Effectively Tackling Reinsurance Problems by Using Evolutionary and Swarm Intelligence Algorithms

This paper is focused on solving different hard optimization problems that arise in the field of insurance and, more specifically, in reinsurance problems. In this area, the complexity of the models and assumptions considered in the definition of the reinsurance rules and conditions produces hard black-box optimization problems -problems in which the objective function does not have an algebraic expression, but it is the output of a system - usually a computer program, which must be solved in order to obtain the optimal output of the reinsurance. The application of traditional optimization approaches is not possible in this kind of mathematical problem, so new computational paradigms must be applied to solve these problems. In this paper, we show the performance of two evolutionary and swarm intelligence techniques -evolutionary programming and particle swarm optimization-. We provide an analysis in three black-box optimization problems in reinsurance, where the proposed approaches exhibit an excellent behavior, finding the optimal solution within a fraction of the computational cost used by inspection or enumeration methods

On the Subgroup Distance Problem

We investigate the computational complexity of finding an element of a permutation group H subset S_n with a minimal distance to a given pi in S_n , for different metrics on S_n . We assume that H is given by a set of generators, such that the problem cannot be solved in polynomial time by exhaustive enumeration. For the case of the Cayley Distance, this problem has been shown to be NP-hard, even if H is abelian of exponent two Pinch, 2006. We present a much simpler proof for this result, which also works for the Hamming Distance, the l\_p distance, Lee's Distance, Kendall's tau, and Ulam's Distance. Moreover, we give an NP-hardness proof for the l\_oo distance using a different reduction idea. Finally, we settle the complexity of the corresponding fixed-parameter and maximization problems

Generation of random chordal graphs using subtrees of a tree

Chordal graphs form one of the most studied graph classes. Several graph problems that are NP-hard in general become solvable in polynomial time on chordal graphs, whereas many others remain NP-hard. For a large group of problems among the latter, approximation algorithms, parameterized algorithms, and algorithms with moderately exponential or sub-exponential running time have been designed. Chordal graphs have also gained increasing interest during the recent years in the area of enumeration algorithms. Being able to test these algorithms on instances of chordal graphs is crucial for understanding the concepts of tractability of hard problems on graph classes. Unfortunately, only few published papers give algorithms for generating chordal graphs. Even in these papers, only very few methods aim for generating a large variety of chordal graphs. Surprisingly, none of these methods is directly based on the “intersection of subtrees of a tree” characterization of chordal graphs. In this paper, we give an algorithm for generating chordal graphs, based on the characterization that a graph is chordal if and only if it is the intersection graph of subtrees of a tree. Upon generating a random host tree, we give and test various methods that generate subtrees of the host tree. We compare our methods to one another and to existing ones for generating chordal graphs. Our experiments show that one of our methods is able to generate the largest variety of chordal graphs in terms of maximal clique sizes. Moreover, two of our subtree generation methods result in an overall complexity of our generation algorithm that is the best possible time complexity for a method generating the entire node set of subtrees in an “intersection of subtrees of a tree” representation. The instances corresponding to the results presented in this paper, and also a set of relatively small-sized instances are made available online.publishedVersio

Paradigms for Parameterized Enumeration

The aim of the paper is to examine the computational complexity and algorithmics of enumeration, the task to output all solutions of a given problem, from the point of view of parameterized complexity. First we define formally different notions of efficient enumeration in the context of parameterized complexity. Second we show how different algorithmic paradigms can be used in order to get parameter-efficient enumeration algorithms in a number of examples. These paradigms use well-known principles from the design of parameterized decision as well as enumeration techniques, like for instance kernelization and self-reducibility. The concept of kernelization, in particular, leads to a characterization of fixed-parameter tractable enumeration problems.Comment: Accepted for MFCS 2013; long version of the pape

Kernelization and Enumeration: New Approaches to Solving Hard Problems

NP-Hardness is a well-known theory to identify the hardness of computational problems. It is believed that NP-Hard problems are unlikely to admit polynomial-time algorithms. However since many NP-Hard problems are of practical significance, different approaches are proposed to solve them: Approximation algorithms, randomized algorithms and heuristic algorithms. None of the approaches meet the practical needs. Recently parameterized computation and complexity has attracted a lot of attention and been a fruitful branch of the study of efficient algorithms. By taking advantage of the moderate value of parameters in many practical instances, we can design efficient algorithms for the NP-Hard problems in practice. In this dissertation, we discuss a new approach to design efficient parameterized algorithms, kernelization. The motivation is that instances of small size are easier to solve. Roughly speaking, kernelization is a preprocess on the input instances and is able to significantly reduce their sizes. We present a 2k kernel for the cluster editing problem, which improves the previous best kernel of size 4k; We also present a linear kernel of size 7k 2d for the d-cluster editing problem, which is the first linear kernel for the problem. The kernelization algorithm is simple and easy to implement. We propose a quadratic kernel for the pseudo-achromatic number problem. This implies that the problem is tractable in term of parameterized complexity. We also study the general problem, the vertex grouping problem and prove it is intractable in term of parameterized complexity. In practice, many problems seek a set of good solutions instead of a good solution. Motivated by this, we present the framework to study enumerability in term of parameterized complexity. We study three popular techniques for the design of parameterized algorithms, and show that combining with effective enumeration techniques, they could be transferred to design efficient enumeration algorithms

Complexity classifications for different equivalence and audit problems for Boolean circuits

We study Boolean circuits as a representation of Boolean functions and consider different equivalence, audit, and enumeration problems. For a number of restricted sets of gate types (bases) we obtain efficient algorithms, while for all other gate types we show these problems are at least NP-hard.Comment: 25 pages, 1 figur

Efficient enumeration of solutions produced by closure operations

In this paper we address the problem of generating all elements obtained by the saturation of an initial set by some operations. More precisely, we prove that we can generate the closure of a boolean relation (a set of boolean vectors) by polymorphisms with a polynomial delay. Therefore we can compute with polynomial delay the closure of a family of sets by any set of "set operations": union, intersection, symmetric difference, subsets, supersets $\dots$). To do so, we study the $Membership_{\mathcal{F}}$ problem: for a set of operations $\mathcal{F}$, decide whether an element belongs to the closure by $\mathcal{F}$ of a family of elements. In the boolean case, we prove that $Membership_{\mathcal{F}}$ is in P for any set of boolean operations $\mathcal{F}$. When the input vectors are over a domain larger than two elements, we prove that the generic enumeration method fails, since $Membership_{\mathcal{F}}$ is NP-hard for some $\mathcal{F}$. We also study the problem of generating minimal or maximal elements of closures and prove that some of them are related to well known enumeration problems such as the enumeration of the circuits of a matroid or the enumeration of maximal independent sets of a hypergraph. This article improves on previous works of the same authors.Comment: 30 pages, 1 figure. Long version of the article arXiv:1509.05623 of the same name which appeared in STACS 2016. Final version for DMTCS journa

Reductions for Frequency-Based Data Mining Problems

Studying the computational complexity of problems is one of the - if not the - fundamental questions in computer science. Yet, surprisingly little is known about the computational complexity of many central problems in data mining. In this paper we study frequency-based problems and propose a new type of reduction that allows us to compare the complexities of the maximal frequent pattern mining problems in different domains (e.g. graphs or sequences). Our results extend those of Kimelfeld and Kolaitis [ACM TODS, 2014] to a broader range of data mining problems. Our results show that, by allowing constraints in the pattern space, the complexities of many maximal frequent pattern mining problems collapse. These problems include maximal frequent subgraphs in labelled graphs, maximal frequent itemsets, and maximal frequent subsequences with no repetitions. In addition to theoretical interest, our results might yield more efficient algorithms for the studied problems.Comment: This is an extended version of a paper of the same title to appear in the Proceedings of the 17th IEEE International Conference on Data Mining (ICDM'17