We describe an exact algorithm for finding the best 2-OPT move that, experimentally, was observed to be much faster than the standard quadratic approach for a large part of a best-improvement local search convergence starting at a random tour. To analyze its average-case complexity, we introduce a family of heuristic procedures and discuss their complexity when applied to a random tour in graphs whose edge costs are either uniform random numbers in [0, 1] or Euclidean distances between random points in the plane. We prove that, for any probability p, there is a heuristic in the family that can find the best 2-OPT move with probability at least p in average-time [Formula: see text]) for uniform instances and [Formula: see text] for Euclidean instances. The exact algorithm is then proved to be even faster in the sense that in those instances in which a heuristic finds the best move, the exact algorithm finds it in a smaller time. We give empirical evidence that a slight variant of our algorithm finds the best move in [Formula: see text] time on both types of instances, achieving the best possible performance for this particular problem. Computational experiments are reported to show the effectiveness of our algorithms, both in best-improvement and in first-improvement 2-OPT local search. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms–Discrete. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2023.0169 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2023.0169 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Haplotype data play a relevant role in several genetic studies, e.g., mapping of complex disease genes, drug design, and evolutionary studies on populations. However, the experimental determination of haplotypes is expensive and time-consuming. This motivates the increasing interest in techniques for inferring haplotype data from genotypes, which can instead be obtained quickly and economically. Several such techniques are based on the maximum parsimony principle, which has been justified by both experimental results and theoretical arguments. However, the problem of haplotype inference by parsimony was shown to be NP-hard, thus limiting the applicability of exact parsimony-based techniques to relatively small data sets. In this paper, we introduce collapse rule, a generalization of the well-known Clark's rule, and describe a new heuristic algorithm for haplotype inference (implemented in a program called CollHaps), based on parsimony and the iterative application of collapse rules. The performance of CollHaps is tested on several data sets. The experiments show that CollHaps enables the user to process large data sets obtaining very "parsimonious" solutions in short processing times. They also show a correlation, especially for large data sets, between parsimony and correct reconstruction, supporting the validity of the parsimony principle to produce accurate solutions.

We introduce an exact algorithm, based on integer linear programming (ILP), for the parsimony haplotyping problem (PHP). The PHP uses molecular data and is aimed at the determination of a smallest set of haplotypes that explain a given set of genotypes. Our approach is based on a set-covering formulation of the problem, solved by branch and bound with both column and row generation. Existing ILP methods for the PHP suffer from the large size of the solution space, when the genotypes are long and with many heterozygous sites. Our approach, on the other hand, is based on an effective implicit representation of the solution space, and allows the solution of both real data and simulated instances, which are very hard to solve for other ILPs.

We consider a combinatorial problem derived from haplotyping a population with respect to a genetic disease, either recessive or dominant. Given a set of individuals, partitioned into healthy and diseased, and the corresponding sets of genotypes, we want to infer "bad'' and "good'' haplotypes to account for these genotypes and for the disease. Assume e.g. the disease is recessive. Then, the resolving haplotypes must consist of bad and good haplotypes, so that (i) each genotype belonging to a diseased individual is explained by a pair of bad haplotypes and (ii) each genotype belonging to a healthy individual is explained by a pair of haplotypes of which at least one is good. We prove that the associated decision problem is NP-complete. However, we also prove that there is a simple solution, provided the data satisfy a very weak requirement.

The synergies among biology, computing and other formal disciplines continue to produce a unique blend of domain-specific and methodological advances that is shaping the very fabrics of the new scientific method. Among the many examples of this phenomenon, the one offered by the unrelenting growth of bioinformatics and computational biology is unique in that nowhere else is the native lexicon of a natural science more directly conducive to digital representation and manipulation. The research articles contained in this Supplement originate from posters presented at the Tenth Annual International Conference on Research in Computational Molecular Biology (RECOMB 2006), which was held in Venice, Italy, on April 2–5, 2006. The RECOMB conference series was started in 1997 by Sorin Istrail, Pavel Pevzner and Michael Waterman. The previous meetings were held in Santa Fe, NM (USA); New York, NY (USA); Lyon, France; Tokyo, Japan; Montreal, Canada; Washington, DC (USA); Berlin, Germany; San Diego, CA (USA); and Boston, MA (USA). RECOMB 2006 was hosted by University of Padova in the Venice Convention Center at Cinema Palace, Venice Lido, Italy. The Tenth Edition of RECOMB was special in several ways. For one, the Program Committee, consisting of 38 specialists of the highest distinction in the field, included all past Committee and Conference Chairs as well as the Members of the Steering Committee. The Committee selected 40 papers out of the received submissions of well over 200. Some of the accepted papers were further expanded and refereed in order to appear in the special issue traditionally devoted to the Conference. For the Tenth Edition, however, in view of the high quality of the contributions submitted for poster presentation, it was felt that another Special Issue, devoted to expanded and duly refereed versions of poster submissions was also warranted. Thus, the present Supplement constitutes one more innovation brought about by the Tenth Anniversary of RECOMB. The eight enclosed papers emerged at the outset of a rigid selection and review and represent a vivid snapshot of work mature enough to be reported within vibrant frameworks still in the making. We hope that they will start one more tradition for RECOMB. This Issue was made possible thanks to the effort of many, in particular, the special task force set up for handling posters, which consisted of Luca Bortolussi (University of Udine, Italy), Giovanni Ciriello (University of Padova, Italy), Matteo Comin (University of Padova, Italy), Claudio Garrutti (University of Padova, Italy), Giosue Lo Bosco (University of Palermo, Italy), Sabrina Mantaci (University of Palermo, Italy) Cinzia Pizzi (University of Padova, Italy, and Helsinki, Finland), Simone Scalabrin (University of Udine, Italy), and Nicola Vitacolonna (University of Udine, Italy). We are also grateful to the external reviewers, the members of the Steering Committee and other colleagues who helped in the process. Finally, we express our thanks to the institutions and corporations who provided financial support for the conference: Broad Institute of MIT and Harvard, USA; College of Computing, Georgia Tech., USA; Department of Energy, USA; Department of Information Engineering, University of Padova, Italy; IBM Corporation, USA; ISMB, International Society for Computational Biology; AICA, Italian Association for Informatics and Automatic Computation; National Science Foundation, USA; University of Padova, Italy.

This is a survey designed for mathematical programming people who do not know molecular biology and want to learn the kinds of combinatorial optimization problems that arise. After a brief introduction to the biology, we present optimization models pertaining to sequencing, evolutionary explanations, structure prediction, and recognition. Additional biology is given in the context of the problems, including some motivation for disease diagnosis and drug discovery. Open problems are cited with an extensive bibliography, and we offer a guide to getting started in this exciting frontier.

In this paper we address the pure parsimony haplotyping problem: Find a minimum number of haplotypes that explains a given set of genotypes. We prove that the problem is APX-hard and present a 2k− 1-approximation algorithm for the case in which each genotype has at most k ambiguous positions. We further give a new integer-programming formulation that has (for the first time) a polynomial number variables and constraints. Finally, we give approximation algorithms, not based on linear programming, whose running times are almost linear in the input size.

We illustrate a new approach to the Contact Map Overlap problem for the comparison of protein structures. The approach is based on formulating the problem as an integer linear program and then relaxing in a Lagrangian way a suitable set of constraints. This relaxation is solved by computing a sequence of simple alignment problems, each in quadratic time, and near--optimal Lagrangian multipliers are found by subgradient optimization. By our approach we achieved a substantial speedup over the best existing methods. We were able to solve optimally for the first time instances for PDB proteins with about 1000 residues and 2000 contacts. Moreover, within a few hours we compared 780 pairs in a testbed of 40 large proteins, finding the optimal solution in 150 cases. Finally, we compared 10,000 pairs of proteins from a test set of 269 proteins in the literature, which took a couple of days on a PC.

With the consensus human genome sequenced and many other sequencing projects at varying stages of completion, greater attention is being paid to the genetic differences among individuals and the abilities of those differences to predict phenotypes. A significant obstacle to such work is the difficulty and expense of determining haplotypes--sets of variants genetically linked because of their proximity on the genome--for large numbers of individuals for use in association studies. This paper presents some algorithmic considerations in a new approach for haplotype determination: inferring haplotypes from localised polymorphism data gathered from short genome 'fragments.' Formalised models of the biological system under consideration are examined, given a variety of assumptions about the goal of the problem and the character of optimal solutions. Some theoretical results and algorithms for handling haplotype assembly given the different models are then sketched. The primary conclusion is that some important simplified variants of the problem yield tractable problems while more general variants tend to be intractable in the worst case.

Abstract Given a set of points and distances between them, a basic problem in network design calls for selecting a graph connecting them at a minimum total routing cost, that is, the sum over all pairs of points of the length of their shortest path in the graph. In this paper, we describe some branch‐and‐bound algorithms for the exact solution of a relevant special case arising when the graph has to be a tree. One of the enhancements to our algorithms is the use of “LP shortcutting,” which we introduce as a general‐purpose technique for speeding up the search. Besides network design, we show how trees of small routing cost find useful application in computational biology, where they can be used to determine good alignments of genomic sequences. This leads to a novel alignment heuristic that we analyze in our computational section. © 2002 Wiley Periodicals, Inc.

Structure comparison is a fundamental problem for structural genomics. A variety of structure comparison methods were proposed and several protein structure classification servers e.g., SCOP, DALI, CATH, were designed based on them, and are extensively used in practice. This area of research continues to be very active, being energized bi-annually by the CASP folding competitions, but despite the extraordinary international research effort devoted to it, progress is slow. A fundamental dimension of this bottleneck is the absence of rigorous algorithmic methods. A recent excellent survey on structure comparison by Taylor et.al. [23] records the state of the art of the area: In structure comparison, we do not even have an algorithm that guarantees an optimal answer for pairs of structures …

We describe an exact algorithm for the problem of sorting a permutation by the minimum number of reversals, originating from evolutionary studies in molecular biology. Our approach is based on an integer linear programming formulation of a graph-theoretic relaxation of the problem, calling for a decomposition of the edge set of a bicolored graph into the maximum number of alternating cycles. The formulation has one variable for each alternating cycle, and the associated linear programming relaxation is solved by column generation. A major advantage with respect to previous approaches is that the subproblem to face in the column-generation phase no longer requires the solution of min-cost general matching problems, but of min-cost bipartite matching problems. Experiments show that there is a tremendous speed-up in going from general matching to bipartite matching, although the best-known algorithms for the two problems have the same theoretical worst-case complexity. We also show the worst-case ratio between the lower bound value obtained by our new method and previous ones. We illustrate the effectiveness of our approach through extensive computational experiments. In particular, we can solve to proven optimality the largest real-world instances from the literature in a few seconds, and the other (smaller) real-world instances within a few milliseconds on a workstation. Moreover, we can solve to optimality random instances with n + 100 within 3 seconds, and with n + 200 within 15 minutes, where n is the size of the permutation, whereas the size of the instances solvable by previous approaches was at most 100. We also describe a polynomial-time heuristic algorithm that consistently finds solutions within 2% of the optimum for random instances with n up to 1000.

Given an undirected graph with nonnegative costs on the edges, the routing cost of any of its spanning trees is the sum over all pairs of vertices of the cost of the path between the pair in the tree. Finding a spanning tree of minimum routing cost is NP-hard, even when the costs obey the triangle inequality. We show that the general case is in fact reducible to the metric case and present a polynomial-time approximation scheme valid for both versions of the problem. In particular, we show how to build a spanning tree of an n-vertex weighted graph with routing cost at most $(1+\epsilon)$ of the minimum in time $O(n^{O({\frac{1}{\epsilon}}% )})$. Besides the obvious connection to network design, trees with small routing cost also find application in the construction of good multiple sequence alignments in computational biology. The communication cost spanning tree problem is a generalization of the minimum routing cost tree problem where the routing costs of different pairs are weighted by different requirement amounts. We observe that a randomized O(log n log log n)-approximation for this problem follows directly from a recent result of Bartal, where n is the number of nodes in a metric graph. This also yields the same approximation for the generalized sum-of-pairs alignment problem in computational biology.