31 research outputs found

    The Worst-Case Complexity of Symmetric Strategy Improvement

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    Symmetric strategy improvement is an algorithm introduced by Schewe et al. (ICALP 2015) that can be used to solve two-player games on directed graphs such as parity games and mean payoff games. In contrast to the usual well-known strategy improvement algorithm, it iterates over strategies of both players simultaneously. The symmetric version solves the known worst-case examples for strategy improvement quickly, however its worst-case complexity remained open. We present a class of worst-case examples for symmetric strategy improvement on which this symmetric version also takes exponentially many steps. Remarkably, our examples exhibit this behaviour for any choice of improvement rule, which is in contrast to classical strategy improvement where hard instances are usually hand-crafted for a specific improvement rule. We present a generalized version of symmetric strategy iteration depending less rigidly on the interplay of the strategies of both players. However, it turns out it has the same shortcomings

    Tropical Carathéodory with Matroids

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    Bárány’s colorful generalization of Carathéodory’s Theorem combines geometrical and combinatorial constraints. Kalai–Meshulam (2005) and Holmsen (2016) generalized Bárány’s theorem by replacing color classes with matroid constraints. In this note, we obtain corresponding results in tropical convexity, generalizing the Tropical Colorful Carathéodory Theorem of Gaubert–Meunier (2010). Our proof is inspired by geometric arguments and is reminiscent of matroid intersection. Moreover, we show that the topological approach fails in this setting. We also discuss tropical colorful linear programming and show that it is NP-complete. We end with thoughts and questions on generalizations to polymatroids, anti-matroids as well as examples and matroid simplicial depth

    Evolution on coral reefs, with systematic treatments of the pencil and fairy wrasses (Teleostei: Labridae: Pseudojuloides & Cirrhilabrus)

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    Fishes represent half of all of the living species of vertebrates that have been described to date. Most species of fish are marine, with at least a third found exclusively in coral reefs. Yet the phylogenetic relationships of coral reef fishes and the drivers of their diversification remain poorly understood, particularly for taxonomic groups at the incipient stages of speciation. Despite the increasing application of high-throughput sequencing techniques to other taxonomic groups, much of the research conducted on coral reef fishes still relies on more traditional sources of information, such as morphology and mitochondrial sequence data. These methods are unreliable in resolving taxonomically problematic groups such as the Labridae, where many groups are still rapidly radiating, and the processes driving this are not well understood. It is therefore prudent to use a combined, integrative approach using both morphological and high-throughput sequencing techniques. This thesis uses the aforementioned techniques, integrated with morphological studies, to tease apart the relationships for members of the Labridae, in particular the fairy and pencil wrasses (Cirrhilabrus and Pseudojuloides respectively). Additionally, it includes taxonomic descriptions of eight new species, as well as investigations into general themes on coral reefs, including, but not limited to hybridisation, deep reef communities, and historical biogeography

    Tropical complementarity problems and Nash equilibria

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    Linear complementarity programming is a generalization of linear programming which encompasses the computation of Nash equilibria for bimatrix games. While the latter problem is PPAD-complete, we show that the tropical analogue of the complementarity problem associated with Nash equilibria can be solved in polynomial time. Moreover, we prove that the Lemke--Howson algorithm carries over the tropical setting and performs a linear number of pivots in the worst case. A consequence of this result is a new class of (classical) bimatrix games for which Nash equilibria computation can be done in polynomial time

    Valued Constraint Satisfaction Problems over Infinite Domains

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    The object of the thesis is the computational complexity of certain combinatorial optimisation problems called \emph{valued constraint satisfaction problems}, or \emph{VCSPs} for short. The requirements and optimisation criteria of these problems are expressed by sums of \emph{(valued) constraints} (also called \emph{cost functions}). More precisely, the input of a VCSP consists of a finite set of variables, a finite set of cost functions that depend on these variables, and a cost uu; the task is to find values for the variables such that the sum of the cost functions is at most uu. By restricting the set of possible cost functions in the input, a great variety of computational optimisation problems can be modelled as VCSPs. Recently, the computational complexity of all VCSPs for finite sets of cost functions over a finite domain has been classified. Many natural optimisation problems, however, cannot be formulated as VCSPs over a finite domain. We initiate the systematic investigation of infinite-domain VCSPs by studying the complexity of VCSPs for piecewise linear (PL) and piecewise linear homogeneous (PLH) cost functions. The VCSP for a finite set of PLH cost functions can be solved in polynomial time if the cost functions are improved by fully symmetric fractional operations of all arities. We show this by (polynomial-time many-one) reducing the problem to a finite-domain VCSP which can be solved using a linear programming relaxation. We apply this result to show the polynomial-time tractability of VCSPs for {\it submodular} PLH cost functions, for {\it convex} PLH cost functions, and for {\it componentwise increasing} PLH cost functions; in fact, we show that submodular PLH functions and componentwise increasing PLH functions form maximally tractable classes of PLH cost functions. We define the notion of {\it expressive power} for sets of cost functions over arbitrary domains, and discuss the relation between the expressive power and the set of fractional operations improving the same set of cost functions over an arbitrary countable domain. Finally, we provide a polynomial-time algorithm solving the restriction of the VCSP for {\it all} PL cost functions to a fixed number of variables

    Humanity from African Naissance to Coming Millennia

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    Humanity From African Naissance to Coming Millennia arises out of the world's first Dual Congress that was held at Sun City (South Africa) in 1998 that refers to a conjoint, integrated meeting of two international scientific associations, the International Association for the Study of Human Palaeontology - IV Congress - and the International Association of Human Biologists. The volume includes 39 refereed papers covering a wide range of topics, from Human Biology, Human Evolution (Emerging Homo, Evolving Homo, Early Modern Humans), Dating, Taxonomy and Systematics, Diet, Brain Evolution, offering the most recent analyses and interpretations in different areas of evolutionary anthropology.Humanity From African Naissance to Coming Millennia arises out of the world's first Dual Congress that was held at Sun City (South Africa) in 1998 that refers to a conjoint, integrated meeting of two international scientific associations, the International Association for the Study of Human Palaeontology - IV Congress - and the International Association of Human Biologists. The volume includes 39 refereed papers covering a wide range of topics, from Human Biology, Human Evolution (Emerging Homo, Evolving Homo, Early Modern Humans), Dating, Taxonomy and Systematics, Diet, Brain Evolution, offering the most recent analyses and interpretations in different areas of evolutionary anthropology

    Non-acyclicity of coset lattices and generation of finite groups

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