Algorithmic approaches for the single individual haplotyping problem

Since its introduction in 2001, the Single Individual Haplotyping problem has received an ever-increasing attention from the scientific community. In this paper we survey, in the form of an annotated bibliography, the developments in the study of the problem from its origin until our days

FASTSET: A Fast Data Structure for the Representation of Sets of Integers

We describe a simple data structure for storing subsets of { 0 ,..., N - 1 } , with N a given integer, which has optimal time performance for all the main set operations, whereas previous data structures are non-optimal for at least one such operation. We report on the comparison of a Java implementation of our structure with other structures of the standard Java Collections

Finding the Best 3-{OPT} Move in Subcubic Time

Given a Traveling Salesman Problem solution, the best 3-OPT move requires us to remove three edges and replace them with three new ones so as to shorten the tour as much as possible. No worst-case algorithm better than the \u398(n3 ) enumeration of all triples is likely to exist for this problem, but algorithms with average case O(n3 12\u25b ) are not ruled out. In this paper we describe a strategy for 3-OPT optimization which can find the best move by looking only at a fraction of all possible moves. We extend our approach also to some other types of cubic moves, such as some special 6-OPT and 5-OPT moves. Empirical evidence shows that our algorithm runs in average subcubic time (upper bounded by O(n2.5 )) on a wide class of random graphs as well as Traveling Salesman Problem Library (TSPLIB) instances

Estimating the strength of poker hands by integer linear programming techniques

We illustrate how Integer Linear Programming techniques can be applied to the popular game of poker Texas Hold'em in order to evaluate the strength of a hand. In particular, we give models aimed at (i) minimizing the number of features that a player should look at when estimating his winning probability (called his {em equity}); (ii) giving weights to such features so that the equity is approximated by the weighted sum of the selected features. We show that ten features or less are enough to estimate the equity of a hand with high precision

NEW IDEAS TO SPEED-UP FLOYD-WARSHALL SHORTEST PATHS ALGORITHM

Floyd and Warshall’s algorithm for the all-pairs shortest path problem is a Θ(n3) procedure which revisits n times all the cells of an n×n distance matrix. At each pass, all the cells are checked but only some of them get updated. In this paper, we report some preliminary results on a new version of the algorithm, designed to avoid checking cells which will not be updated, in order to reduce the overall time. Our procedure uses heaps to quickly identify which cells can be good candidates for an update. The new version improves over Floyd-Warshall’s original for those input graphs in which the number of cells updated over all passes is substantially smaller than the number of checks. However, our procedure is worse than the original if the ratio between cell checks and updates is not large enough. To obtain an improvement independently of the particular instance type, we propose a hybrid combination of the two approaches, which starts with the original Floyd and Warshall version and then switches to the new one after some iterations. Preliminary experiments show the effectiveness of this strateg

Finding the largest triangle in a graph in expected quadratic time

Finding the largest triangle in an n-nodes edge-weighted graph belongs to a set of problems all equivalent under subcubic reductions. Namely, a truly subcubic algorithm for any one of them would imply that they are all subcubic. A recent strong conjecture states that none of them can be solved in less than \u398(n3) time, but this negative result does not rule out the possibility of algorithms with average, rather than worst-case, subcubic running time. Indeed, in this work we describe the first truly-subcubic average complexity procedure for this problem for graphs whose edge lengths are uniformly distributed in [0,1]. Our procedure finds the largest triangle in average quadratic time, which is the best possible complexity of any algorithm for this problem. We also give empirical evidence that the quadratic average complexity holds for many other random distributions of the edge lengths. A notable exception is when the lengths are distances between random points in Euclidean space, for which the algorithm takes average cubic time

Algorithmic strategies for finding the best TSP 2-OPT move in average sub-quadratic time

We describe an exact algorithm for finding the best 2-OPT move which, experimentally, was observed to be much faster than the standard quadratic approach. 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: (i) there is a heuristic in the family which can find the best move with probability at least p in average-time O(n^3/2) for uniform instances and O(n) for Euclidean instances; (ii) the exact algorithm take lesser time then the above heuristic on all instances on which the heuristic finds the best move. During local search, while the tour becomes less and less random, the speed of our algorithm worsens until it becomes quadratic. We then discuss how to fine tune a successful hybrid approach, made of our algorithm in the beginning followed by the usual quadratic enumeration

Local search inequalities

3noWe describe a general method for deriving new inequalities for integer programming formulations of combinatorial optimization problems. The inequalities, motivated by local search algorithms, are valid for all optimal solutions but not necessarily for all feasible solutions. These local search inequalities can help in either pruning the search tree at some nodes or in improving the bound of the LP relaxations.openopenLancia, Giuseppe; Rinaldi, Franca; Serafini, PaoloLancia, Giuseppe; Rinaldi, Franca; Serafini, Paol

A Facility Location Model for Air Pollution Detection

We describe mathematical models and practical algorithms for a problem concerned with monitoring the air pollution in a large city. We have worked on this problem within a project for assessing the air quality in the city of Rome by placing a certain number of sensors on some of the city buses. We cast the problem as a facility location model. By reducing the large number of data variables and constraints, we were able to solve to optimality the resulting MILP model within minutes. Furthermore, we designed a genetic algorithm whose solutions were on average very close to the optimal ones. In our computational experiments we studied the placement of sensors on 187 candidate bus routes. We considered the coverage provided by 10 up to 60 sensors