On the minimum common integer partition problem.

We introduce a new combinatorial optimization problem in this paper, called the Minimum Common Integer Partition (MCIP) problem, which was inspired by computational biology applications including ortholog assignment and DNA fingerprint assembly. A partition of a positive integer n is a multiset of positive integers that add up to exactly n, and an integer partition of a multiset S of integers is defined as the multiset union of partitions of integers in S. Given a sequence of multisets S1, · · · , S k of integers, where k ≥ 2, we say that a multiset is a common integer partition if it is an integer partition of every multiset Si, 1 ≤ i ≤ k. The MCIP problem is thus defined as to find a common integer partition of S1, · · · , S k with the minimum cardinality. It is easy to see that the MCIP problem is NP-hard since it generalizes the wellknown Set Partition problem. We can in fact show that it is APX-hard. We will also present a -approximation algorithm for k ≥ 3

Computational Performance Evaluation of Two Integer Linear Programming Models for the Minimum Common String Partition Problem

In the minimum common string partition (MCSP) problem two related input strings are given. "Related" refers to the property that both strings consist of the same set of letters appearing the same number of times in each of the two strings. The MCSP seeks a minimum cardinality partitioning of one string into non-overlapping substrings that is also a valid partitioning for the second string. This problem has applications in bioinformatics e.g. in analyzing related DNA or protein sequences. For strings with lengths less than about 1000 letters, a previously published integer linear programming (ILP) formulation yields, when solved with a state-of-the-art solver such as CPLEX, satisfactory results. In this work, we propose a new, alternative ILP model that is compared to the former one. While a polyhedral study shows the linear programming relaxations of the two models to be equally strong, a comprehensive experimental comparison using real-world as well as artificially created benchmark instances indicates substantial computational advantages of the new formulation.Comment: arXiv admin note: text overlap with arXiv:1405.5646 This paper version replaces the one submitted on January 10, 2015, due to detected error in the calculation of the variables involved in the ILP model

Isomorphism and Similarity for 2-Generation Pedigrees

We consider the emerging problem of comparing the similarity between (unlabeled) pedigrees. More specifically, we focus on the simplest pedigrees, namely, the 2-generation pedigrees. We show that the isomorphism testing for two 2-generation pedigrees is GI-hard. If the 2-generation pedigrees are monogamous (i.e., each individual at level-1 can mate with exactly one partner) then the isomorphism testing problem can be solved in polynomial time. We then consider the problem by relaxing it into an NP-complete decomposition problem which can be formulated as the Minimum Common Integer Pair Partition (MCIPP) problem, which we show to be FPT by exploiting a property of the optimal solution. While there is still some difficulty to overcome, this lays down a solid foundation for this research

Acoplamientos óptimos de caminos de longitud dos

96 páginas. Maestría en Optimización.Let P be a set of 3k points in the Euclidean plane. A 3-matching is a partition of P into k subsets of 3 points each, called triplets. The cost of each triplet fa; b; cg is given by minfjabj + jbcj; jbcj + jcaj; jcaj + jabjg, and the cost of the 3-matching is the sum of the costs of its triplets. The Euclidean 3-matching problem consists on finding a minimum cost 3-matching of P under the Euclidean metric. In the usual formulation of the Euclidean 3- matching problem we need to find a minimum cost 3-matching of P. This problem has several applications, especially in the insertion of components on a printed circuit board. Johnsson, Magyar, and Nevalainen introduced two integer programming formulations for this problem, and proved that its decision version is NP-complete if each triplet has an arbitrary positive cost (i.e., not necessarily Euclidean). The problem remains NP-complete even if the points of P correspond to vertices of a unit distance graph (a metric cost function). In this work, we prove that the linear programming relaxations of these two models are equivalent. Then we introduce three new integer programming models that use fewer variables than those from Johnsson, Magyar, and Nevalainen. We also compare the linear programming relaxations of the models. Besides the minimization problem, we are also interested in a similar maximization problem: finding a maximum cost non-crossing Euclidean 3-matching of P, where non-crossing means that no two segments intersect in a common interior point. Both problems, minimum cost and maximum cost non-crossing, are challenging, and we believe that both are NP-hard. Exact solutions to both problems can be attained through integer programming; however, in order to obtain good solutions in feasible times, we fix our attention to heuristics. We present three heuristics specially designed for our problems and compare their solutions and execution times against solving the exact models

An Integer Programming Formulation of the Minimum Common String Partition problem

We consider the problem of finding a minimum common partition of two strings (MCSP). The problem has its application in genome comparison. MCSP problem is proved to be NP-hard. In this paper, we develop an Integer Programming (IP) formulation for the problem and implement it. The experimental results are compared with the previous state-of-the-art algorithms and are found to be promising.Comment: arXiv admin note: text overlap with arXiv:1401.453

Bandwidth theorem for random graphs

A graph $G$ is said to have \textit{bandwidth} at most $b$, if there exists a labeling of the vertices by $1,2,..., n$, so that $|i - j| \leq b$ whenever $\{i,j\}$ is an edge of $G$. Recently, B\"{o}ttcher, Schacht, and Taraz verified a conjecture of Bollob\'{a}s and Koml\'{o}s which says that for every positive $r,\Delta,\gamma$, there exists $\beta$ such that if $H$ is an $n$-vertex $r$-chromatic graph with maximum degree at most $\Delta$ which has bandwidth at most $\beta n$, then any graph $G$ on $n$ vertices with minimum degree at least $(1 - 1/r + \gamma)n$ contains a copy of $H$ for large enough $n$. In this paper, we extend this theorem to dense random graphs. For bipartite $H$, this answers an open question of B\"{o}ttcher, Kohayakawa, and Taraz. It appears that for non-bipartite $H$ the direct extension is not possible, and one needs in addition that some vertices of $H$ have independent neighborhoods. We also obtain an asymptotically tight bound for the maximum number of vertex disjoint copies of a fixed $r$-chromatic graph $H_0$ which one can find in a spanning subgraph of $G(n,p)$ with minimum degree $(1-1/r + \gamma)np$.Comment: 29 pages, 3 figure