attention not only in theoretical studies but also in manufacturing industry. In recent years, a lot of work has been done and many heuristics and metaheuristics have been proposed to tackle the DPFSP. Unfortunately, all the existing algorithms are centralized despite the fact that the distributed approaches are known to be more practical for the complex scheduling problems ones. Thus, we argue that distributed artificial intelligence techniques, namely Multi-Agent Systems (MAS), offer an appropriate tool to tackle problems of a distributed nature when they are properly designed and implemented. Thanks to their flexibility, adaptively and extensibility; MAS represents a promising variant to achieve a better performance. In this study, by combining the population-based evolutionary searching abilities of Chemical Reaction Optimization (CRO) metaheuristic with the capabilities of MAS in modeling hard combinatorial problems, we suggest an agent-based evolutionary algorithm called CROMAS to effectively solve the DPFSP. We tested our algorithm on well-known benchmark instances and compared its performance with respect to other recent methods. Experiments reveal that CROMAS is very effective and able to provide competitive results.

The Configurable SAT Solver Challenge (CSSC)

It is well known that different solution strategies work well for different types of instances of hard combinatorial problems. As a consequence, most solvers for the propositional satisfiability problem (SAT) expose parameters that allow them to be customized to a particular family of instances. In the international SAT competition series, these parameters are ignored: solvers are run using a single default parameter setting (supplied by the authors) for all benchmark instances in a given track. While this competition format rewards solvers with robust default settings, it does not reflect the situation faced by a practitioner who only cares about performance on one particular application and can invest some time into tuning solver parameters for this application. The new Configurable SAT Solver Competition (CSSC) compares solvers in this latter setting, scoring each solver by the performance it achieved after a fully automated configuration step. This article describes the CSSC in more detail, and reports the results obtained in its two instantiations so far, CSSC 2013 and 2014.

Main Architectures Used in ETL as a Tool, Necessary for the Integration of Data in Large Volumes - a Task in the Field of Digital Preservation of Cultural Heritage

The digital recording of Cultural Heritage (CH) and its metadata relates to the concept for data warehousing systems and their lifecycle. It uses models and development tools of ETL packages. The tasks used to extract metadata from raw digitized CH data often include hard computing problems, including NP-complete problems. This paper describes the current state of digital preservation of Cultural Heritage with an accent on data integration and performance optimization of processes. We show our vision for further solutions for automated optimization of resource utilization in ETL jobs cutting development cost. In parallel, we describe our experiment that uses quantum computation (in simulation mode) to solve hard combinatorial problems as multiple query optimization.

A Column Generation Method for Constructing and Scheduling Multiple Forwarding Trees in Wireless Sensor Networks

This paper considers the problem of jointly constructing and scheduling forwarding trees in a wireless sensor network, each to collect measurements from a group of sensor nodes at a single sink node. The goal is to construct such trees that gather measurements in the most energy efficient manner and with minimal gathering latency. We assume transmissions (carrying measurements) on wireless links interfere with one another, and thus, appropriate link scheduling is required to manage interference. We refer to this problem as forwarding tree construction and scheduling (FTCS). Each tree may be constructed independently, and then, its links are scheduled. However, when all trees are combined together, the shortest and energy efficient schedule may not be guaranteed. Furthermore, a large number of possible forwarding trees for each group of sensors may be considered. Both problems of enumerating forwarding trees and scheduling links for those trees are hard combinatorial problems. This is compounded by the fact that the two problems must be solved jointly, to guarantee the selection of the best forwarding trees that, when their links are scheduled, guarantee a shortest energy efficient schedule. After highlighting the complexity of the FTCS problem, we present a novel primal-dual decomposition method using column generation. We also highlight several challenges we faced when solving the decomposed problem and present efficient techniques for mitigating those challenges. One major advantage of this paper is that it can serve as a benchmark for evaluating the performance of any low complexity method for solving the FTCS problem for larger network instances, where no known exact solutions can be found.

A Hyper-Heuristic Based On An Adapter Layer For Transportation Combinatorial Problems

Hyper-heuristics are optimization techniques for solving hard combinatorial problems. Their main feature is that their design involves an important decoupling of the search components from the problem domain ones. This allows them to extend their applicability to different problem domains without major redesign, unlike traditional methods such as metaheuristics. In this work, a hyper-heuristic is evaluated for a transportation problem. The implemented hyper-heuristic uses a greedy operator, and it implements an adapter layer that would allow it to be used in other similar problems. Experimental results shows balanced solution quality and CPU time performance, regarding other metaheuristics in literature.

Large Sets of Orthogonal Sequences Suitable for Applications in CDMA Systems

In this paper, we employ the so-called semi-bent functions to achieve significant improvements over currently known methods regarding the number of orthogonal sequences per cell that can be assigned to a regular tessellation of hexagonal cells, typical for certain code-division multiple-access (CDMA) systems. Our initial design method generates a large family of orthogonal sets of sequences derived from vectorial semi-bent functions. A modification of the original approach is proposed to avoid a hard combinatorial problem of allocating several such orthogonal sets to a single cell of a regular hexagonal network, while preserving the orthogonality to adjacent cells. This modification increases the number of users per cell by starting from shorter codewords and then extending the length of these codewords to the desired length. The specification and assignment of these orthogonal sets to a regular tessellation of hexagonal cells have been solved regardless of the parity and size of $m$ (where $2^m$ is the length of the codewords). In particular, when the re-use distance is $D=4$ the number of users per cell is $2^{m-2}$ for almost all $m$, which is twice as many as can be obtained by the best known methods.

Parallelization of the Branch-and-Bound Algorithm in Transportation Discrete Network Design

Transportation Discrete Network Design Problem (TDNDP) aims at choosing a subset of proposed projects to minimize the users’ total travel time with respect to budget constraint. Because TDNDP is a hard combinatorial problem, recent research has widely addressed heuristic approaches and ignored the exact solution. This paper is going to explore how application of parallel computation can affect the performance of an exact algorithm in TDNDP. First, we show that the Branch-and-Bound (B&B) algorithm proposed by LeBlanc is well adapted to a parallel design with synchronized Master-Slave (MS) paradigm. Then we develop a parallel B&B algorithm and implement it with two search strategies of Depth-First-Search (DFS) and Best-First-Search (BFS). Detailed results over up to 16 processing cores are reported and discussed in an illustrative example of the Chicago Sketch network. The results suggest an almost linear speedup for both strategies which slightly drops as more processing cores are added. When using 16 processing cores the speedup values of 11.80 and 12.20 are achieved for DFS and BFS strategies respectively. Furthermore, the BFS strategy reveals a very fast parallel performance by finding the optimal solution via the minimum computational effort.

Multi-objective Truss Optimization Using Different Types of the BB-BC Algorithm

Optimum design of truss structures is considered as a benchmark problem in the field of the structural optimization. In order to solve this hard combinatorial problem, it is necessary to implement adequate optimization tool that would provide sufficiently wide range of possible solutions within a reasonable time as well as to obtain good exploration and exploitation of search space. The aim of presented study was to compare efficiency of different multi-objective algorithms in solving this task.

Multi-objective Truss Optimization Using Different Types of the BB-BC Algorithm

Optimum design of truss structures is considered as a benchmark problem in the field of the structural optimization. In order to solve this hard combinatorial problem, it is necessary to implement adequate optimization tool that would provide sufficiently wide range of possible solutions within a reasonable time as well as to obtain good exploration and exploitation of search space. The aim of presented study was to compare efficiency of different multi-objective algorithms in solving this task.