New Integrality Gap Results for the Firefighters Problem on Trees

The firefighter problem is NP-hard and admits a $(1-1/e)$ approximation based on rounding the canonical LP. In this paper, we first show a matching integrality gap of $(1-1/e+\epsilon)$ on the canonical LP. This result relies on a powerful combinatorial gadget that can be used to prove integrality gap results for many problem settings. We also consider the canonical LP augmented with simple additional constraints (as suggested by Hartke). We provide several evidences that these constraints improve the integrality gap of the canonical LP: (i) Extreme points of the new LP are integral for some known tractable instances and (ii) A natural family of instances that are bad for the canonical LP admits an improved approximation algorithm via the new LP. We conclude by presenting a $5/6$ integrality gap instance for the new LP.Comment: 22 page

Super qq-Howe duality and web categories

We use super $q$-Howe duality to provide diagrammatic presentations of an idempotented form of the Hecke algebra and of categories of $\mathfrak{gl}_N$-modules (and, more generally, $\mathfrak{gl}_{N|M}$-modules) whose objects are tensor generated by exterior and symmetric powers of the vector representations. As an application, we give a representation theoretic explanation and a diagrammatic version of a known symmetry of colored HOMFLY--PT polynomials.Comment: 38 pages, many colored figures, extra section containing new results, added suggestions of two referees, comments welcom

Approximation algorithms for network design and cut problems in bounded-treewidth

This thesis explores two optimization problems, the group Steiner tree and firefighter problems, which are known to be NP-hard even on trees. We study the approximability of these problems on trees and bounded-treewidth graphs. In the group Steiner tree, the input is a graph and sets of vertices called groups; the goal is to choose one representative from each group and connect all the representatives with minimum cost. We show an O(log^2 n)-approximation algorithm for bounded-treewidth graphs, matching the known lower bound for trees, and improving the best possible result using previous techniques. We also show improved approximation results for group Steiner forest, directed Steiner forest, and a fault-tolerant version of group Steiner tree. In the firefighter problem, we are given a graph and a vertex which is burning. At each time step, we can protect one vertex that is not burning; fire then spreads to all unprotected neighbors of burning vertices. The goal is to maximize the number of vertices that the fire does not reach. On trees, a classic (1-1/e)-approximation algorithm is known via LP rounding. We prove that the integrality gap of the LP matches this approximation, and show significant evidence that additional constraints may improve its integrality gap. On bounded-treewidth graphs, we show that it is NP-hard to find a subpolynomial approximation even on graphs of treewidth 5. We complement this result with an O(1)-approximation on outerplanar graphs.Diese Arbeit untersucht zwei Optimierungsprobleme, von welchen wir wissen, dass sie selbst in Bäumen NP-schwer sind. Wir analysieren Approximationen für diese Probleme in Bäumen und Graphen mit begrenzter Baumweite. Im Gruppensteinerbaumproblem, sind ein Graph und Mengen von Knoten (Gruppen) gegeben; das Ziel ist es, einen Knoten von jeder Gruppe mit minimalen Kosten zu verbinden. Wir beschreiben einen O(log^2 n)-Approximationsalgorithmus für Graphen mit beschränkter Baumweite, dies entspricht der zuvor bekannten unteren Schranke für Bäume und ist zudem eine Verbesserung über die bestmöglichen Resultate die auf anderen Techniken beruhen. Darüber hinaus zeigen wir verbesserte Approximationsresultate für andere Gruppensteinerprobleme. Im Feuerwehrproblem sind ein Graph zusammen mit einem brennenden Knoten gegeben. In jedem Zeitschritt können wir einen Knoten der noch nicht brennt auswählen und diesen vor dem Feuer beschützen. Das Feuer breitet sich anschließend zu allen Nachbarn aus. Das Ziel ist es die Anzahl der Knoten die vom Feuer unberührt bleiben zu maximieren. In Bäumen existiert ein lang bekannter (1-1/e)-Approximationsalgorithmus der auf LP Rundung basiert. Wir zeigen, dass die Ganzzahligkeitslücke des LP tatsächlich dieser Approximation entspricht, und dass weitere Einschränkungen die Ganzzahligkeitslücke möglicherweise verbessern könnten. Für Graphen mit beschränkter Baumweite zeigen wir, dass es NP-schwer ist, eine sub-polynomielle Approximation zu finden

Handlebody diagram algebras

In this paper we study handlebody versions of some classical diagram algebras, most prominently, handlebody versions of Temperley-Lieb, blob, Brauer, BMW, Hecke and Ariki-Koike algebras. Moreover, motivated by Green-Kazhdan-Lusztig's theory of cells, we reformulate the notion of (sandwich, inflated or affine) cellular algebras. We explain this reformulation and how all of the above algebras are part of this theory.Comment: 45 pages, lots of figures, revision following referees comments, to appear in Rev. Mat. Iberoam, comments welcom

Inversión bajo incertidumbre

The aim of this paper is to relate the General Equilibrium Theory and modern Finance Theory. We work with an infinite dimensional economy with a market for goods an actives, and where the number of agents is finite. Each agent has an intertemporal utility function with uncertainty on the states of the world. We characterize the equilibrium with actives and spot goods and analyze sufficient conditions for the existence of equilibrium in this case. We show that it is possible to extend to infinite dimensional models the Chichilnisky theorem on uniqueness of equilibrium. Finally we analyze payoffs and prices of actives that follow Ito processes in a CCAPM frame.

Editorial

[Editorial

A note on the ϵ-indicator subset selection

The ϵ-indicator subset selection selects a subset of a nondominated point set that is as close as possible to a reference point set with respect to the ϵ-indicator. This selection procedure is used by population-based heuristic approaches for multiobjective optimization problems. Given that this procedure is called very often during the run of the heuristic approach, efficient ways of computing the optimal subset are strongly required. In this note, we give a correctness proof of the ϵ-indicator subset selection algorithm proposed by Ponte et al. (2012) [1] for the bidimensional case as well as several algorithmic improvements in terms of time complexity. Extensions to larger dimension are also discussed

Desenvolvimento e implementação de um conceito de restauração comercial

Trabalho de natureza profissional para obtenção do Título de Especialista do Instituto Politécnico do Porto, na área de Hotelaria e Restauração, defendido a 07-05-2015.O trabalho agora apresentado – Desenvolvimento e implementação de um conceito de restauração comercial, no âmbito da candidatura à atribuição do Título de Especialista em Hotelaria e Restauração (área de formação 811), tem por base um projeto do Departamento de Desenvolvimento da empresa Ibersol, do qual o candidato era, há data, responsável. O projeto em questão foi desenvolvido no período compreendido entre Junho e Dezembro de 2006 sendo representadas neste trabalho as diferentes fases do processo de desenvolvimento e implementação de um Conceito de Restauração Comercial no âmbito do Portfolio de Conceitos da Empresa