Algorithms For Combinatorial Optimization Problems Research Articles

Machine Learning Approaches to Learning Heuristics for Combinatorial Optimization Problems

This paper examines various ways of utilizing machine learning to enhance or develop heuristic and metaheuristic algorithms for combinatorial optimization problems (COPs). A general agent-based framework is introduced, which enables learning both constructive and perturbative heuristics for a wide range of COPs. The framework consists of an agent-based generalized solution algorithm (GSA) to COPs, along with a machine learning architecture and training procedure that would inject intelligence via learning to the agents. A GSA in our framework is an assembly of k agents working in parallel, generating the solution to a given problem one step at a time, either by improving a known solution or by constructing one from scratch. The generated partial solution is then implemented in a simulated or real system that represents the environment, with state changes being fed back to the k agents. In the proposed framework, intelligence can be injected into the agents individually or in groups, via learning from the correlation of their solution decisions with a final objective value. A discussion is also provided on what machine learning approaches to use depending on the availability of training data, which in turn depends on the choice of algorithm architecture.

Self-configuring Nature Inspired Algorithms for Combinatorial Optimization Problems

Self-configuring Nature Inspired Algorithms for Combinatorial Optimization Problems

LP-Based Algorithms for Capacitated Facility Location

Linear programming (LP) has played a key role in the study of algorithms for combinatorial optimization problems. In the field of approximation algorithms, this is well illustrated by the uncapacitated facility location problem. A variety of algorithmic methodologies, such as LP-rounding and the primal-dual method, have been applied to and evolved from algorithms for this problem. Unfortunately, this collection of powerful algorithmic techniques had not yet been applicable to the more general capacitated facility location problem. In fact, all of the known algorithms with good performance guarantees were based on a single technique, local search, and no LP relaxation was known to efficiently approximate the problem. In this paper, we present an LP relaxation with a constant integrality gap for the capacitated facility location. We demonstrate that the fundamental theories of multicommodity flows and matchings provide key insights that lead to the strong relaxation. Our algorithmic proof of integrality gap...

Single Pass Spectral Sparsification in Dynamic Streams

We present the first single pass algorithm for computing spectral sparsifiers for graphs in the dynamic semi-streaming model. Given a single pass over a stream containing insertions and deletions of edges to a graph $G$, our algorithm maintains a randomized linear sketch of the incidence matrix of $G$ into dimension $O (\frac{1}{\epsilon^2}n{polylog} (n))$. Using this sketch, at any point, the algorithm can output a $(1 \pm \epsilon)$ spectral sparsifier for $G$ with high probability. While $O (\frac{1}{\epsilon^2} n{polylog}(n))$ space algorithms are known for computing cut sparsifiers in dynamic streams [K. J. Ahn, S. Guha, and A. McGregor, in Proceedings of the 31st ACM Symposium on Principles of Database Systems, 2012, pp. 5--14; A. Goel, M. Kapralov, and I. Post, \hrefhttp://arXiv.org/abs/1203.4900 arXiv:1203.4900, 2002] and spectral sparsifiers in insertion-only streams [J. A. Kelner and A. Levin, Theory Comput. Syst., 53 (2013), pp. 243--262], prior to our work, the best known single pass algorithm for maintaining spectral sparsifiers in dynamic streams required sketches of dimension $\Omega (\frac{1}{\epsilon^2}n^{5/3})$ [K. J. Ahn, S. Guha, and A. McGregor, in Proceedings of the 16th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, 2013, pp. 1--10]. To achieve our result, we show that using a coarse sparsifier for $G$ and a linear sketch of $G$'s incidence matrix, it is possible to sample edges by effective resistance, obtaining a spectral sparsifier of arbitrary precision. Sampling from the sketch requires a novel application of $\ell_2/\ell_2$ sparse recovery, a natural extension of the $\ell_0$ methods used for cut sparsifiers in [K. J. Ahn, S. Guha, and A. McGregor, in Proceedings of the 31st ACM Symposium on Principles of Database Systems, 2012, pp. 5--14]. Recent work on row sampling for matrix approximation gives a recursive approach for obtaining the required coarse sparsifiers [G. L. Miller and R. Peng, ŭlhttp://arXiv.org/abs/1211.2713v1, 2012]. Under certain restrictions, our approach also extends to the problem of maintaining a spectral approximation for a general matrix $A^\top A$ given a stream of updates to rows in $A$.

Improved NP-Inapproximability for $2$-Variable Linear Equations

$ \newcommand{\twolin}{2\textnormal{-Lin}(2)\xspace} \newcommand{\eps}{\epsilon} \newcommand{\NP}{\textsf{NP}} $ An instance of the $2-\text{Lin}(2)$ problem is a system of equations of the form “$x_i + x_j = b \pmod{2}$.” Given such a system in which it is possible to satisfy all but an $\eps$ fraction of the equations, we show it is $\NP$-hard to satisfy all but a $C \eps$ fraction of equations, for any $C < {11}/{8} = 1.375$ (and any $0 < \eps \leq 1/8$). The previous best result, standing for over $15$ years, had ${5}/{4}$ in place of ${11}/{8}$. Our result provides the best known $\NP$-hardness even for the $\text{Unique-Games}$ problem, and it also holds for the special case of $\text{Max-Cut}$. The precise factor ${11}/{8}$ is unlikely to be best possible; we also give a conjecture concerning analysis of Boolean functions which, if true, would yield a larger hardness factor of ${3}/{2}$. Our proof is by a modified gadget reduction from a pairwise-independent predicate. We also show an inherent limitation to this type of gadget reduction. In particular, any such reduction can never establish a hardness factor $C$ greater than $2.54$. Previously, no such limitations on gadget reductions was known. A preliminary version of this paper appeared in the Proceedings of the 18th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX 2015).

Parameter Selection of Discrete Particle Swarm Optimization Algorithm for the Quadratic Assignment Problems

Quadratic Assignment Problems (QAPs) are the hardest of combinatorial optimization problems, with some problems of sizes of the order of 30 still remaining unsolved optimally. Solving these problems with exact optimization methods is cumbersome and hence the use of non-traditional optimization methods is advisable. The discrete version of Particle Swarm Optimization (DPSO) is a widely accepted non-conventional optimization algorithm for combinatorial optimization problems. This paper uses Taguchi's robust design approach to determine the best combination of parameters for discrete particle swarm optimization algorithm for solving quadratic assignment problems.

A Multiple Pheromone Table Based Ant Colony Optimization for Clustering

Ant colony optimization (ACO) is an efficient heuristic algorithm for combinatorial optimization problems, such as clustering. Because the search strategy of ACO is similar to those of other well-known heuristics, the probability of searching particular regions will be increased if better results are found and kept. Although this kind of search strategy may find a better approximate solution, it also has a high probability of losing the potential search directions. To prevent the ACO from losing too many potential search directions at the early iterations, a novel pheromone updating strategy is presented in this paper. In addition to the “original” pheromone table used to keep track of thepromisinginformation, a second pheromone table is added to the proposed algorithm to keep track of theunpromisinginformation so as to increase the probability of searching directions worse than the current solutions. Several well-known clustering datasets are used to evaluate the performance of the proposed method in this paper. The experimental results show that the proposed method can provide better results than ACO and other clustering algorithms in terms of quality.

Adaptive Quantum Inspired Genetic Algorithm for Combinatorial Optimization Problems

development in the field of quantum computing gives us a significant edge over classical computing in terms of time and efficiency. This is particularly useful for NP-hard problems such as graph layout problems. Since many real world problems are effectively solved by genetic algorithm (GA) and the performance of GA highly depends upon the setting of its parameters, therefore this paper focuses on a Quantum Inspired Genetic Algorithm (QIGA) and develops and evaluates adaptive strategies for the same. QIGA adapts ideas of Q-bits, superposition of Q-bits from quantum computing. The effectiveness and the applicability of adaptive QIGA is demonstrated by experimental results on the benchmark Knapsack, Maxcut and Onemax combinatorial optimization problems. The results show that adaptive QIGA is superior to QIGAs. Keywordsinspired genetic algorithm, Parameter control, adaptive QIGA.

Optimization of integrated polarization filters.

This study reports on the design of small footprint, integrated polarization filters based on engineered photonic lattices. Using a rods-in-air lattice as a basis for a TE filter and a holes-in-slab lattice for the analogous TM filter, we are able to maximize the degree of polarization of the output beams up to 98% with a transmission efficiency greater than 75%. The proposed designs allow not only for logical polarization filtering, but can also be tailored to output an arbitrary transverse beam profile. The lattice configurations are found using a recently proposed parallel tabu search algorithm for combinatorial optimization problems in integrated photonics.

Optimization-based features extraction for K-complex detection

The K-complex is a transient electroencephalogram (EEG, brain activity) waveform that contributes to sleep stage scoring. An automated detection of K-complexes is an important component of sleep stage monitoring. This automation is difficult due to the stochastic nature of brain signals, presence of noise, complexity, and extreme size of data. We develop an optimization model, based on solving a sequence of linear least squares problems, to extract key features of EEG signals. The proposed approach significantly reduces the dimension of the problem and the computational time while the classification accuracy is enhanced in most cases. Numerical results show that this procedure is efficient in detecting K-complexes. References R. Agarwal and J. Gotman. Digital tools in polysomnography. J. Clin. Neurophysiol., 19(2):136–143, 2002. http://journals.lww.com/clinicalneurophys/Abstract/2002/03000/Digital_Tools_in_Polysomnography.4.aspx . J. L. Barlow. Numerical aspects of solving linear least squares problems. Technical report, Computer Science Department, The Pennsylvania State University, University Park, PA, USA, January 1999. www.cse.psu.edu/ barlow/book.ps . A. Bjorck. Numerical Methods for Least Squares Problems. Handbook of Numerical Analysis. SIAM, 1996. doi:10.1137/1.9781611971484 . S. Boyd and L. Vandenberghe. Convex Optimization. Cambridge University Press, New York, NY, USA, 2010. http://www.cambridge.org/au/academic/subjects/statistics-probability/optimization-or-and-risk/convex-optimization?format=HB . G. Bremer, J. R. Smith, and I. Karacan. Automatic detection of the K-complex in sleep electroencephalograms. IEEE T. Bio-Med. Eng., 17(4):314–323, 1970. doi:10.1109/TBME.1970.4502759 . P. R. Carney, R. B. Berry, and J. D. Geyer. Clinical Sleep Disorders. LWW medical book collection. Lippincott Williams and Wilkins, 2005. http://www.lww.com/Product/9780781786928 . G. H. Golub and C. F. Van Loan. Matrix Computations. Johns Hopkins Studies in the Mathematical Sciences. Johns Hopkins University Press, Baltimore, MD, USA, 1996. http://portal.acm.org/citation.cfm?id=248979 . C. Iber, S. Ancoli-Israel, A. L. Chesson, and S. F. Quani. AASM manual for the scoring of sleep and associated events. Rules, technology and technical specifications. Technical report, AASM, Westchester, IL, 2007. http://www.aasmnet.org/scoringmanual/v2.0.2/html/index.html?IXDevelopmentProcess.html . B. H. Jansen. Artificial neural nets for K-complex detection. IEEE Eng. Med. Biol., 9(3):50–52, 1990. doi:10.1109/51.59213 . A. Kales, A. Rechtschaffen, Los Angeles University of California, and NINDB Neurological Information Network (U.S.). A manual of standardized terminology, techniques and scoring system for sleep stages of human subjects. U. S. National Institute of Neurological Diseases and Blindness, Neurological Information Network Bethesda, Md, 1968. http://nla.gov.au/nla.cat-vn823711 . J. E. Mitchell. Branch-and-cut algorithms for combinatorial optimization problems. In P. M. Pardalos and M. G. C. Resende, editors, Handbook of Applied Optimization, pages 65–77. Oxford University Press, 2002. http://global.oup.com/academic/product/handbook-of-applied-optimization-9780195125948?cc=au&lang=en& . D. Moloney, N. Sukhorukova, P. Vamplew, J. Ugon, G. Li, G. Beliakov, C. Philippe, H. Amiel, and A. Ugon. Detecting K-complexes for sleep stage identification using nonsmooth optimization. ANZIAM J., 52:319–332, 2011. doi:10.1017/S1446181112000016 . O. Sheriff, B. Pagnrek, S. Mamouhd, and R. Broughton. Automatic detection of K-complex in sleep EEG. Int. Electrical Electronic Conf. Exp., 81, 1977. Z. Tang and N. Ishii. Detection of the K-complex using a new method of recognizing waveform based on the discrete wavelet transform. IEICE T. Inf. Syst., E78–D(1):77–85, 1995. http://search.ieice.org/bin/summary.php?id=e78-d_1_77 . L. N. Trefethen and D. Bau. Numerical Linear Algebra. SIAM, 1997. http://bookstore.siam.org/ot50/ . Weka web site. www.cs.waikato.ac.nz/ml/weka/ .

A Survey of Quantum Genetic Algorithm for Combinatorial Optimization Problems

Combined with quantum computing and genetic algorithm, quantum genetic algorithm (QGA) shows considerable ability of parallelism. Experiments have shown that QGA performs quite well on TSP, job shop problem and some other typical combinatorial optimization problems. The other problems like nutritional diet which can be transformed into specific combinational optimization problem also can be solved through QGA smoothly. This paper sums up the main points of QGA for general combinatorial optimization problems. These points such as modeling of the problem, qubit decoding and rotation strategy are useful to enhance the convergence speed of QGA and avoid premature at the same time.

A Self-Adaptive Heuristic Algorithm for Combinatorial Optimization Problems

AbstractThis paper introduces a new self-tuning mechanism to the local search heuristic for solving of combinatorial optimization problems. Parameter tuning of heuristics makes them difficult to apply, as parameter tuning itself is an optimization problem. For this purpose, a modified local search algorithm free from parameter tuning, called Self-Adaptive Local Search (SALS), is proposed for obtaining qualified solutions to combinatorial problems within reasonable amount of computer times. SALS is applied to several combinatorial optimization problems, namely, classical vehicle routing, permutation flow-shop scheduling, quadratic assignment, and topological design of networks. It is observed that self-adaptive structure of SALS provides implementation simplicity and flexibility to the considered combinatorial optimization problems. Detailed computational studies confirm the performance of SALS on the suit of test problems for each considered problem type especially in terms of solution quality.

Optimal recombination in genetic algorithms for combinatorial optimization problems: Part II

This paper surveys results on complexity of the optimal recombination problem (ORP), which consists in finding the best possible offspring as a result of a recombination operator in a genetic algorithm, given two parent solutions. In Part II, we consider the computational complexity of ORPs arising in genetic algorithms for problems on permutations: the Travelling Salesman Problem, the Shortest Hamilton Path Problem and the Makespan Minimization on Single Machine and some other related problems. The analysis indicates that the corresponding ORPs are NP-hard, but solvable by faster algorithms, compared to the problems they are derived from.

Optimal recombination in genetic algorithms for combinatorial optimization problems: Part I

This paper surveys results on complexity of the optimal recombination problem (ORP), which consists in finding the best possible offspring as a result of a recombination operator in a genetic algorithm, given two parent solutions. We consider efficient reductions of the ORPs, allowing to establish polynomial solvability or NP-hardness of the ORPs, as well as direct proofs of hardness results. Part I presents the basic principles of optimal recombination with a survey of results on Boolean Linear Programming Problems. Part II (to appear in a subsequent issue) is devoted to the ORPs for problems which are naturally formulated in terms of search for an optimal permutation.

개미군집 최적화 알고리즘을 이용한 상수도관망 시스템의 최저비용설계 모델의 현장 적용

In this study, Ant Colony Algorithm(ACO) was used for optimal model. ACO which are metaheuristic algorithm for combinatorial optimization problem are inspired by the fact that ants are able to find the shortest route between their nest and food source. For applying the model to water distribution systems, pipes, tanks(reservoirs), pump construction and pump operation cost were considered as object function and pressure at each node and reservoir level were considered as constraints. Modified model from Ostfeld and Tubaltzev(2008) was verified by applying 2-Looped, Hanoi and Ostfeld’s networks. And sensitivity analysis about ant number, number of ants in a best group and pheromone decrease rate was accomplished. After the verification, it was applied to real water network from S water treatment plant. As a result of the analysis, in the Two-looped network, the best design cost was found to $419,000 and in the Hanoi network, the best design cost was calculated to $6,164,384, and in the Ostfeld’s network, the best design cost was found to $3,525,096. These are almost equal or better result compared with previous researches. Last, the cost of optimal design for real network, was found for 66 billion dollar that is 8.8 % lower than before. In addition, optimal diameter for aged pipes was found in this study and the 5 of 8 aged pipes were changed the diameter. Through this result, pipe construction cost reduction was found to 11 percent lower than before. And to conclusion, The least cost design model on water distribution system was developed and verified successfully in this study and it will be very useful not only optimal pipe change plan but optimization plan for whole water distribution system.

Hardness of Vertex Deletion and Project Scheduling

Assuming the Unique Games Conjecture, we show strong inapproximability results for two natural vertex deletion problems on directed graphs: for any integer $k\geq 2$ and arbitrary small $\epsilon > 0$, the Feedback Vertex Set problem and the DAG Vertex Deletion problem are inapproximable within a factor $k-\epsilon$ even on graphs where the vertices can be almost partitioned into $k$ solutions. This gives a more structured and yet simpler (albeit using the “It Ain't Over Till It's Over” theorem) UG-hardness result for the Feedback Vertex Set problem than the previous hardness result. In comparison to the classical Feedback Vertex Set problem, the DAG Vertex Deletion problem has received little attention and, although we think it is a natural and interesting problem, the main motivation for our inapproximability result stems from its relationship with the classical Discrete Time-Cost Tradeoff Problem. More specifically, our results imply that the deadline version is UG-hard to approximate within any constant. This explains the difficulty in obtaining good approximation algorithms for that problem and further motivates previous alternative approaches such as bicriteria approximations. An extended abstract of this paper appeared in the Proceedings of the 15th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, 2012 (APPROX'12).

Constant Approximation Algorithms for Embedding Graph Metrics into Trees and Outerplanar Graphs

In this paper, we present a simple factor 6 algorithm for approximating the optimal multiplicative distortion of embedding a graph metric into a tree metric (thus improving and simplifying the factor 100 and 27 algorithms of Bǎdoiu et al. (Proceedings of the eighteenth annual ACM–SIAM symposium on discrete algorithms (SODA 2007), pp. 512–521, 2007) and Bǎdoiu et al. (Proceedings of the 11th international workshop on approximation algorithms for combinatorial optimization problems (APPROX 2008), Springer, Berlin, pp. 21–34, 2008)). We also present a constant factor algorithm for approximating the optimal distortion of embedding a graph metric into an outerplanar metric. For this, we introduce a general notion of a metric relaxed minor and show that if G contains an α-metric relaxed H-minor, then the distortion of any embedding of G into any metric induced by a H-minor free graph is ≥α. Then, for H=K 2,3, we present an algorithm which either finds an α-relaxed minor, or produces an O(α)-embedding into an outerplanar metric.

Bipartite Matching in the Semi-streaming Model

We present the first deterministic 1+ε approximation algorithm for finding a large matching in a bipartite graph in the semi-streaming model which requires only O((1/ε)5) passes over the input stream. In this model, the input graph G=(V,E) is given as a stream of its edges in some arbitrary order, and storage of the algorithm is bounded by O(npolylog n) bits, where \(n = \lvert {V}\rvert \). The only previously known arbitrarily good approximation for general graphs is achieved by the randomized algorithm of McGregor (Proceedings of the International Workshop on Approximation Algorithms for Combinatorial Optimization Problems and Randomization and Computation, Berkeley, CA, USA, pp. 170–181, 2005), which uses Ω((1/ε)1/ε) passes. We show that even for bipartite graphs, McGregor’s algorithm needs Ω(1/ε)Ω(1/ε) passes, thus it is necessarily exponential in the approximation parameter. The design as well as the analysis of our algorithm require the introduction of some new techniques. A novelty of our algorithm is a new deterministic assignment of matching edges to augmenting paths which is responsible for the complexity reduction, and gets rid of randomization.We repeatedly grow an initial matching using augmenting paths up to a length of 2k+1 for k=⌈2/ε⌉. We terminate when the number of augmenting paths found in one iteration falls below a certain threshold also depending on k, that guarantees a 1+ε approximation. The main challenge is to find those augmenting paths without requiring an excessive number of passes. In each iteration, using multiple passes, we grow a set of alternating paths in parallel, considering each edge as a possible extension as it comes along in the stream. Backtracking is used on paths that fail to grow any further. Crucial are the so-called position limits: when a matching edge is the ith matching edge in a path and it is then removed by backtracking, it will only be inserted into a path again at a position strictly lesser than i. This rule strikes a balance between terminating quickly on the one hand and giving the procedure enough freedom on the other hand.

Contributors

Contributors

A hybrid approach using chaotic dynamics and global search algorithms for combinatorial optimization problems

Chaotic dynamics have been effectively applied to improve various heuristic algorithms for combinatorial optimization problems in many studies. Currently, the most used chaotic optimization scheme is to drive heuristic solution search algorithms applicable to large-scale problems by chaotic neurodynamics including the tabu effect of the tabu search. Alternatively, meta-heuristic algorithms are used for combinatorial optimization by combining a neighboring solution search algorithm, such as tabu, gradient, or other search method, with a global search algorithm, such as genetic algorithms (GA), ant colony optimization (ACO), or others. In these hybrid approaches, the ACO has effectively optimized the solution of many benchmark problems in the quadratic assignment problem library. In this paper, we propose a novel hybrid method that combines the effective chaotic search algorithm that has better performance than the tabu search and global search algorithms such as ACO and GA. Our results show that the proposed chaotic hybrid algorithm has better performance than the conventional chaotic search and conventional hybrid algorithms. In addition, we show that chaotic search algorithm combined with ACO has better performance than when combined with GA.