Nicola Yanev

A dynamic programming based reduction procedure for the multidimensional 0–1 knapsack problem

Abstract This paper presents a preprocessing procedure for the 0–1 multidimensional knapsack problem. First, a non-increasing sequence of upper bounds is generated by solving LP-relaxations. Then, a non-decreasing sequence of lower bounds is built using dynamic programming. The comparison of the two sequences allows either to prove that the best feasible solution obtained is optimal, or to fix a subset of variables to their optimal values. In addition, a heuristic solution is obtained. Computational experiments with a set of large-scale instances show the efficiency of our reduction scheme. Particularly, it is shown that our approach allows to reduce the CPU time of a leading commercial software.

A hybrid algorithm for the unbounded knapsack problem

This paper presents a new approach for exactly solving the Unbounded Knapsack Problem (UKP) and proposes a new bound that was proved to dominate the previous bounds on a special class of UKP instances. Integrating bounds within the framework of sparse dynamic programming led to the creation of an efficient and robust hybrid algorithm, called EDUK2. This algorithm takes advantage of the majority of the known properties of UKP, particularly the diverse dominance relations and the important periodicity property. Extensive computational results show that, in all but a very few cases, EDUK2 significantly outperforms both MTU2 and EDUK, the currently available UKP solvers, as well the well-known general purpose mathematical programming optimizer CPLEX of ILOG. These experimental results demonstrate that the class of hard UKP instances needs to be redefined, and the authors offer their insights into the creation of such instances.

Maximum Contact Map Overlap Revisited

Among the measures for quantifying the similarity between three-dimensional (3D) protein structures, maximum contact map overlap (CMO) received sustained attention during the past decade. Despite this, the known algorithms exhibit modest performance and are not applicable for large-scale comparison. This article offers a clear advance in this respect. We present a new integer programming model for CMO and propose an exact branch-and-bound algorithm with bounds obtained by a novel Lagrangian relaxation. The efficiency of the approach is demonstrated on a popular small benchmark (Skolnick set, 40 domains). On this set, our algorithm significantly outperforms the best existing exact algorithms. Many hard CMO instances have been solved for the first time. To further assess our approach, we constructed a large-scale set of 300 protein domains. Computing the similarity measure for any of the 44850 pairs, we obtained a classification in excellent agreement with SCOP. Supplementary Material is available at www.liebertonline.com/cmb.

A combinatorial approach to the classification problem

We study the two-group classification problem which involves classifying an observation into one of two groups based on its attributes. The classification rule is a hyperplane which misclassifies the fewest number of observations in the training sample. Exact and heuristic algorithms for solving the problem are presented. Computational results confirm the efficiency of this approach.

Maximum cliques in protein structure comparison

Computing the similarity between two protein structures is a crucial task in molecular biology, and has been extensively investigated. Many protein structure comparison methods can be modeled as maximum clique problems in specific k-partite graphs, referred here as alignment graphs. In this paper, we propose a new protein structure comparison method based on internal distances (DAST), which main characteristic is that it generates alignments having RMSD smaller than any previously given threshold. DAST is posed as a maximum clique problem in an alignment graph, and in order to compute DAST’s alignments, we also design an algorithm (ACF) for solving such maximum clique problems. We compare ACF with one of the fastest clique finder, recently conceived by Ostergȧrd. On a popular benchmark (the Skolnick set) we observe that ACF is about 20 times faster in average than the Ostergȧrd’s algorithm. We then successfully use DAST’s alignments to obtain automatic classification in very good agreement with SCOP.

An Efficient Lagrangian Relaxation for the Contact Map Overlap Problem

Among the measures for quantifying the similarity between protein 3-D structures, contact map overlap (CMO) maximization deserved sustained attention during past decade. Despite this large involvement, the known algorithms possess a modest performance and are not applicable for large scale comparison. This paper offers a clear advance in this respect. We present a new integer programming model for CMO and propose an exact B&B algorithm with bounds obtained by a novel Lagrangian relaxation. The efficiency of the approach is demonstrated on a popular small benchmark (Skolnick set, 40 domains). On this set our algorithm significantly outperforms the best existing exact algorithms. Many hard CMO instances have been solved for the first time. To assess furthermore our approach, we constructed a large scale set of 300 protein domains. Computing the similarity measure for any of the 44850 couples, we obtained a classification in excellent agreement with SCOP.

Protein Threading: From Mathematical Models to Parallel Implementations

This paper presents a new network-flow formulation for the problem of predicting 3D protein structures using threading. Several integer-programming models based on this formulation are proposed and compared. These models allow for an efficient decomposition and for the application of a parallel branch-and-cut algorithm, significantly reducing the running time. The efficiency of our approach has been confirmed by extensive computational experiments.

Optimal Orthogonal Tiling

Iteration space tiling is a common strategy used by parallelizing compilers and in performance tuning of parallel codes. We address the problem of determining the tile size that minimizes the total execution time. We restrict our attention to orthogonal tiling—uniform dependency programs with (hyper) parallelepiped shaped iteration domains which can be tiled with hyperplanes parallel to the domain boundaries. Our formulation includes many machine and program models used in the literature, notably the Bsp programming model. We resolve the optimization problem analytically, yielding a closed form solution.

Optimal semi-oblique tiling

For 2-D iteration space tiling, we address the problem of determining the tile parameters that minimize the total execution time under the BSP model. We consider uniform dependency computations, tiled so that (at least) one of the tile boundaries is parallel to the domain boundary. We determine the optimal tile size as a closed form solution. In addition, we determine the optimal number of processors and also the optimal slope of the oblique tile boundary. Our predictions are validated, among other examples, on a sequence alignment problem specialized to similar sequences using Fickets “k-band” algorithm, for which, our optimal semi-oblique tiling yields an improvement over orthogonal tiling by a factor of 2.5. Our optimal solution requires a block-cyclic distribution of tiles to processors. The best one can obtain with only block distribution (as many authors require) is 3 times slower.

Solving the protein threading problem in parallel

We propose a network flow formulation for protein threading and show its equivalence with the shortest path problem on a graph with a very particular structure. The underlying mixed integer programming (MIP) model proves to be very appropriate for the protein threading problem - huge real-life instances have been solved in a reasonable time by using only a mixed integer optimizer instead of a special-purpose branch and bound algorithm. The properties of the MIP model allow decomposition of the main problem on a large number of subproblems (tasks). We show in this paper that a branch and bound-like algorithm can be efficiently applied to solving in parallel these tasks, which leads to a significant reduction in the total running time. Computational experiments with huge problem instances are presented.