A unifying look at the Apostolico–Giancarlo string-matching algorithm

String matching is the problem of finding all the occurrences of a pattern in a text. We present a new method to compute the combinatorial shift function (“matching shift”) of the well-known Boyer–Moore string matching algorithm. This method implies the computation of the length of the longest suffixes of the pattern ending at each position in this pattern. These values constituted an extra-preprocessing for a variant of the Boyer–Moore algorithm designed by Apostolico and Giancarlo. We give here a new presentation of this algorithm that avoids extra preprocessing together with a tight bound of 1.5n character comparisons (where n is the length of the text).     

Average-optimal string matching

The exact string matching problem is to find the occurrences of a pattern of length m from a text of length n symbols. We develop a novel and unorthodox filtering technique for this problem. Our method is based on transforming the problem into multiple matching of carefully chosen pattern subsequences. While this is seemingly more difficult than the original problem, we show that the idea leads to very simple algorithms that are optimal on average. We then show how our basic method can be used to solve multiple string matching as well as several approximate matching problems in average optimal time. The general method can be applied to many existing string matching algorithms. Our experimental results show that the algorithms perform very well in practice.     

Bit-parallel -matching and suffix automata

(δ,γ)-matching is a string matching problem with applications to music retrieval. The goal is, given a pattern P_1…m and a text T_1…n on an alphabet of integers, find the occurrences P^′ of the pattern in the text such that (i) , and (ii) . The problem makes sense for δγδm. Several techniques for (δ,γ)-matching have been proposed, based on bit-parallelism or on skipping characters. We first present an O(mnlog(γ)/w) worst-case time and O(n) average-case time bit-parallel algorithm (being w the number of bits in the computer word). It improves the previous O(mnlog(δm)/w) worst-case time algorithm of the same type. Second, we combine our bit-parallel algorithm with suffix automata to obtain the first algorithm that skips characters using both δ and γ. This algorithm examines less characters than any previous approach, as the others do just δ-matching and check the γ-condition on the candidates. We implemented our algorithms and drew experimental results on real music, showing that our algorithms are superior to current alternatives with high values of δ.     

Exact algorithms for procurement problems under a total quantity discount structure

In this paper, we study the procurement problem faced by a buyer who needs to purchase a variety of goods from suppliers applying a so-called total quantity discount policy. This policy implies that every supplier announces a number of volume intervals and that the volume interval in which the total amount ordered lies determines the discount. Moreover, the discounted prices apply to all goods bought from the supplier, not only to those goods exceeding the volume threshold. We refer to this cost-minimization problem as the total quantity discount (TQD) problem. We give a mathematical formulation for this problem and argue that not only it is NP-hard, but also that there exists no polynomial-time approximation algorithm with a constant ratio (unless P = NP). Apart from the basic form of the TQD problem, we describe four variants. In a first variant, the market share that one or more suppliers can obtain is constrained. Another variant allows the buyer to procure more goods than strictly needed, in order to reach a lower total cost. We also consider a setting where the buyer needs to pay a disposal cost for the extra goods bought. In a third variant, the number of winning suppliers is limited, both in general and per product. Finally, we investigate a multi-period variant, where the buyer not only needs to decide what goods to buy from what supplier, but also when to do this, while considering the inventory costs. We show that the TQD problem and its variants can be solved by solving a series of min-cost flow problems. Finally, we investigate the performance of three exact algorithms (min-cost flow based branch-and-bound, linear programming based branch-and-bound, and branch-and-cut) on randomly generated instances involving 50 suppliers and 100 goods. It turns out that even the large instances of the basic problem are solved to optimality within a limited amount of time. However, we find that different algorithms perform best in terms of computation time for different variants.     

Bit-parallel approximate string matching algorithms with transposition

Using bit-parallelism has resulted in fast and practical algorithms for approximate string matching under Levenshtein edit distance, which permits a single edit operation to insert, delete or substitute a character. Depending on the parameters of the search, currently the fastest non-filtering algorithms in practice are the O(km/wn) algorithm of Wu and Manber, the O((k+2)(m−k)/wn) algorithm of Baeza-Yates and Navarro, and the O(m/wn) algorithm of Myers, where m is the pattern length, n is the text length, k is the error threshold and w is the computer word size. In this paper we discuss a uniform way of modifying each of these algorithms to permit also a fourth type of edit operation: transposing two adjacent characters in the pattern. This type of edit distance is also known as Damerau edit distance. In the end we also present an experimental comparison of the resulting algorithms.     

Solving multiobjective,multiconstraint knapsack problems using mathematical programming and evolutionary algorithms

In this paper, we solve instances of the multiobjective multiconstraint (or multidimensional) knapsack problem (MOMCKP) from the literature, with three objective functions and three constraints. We use exact as well as approximate algorithms. The exact algorithm is a properly modified version of the multicriteria branch and bound (MCBB) algorithm, which is further customized by suitable heuristics. Three branching heuristics and a more general purpose composite branching and construction heuristic are devised. Comparison is made to the published results from another exact algorithm, the adaptive ε-constraint method [Laumanns, M., Thiele, L., Zitzler, E., 2006. An efficient, adaptive parameter variation scheme for Metaheuristics based on the epsilon-constraint method. European Journal of Operational Research 169, 932–942], using the same data sets. Furthermore, the same problems are solved using standard multiobjective evolutionary algorithms (MOEA), namely, the SPEA2 and the NSGAII. The results from the exact case show that the branching heuristics greatly improve the performance of the MCBB algorithm, which becomes faster than the adaptive ε -constraint. Regarding the performance of the MOEA algorithms in the specific problems, SPEA2 outperforms NSGAII in the degree of approximation of the Pareto front, as measured by the coverage metric (especially for the largest instance).     

The capacitated multiple allocation hub location problem: Formulations and algorithms

In this paper we consider and present formulations and solution approaches for the capacitated multiple allocation hub location problem. We present a new mixed integer linear programming formulation for the problem. We also construct an efficient heuristic algorithm, using shortest paths. We incorporate the upper bound obtained from this heuristic in a linear-programming-based branch-and-bound solution procedure. We present the results of extensive computational experience with both the heuristic and the exact methods.     

FFT-based algorithms for the string matching with mismatches problem

The string matching with mismatches problem requires finding the Hamming distance between a pattern P of length m and every length m substring of text T with length n. Fischer and Paterson's FFT-based algorithm solves the problem without error in O(σnlogm), where σ is the size of the alphabet Σ [SIAM–AMS Proc. 7 (1973) 113–125]. However, this in the worst case reduces to O(nmlogm). Atallah, Chyzak and Dumas used the idea of randomly mapping the letters of the alphabet to complex roots of unity to estimate the score vector in time O(nlogm) [Algorithmica 29 (2001) 468–486]. We show that the algorithm's score variance can be substantially lowered by using a bijective mapping, and specifically to zero in the case of binary and ternary alphabets. This result is extended via alphabet remappings to deterministically solve the string matching with mismatches problem with a constant factor of 2 improvement over Fischer–Paterson's method.     

The design of branch and bound algorithms for a class of nonlinear integer programs

This paper is concerned with computational experimentation leading to the design of effective branch and bound algorithms for an important class of nonlinear integer programming problems, namely linearly constrained problems, which are used to model several real-world situations. The main contribution here is a study of the effect of node and branching variable selection and storage reduction strategies on overall computational effort for this class of problems, as well as the generation of a set of adequate test problems. Several node and branching variable strategies are compared in the context of a pure breadth-first enumeration, as well as in a special breadth and depth enumeration combination approach presented herein. Also, the effect of using updated pseudocosts is briefly addressed. Computational experience is presented on a set of eighteen suitably-sized nonlinear test problems, as well as on some random linear integer programs. Some of the new rules proposed are demonstrated to be significantly superior to previously suggested strategies; interestingly, even for linear integer programming problems.     

The design of branch and bound algorithms for a class of nonlinear integer programs

This paper is concerned with computational experimentation leading to the design of effective branch and bound algorithms for an important class of nonlinear integer programming problems, namely linearly constrained problems, which are used to model several real-world situations. The main contribution here is a study of the effect of node and branching variable selection and storage reduction strategies on overall computational effort for this class of problems, as well as the generation of a set of adequate test problems. Several node and branching variable strategies are compared in the context of a pure breadth-first enumeration, as well as in a special breadth and depth enumeration combination approach presented herein. Also, the effect of using updated pseudocosts is briefly addressed. Computational experience is presented on a set of eighteen suitably-sized nonlinear test problems, as well as on some random linear integer programs. Some of the new rules proposed are demonstrated to be significantly superior to previously suggested strategies; interestingly, even for linear integer programming problems.     

A branch and bound algorithm for the matrix bandwidth minimization

In this article, we first review previous exact approaches as well as theoretical contributions for the problem of reducing the bandwidth of a matrix. This problem consists of finding a permutation of the rows and columns of a given matrix which keeps the non-zero elements in a band that is as close as possible to the main diagonal. This NP-complete problem can also be formulated as a labeling of vertices on a graph, where edges are the non-zero elements of the corresponding symmetrical matrix. We propose a new branch and bound algorithm and new expressions for known lower bounds for this problem. Empirical results with a collection of previously reported instances indicate that the proposed algorithm compares favourably to previous methods.     

An acceleration of Erlenkotter-Körkel’s algorithms for the uncapacitated facility location problem

This contribution is focused on an acceleration of branch and bound algorithms for the uncapacitated facility location problem. Our approach is based on the well-known Erlenkotters’ procedures and Körkels’ multi-ascent and multi-adjustment algorithms, which have proved to be the efficient tools for solving the large-sized instances of the uncapacitated facility location problem. These two original approaches were examined and a thorough analysis of their performance revealed how each particular procedure contributes to the computational time of the whole algorithms. These analyses helped us to focus our effort on the most frequent procedures. The unique contribution of this paper is a new dual ascent procedure. This procedure leads to considerable acceleration of the lower bound computation process and reduces the resulting computational time. To demonstrate more efficient performance of amended algorithms we present the results of extensive numerical experiments.     

Effect of the subdivision strategy on convergence and efficiency of some global optimization algorithms

We investigate subdivision strategies that can improve the convergence and efficiency of some branch and bound algorithms of global optimization. In particular, a general class of so called weakly exhaustive simplicial subdivision processes is introduced that subsumes all previously known radial exhaustive processes. This result provides the basis for constructing flexible subdivision strategies that can be adapted to take advantage of various problem conditions.     

Design of planar articulated mechanisms using branch and bound

This paper considers an optimization model and a solution method for the design of two-dimensional mechanical mechanisms. The mechanism design problem is modeled as a nonconvex mixed integer program which allows the optimal topology and geometry of the mechanism to be determined simultaneously. The underlying mechanical analysis model is based on a truss representation allowing for large displacements. For mechanisms undergoing large displacements elastic stability is of major concern. We derive conditions, modeled by nonlinear matrix inequalities, which guarantee that a stable equilibrium is found and that buckling is prevented. The feasible set of the design problem is described by nonlinear differentiable and non-differentiable constraints as well as nonlinear matrix inequalities.To solve the mechanism design problem a branch and bound method based on convex relaxations is developed. To guarantee convergence of the method, two different types of convex relaxations are derived. The relaxations are strengthened by adding valid inequalities to the feasible set and by solving bound contraction sub-problems. Encouraging computational results indicate that the branch and bound method can reliably solve mechanism design problems of realistic size to global optimality.     

一类新的非线性比式和问题的分枝定界算法(英文)

对一类新的非线性比式和问题(SNR)提出分枝定界算法,该问题的研究还很少.首先,通过两层线性化技术,构造一个松弛线性规划,求解该线性规划问题,得到问题(SNR)最优值的下界.其次,介绍新的下界更新技术,证明所给算法的收敛性.数值试验显示了算法的可行性和有效性     

Minimizing flow time in cyclic schedules for identical jobs with acyclic precedence: the bottleneck lower bound

This paper studies minimizing the flow time of a cyclic schedule for repeated identical jobs, where one job is started/completed in each cycle, subject to the schedule achieving maximum throughput. We propose a branch and bound method for a single machine problem, and use this method to derive an improved lower bound for the multiple machine problem.     

Fixed-parameter algorithms for vertex cover P3

Let $P_{\ell }$ denote a path in a graph $G=(V,E)$ with $\ell$ vertices. A vertex cover $P_{\ell }$ set $C$ in $G$ is a vertex subset such that every $P_{\ell }$ in $G$ has at least a vertex in $C$. The Vertex Cover$P_{\ell }$ problem is to find a vertex cover $P_{\ell }$ set of minimum cardinality in a given graph. This problem is NP-hard for any integer $\ell ⩾2$. The parameterized version of Vertex Cover$P_{\ell }$ problem called $k$-Vertex Cover$P_{\ell }$ asks whether there exists a vertex cover $P_{\ell }$ set of size at most $k$ in the input graph. In this paper, we give two fixed parameter algorithms to solve the $k$-Vertex Cover$P_3$ problem. The first algorithm runs in time $O^1\left(1.7964^k\right)$ in polynomial space and the second algorithm runs in time $O^1\left(1.7485^k\right)$ in exponential space. Both algorithms are faster than previous known fixed-parameter algorithms.     

An efficient solver for weighted Max-SAT

We present a new branch and bound algorithm for weighted Max-SAT, called Lazy which incorporates original data structures and inference rules, as well as a lower bound of better quality. We provide experimental evidence that our solver is very competitive and outperforms some of the best performing Max-SAT and weighted Max-SAT solvers on a wide range of instances.     

Comparing branch-and-price algorithms for the Multi-Commodity k-splittable Maximum Flow Problem

The Multi-Commodity k-splittable Maximum Flow Problem consists in routing as much flow as possible through a capacitated network such that each commodity uses at most k paths and the capacities are satisfied. The problem appears in telecommunications, specifically when considering Multi-Protocol Label Switching. The problem has previously been solved to optimality through branch-and-price. In this paper we propose two exact solution methods both based on an alternative decomposition. The two methods differ in their branching strategy. The first method, which branches on forbidden edge sequences, shows some performance difficulty due to large search trees. The second method, which branches on forbidden and forced edge sequences, demonstrates much better performance. The latter also outperforms a leading exact solution method from the literature. Furthermore, a heuristic algorithm is presented. The heuristic is fast and yields good solution values.     

An enumeration procedure for the assembly line balancing problem based on branching by non-decreasing idle time

In this article, we present a new exact algorithm for solving the simple assembly line balancing problem given a determined cycle time (SALBP-1). The algorithm is a station-oriented bidirectional branch-and-bound procedure based on a new enumeration strategy that explores the feasible solutions tree in a non-decreasing idle time order. The procedure uses several well-known lower bounds, dominance rules and a new logical test based on the assimilation of the feasibility problem for a given cycle time and number of stations (SALBP-F) to a maximum-flow problem.