Tropically convex constraint satisfaction

A semilinear relation S is max-closed if it is preserved by taking the componentwise maximum. The constraint satisfaction problem for max-closed semilinear constraints is at least as hard as determining the winner in Mean Payoff Games, a notorious problem of open computational complexity. Mean Payoff Games are known to be in the intersection of NP and co-NP, which is not known for max-closed semilinear constraints. Semilinear relations that are max-closed and additionally closed under translations have been called tropically convex in the literature. One of our main results is a new duality for open tropically convex relations, which puts the CSP for tropically convex semilinaer constraints in general into NP intersected co-NP. This extends the corresponding complexity result for scheduling under and-or precedence constraints, or equivalently the max-atoms problem. To this end, we present a characterization of max-closed semilinear relations in terms of syntactically restricted first-order logic, and another characterization in terms of a finite set of relations L that allow primitive positive definitions of all other relations in the class. We also present a subclass of max-closed constraints where the CSP is in P; this class generalizes the class of max-closed constraints over finite domains, and the feasibility problem for max-closed linear inequalities. Finally, we show that the class of max-closed semilinear constraints is maximal in the sense that as soon as a single relation that is not max-closed is added to L, the CSP becomes NP-hard.Comment: 29 pages, 2 figure

Tractability in Constraint Satisfaction Problems: A Survey

International audienceEven though the Constraint Satisfaction Problem (CSP) is NP-complete, many tractable classes of CSP instances have been identified. After discussing different forms and uses of tractability, we describe some landmark tractable classes and survey recent theoretical results. Although we concentrate on the classical CSP, we also cover its important extensions to infinite domains and optimisation, as well as #CSP and QCSP

First Class Objects and Indexes for Chant Manuscripts

We discuss a crucial part of infrastructure for the Web-delivery of medieval chant resources. Although widely accepted by software professionals, the distributed-content model is sharply opposed by some chant scholars. We advocate for a paradigm of the Web as a massive database where each "first class object" acts like a record; metadata about, and links to such objects are compiled in virtual libraries. Scholarly-edited indexes determine which objects are in libraries, and unreliable content is excluded. Special metadata ontologies can be defined without modifying the primary content. Copyright 2007 ACM

Feedback from nature: simple randomised distributed algorithms for maximal independent set selection and greedy colouring

We propose distributed algorithms for two wellestablished problems that operate efficiently under extremely harsh conditions. Our algorithms achieve state-of-the-art performance in a simple and novel way. Our algorithm for maximal independent set selection operates on a network of identical anonymous processors. The processor at each node has no prior information about the network. At each time step, each node can only broadcast a single bit to all its neighbours, or remain silent. Each node can detect whether one or more neighbours have broadcast, but cannot tell how many of its neighbours have broadcast, or which ones. We build on recent work of Afek et al. which was inspired by studying the development of a network of cells in the fruit fly [2]. However we incorporate for the first time another important feature of the biological system: varying the probability value used at each node based on local feedback from neighbouring nodes. Given any n-node network, our algorithm achieves with high probability the optimal time complexity of O(logn) rounds and the optimal expected message complexity of O(1) single-bit messages broadcast by each node. We also show that the previous approach, without feedback, cannot achieve better than Ω(log2 n) time complexity with high probability, whatever global scheme is used to choose the probabilities. Our algorithm for distributed greedy colouring works under similar harsh conditions: each identical node has no prior information about the network, can only broadcast a single message to all neighbours at each time step representing a desired colour, and can only detect whether at least one neighbour has broadcast each colour value. We show that with high probability our algorithm has a time complexity of O(∆ +logn), where ∆ is the maximum degree of the network, and also has an expected message complexity of O(1) messages broadcast by each node

The Complexity of Valued Constraint Models.

The Valued Constraint Satisfaction Problem ( ) is a general framework encompassing many optimisation problems. We discuss precisely what it means for a problem to be modelled in the framework. Using our analysis, we show that some optimisation problems, such as (s,t)-Min-Cut and Submodular Function Minimisation, can be modelled using a restricted set of valued constraints which are tractable to solve regardless of how they are combined. Hence, these problems can be viewed as special cases of more general problems which include all possible instances using the same forms of valued constraint. However, other, apparently similar, problems such as Min-Cut and Symmetric Submodular Function Minimisation, which also have polynomial-time algorithms, can only be naturally modelled in the framework by using valued constraints which can represent NP-complete problems. This suggests that the reason for tractability in these problems is more subtle; it relies not only on the form of the valued constraints, but also on the precise structure of the problem. Furthermore, our results suggest that allowing constant constraints can significantly alter the complexity of problems in the framework, in contrast to the framework. © 2009 Springer Berlin Heidelberg

E-library of medieval chant manuscript transcriptions

In this paper we present our rationale and design principles for a distributed e-library of medieval chant manuscript transcriptions. We describe the great variety in neumatic notations, in order to motivate a standardised data representation that is lossless and universal with respect to these musical artefacts. We present some details of the data representation and an XML Schema for describing and delivering transcriptions via the Web. We argue against proposed data formats that look simpler, on the grounds that they will inevitably lead to fragmentation of digital libraries. We plan to develop applications software that will allow users to take full advantage of the carefully designed representation we describe, while shielding users from its complexity. We argue that a distributed e-library of this kind will greatly facilitate scholarship, education, and public appreciation of these artefacts. Copyright 2005 ACM

Generalizing constraint satisfaction on trees: Hybrid tractability and variable elimination

The Constraint Satisfaction Problem (CSP) is a central generic problem in artificial intelligence. Considerable progress has been made in identifying properties which ensure tractability in such problems, such as the property of being tree-structured. In this paper we introduce the broken-triangle property, which allows us to define a novel tractable class for this problem which significantly generalizes the class of problems with tree structure. We show that the broken-triangle property is conservative (i.e., it is preserved under domain reduction and hence under arc consistency operations) and that there is a polynomial-time algorithm to determine an ordering of the variables for which the broken-triangle property holds (or to determine that no such ordering exists). We also present a non-conservative extension of the broken-triangle property which is also sufficient to ensure tractability and can also be detected in polynomial time. We show that both the broken-triangle property and its extension can be used to eliminate variables, and that both of these properties provide the basis for preprocessing procedures that yield unique closures orthogonal to value elimination by enforcement of consistency. Finally, we also discuss the possibility of using the broken-triangle property in variable-ordering heuristics. © 2010 Elsevier B.V. All rights reserved

E-library of medieval chant manuscript transcriptions

In this paper we present our rationale and design principles for a distributed e-library of medieval chant manuscript transcriptions. We describe the great variety in neumatic notations, in order to motivate a standardised data representation that is lossless and universal with respect to these musical artefacts. We present some details of the data representation and an XML Schema for describing and delivering transcriptions via the Web. We argue against proposed data formats that look simpler, on the grounds that they will inevitably lead to fragmentation of digital libraries. We plan to develop applications software that will allow users to take full advantage of the carefully designed representation we describe, while shielding users from its complexity. We argue that a distributed e-library of this kind will greatly facilitate scholarship, education, and public appreciation of these artefacts. Copyright 2005 ACM

Galois connections for patterns: an algebra of labelled graphs

A pattern is a generic instance of a binary constraint satisfaction problem (CSP) in which the compatibility of certain pairs of variable-value assignments may be unspecified. The notion of forbidden pattern has led to the discovery of several novel tractable classes for the CSP. However, for this field to come of age it is time for a theoretical study of the algebra of patterns. We present a Galois connection between lattices composed of sets of forbidden patterns and sets of generic instances, and investigate its consequences. We then extend patterns to augmented patterns and exhibit a similar Galois connection. Augmented patterns are a more powerful language than flat (i.e. non-augmented) patterns, as we demonstrate by showing that, for any k≥ 1, instances with tree-width bounded by k cannot be specified by forbidding a finite set of flat patterns but can be specified by a finite set of augmented patterns. A single finite set of augmented patterns can also describe the class of instances such that each instance has a weak near-unanimity polymorphism of arity k (thus covering all tractable language classes).We investigate the power of forbidding augmented patterns and discuss their potential for describing new tractable classes