985 research outputs found
Necessary conditions for tractability of valued CSPs
The connection between constraint languages and clone theory has been a
fruitful line of research on the complexity of constraint satisfaction
problems. In a recent result, Cohen et al. [SICOMP'13] have characterised a
Galois connection between valued constraint languages and so-called weighted
clones. In this paper, we study the structure of weighted clones. We extend the
results of Creed and Zivny from [CP'11/SICOMP'13] on types of weightings
necessarily contained in every nontrivial weighted clone. This result has
immediate computational complexity consequences as it provides necessary
conditions for tractability of weighted clones and thus valued constraint
languages. We demonstrate that some of the necessary conditions are also
sufficient for tractability, while others are provably not.Comment: To appear in SIAM Journal on Discrete Mathematics (SIDMA
An Algebraic Theory of Complexity for Discrete Optimization
International audienceDiscrete optimization problems arise in many different areas and are studied under many different names. In many such problems the quantity to be optimized can be expressed as a sum of functions of a restricted form. Here we present a unifying theory of complexity for problems of this kind. We show that the complexity of a finite-domain discrete optimization problem is determined by certain algebraic properties of the objective function, which we call weighted polymorphisms. We define a Galois connection between sets of rational-valued functions and sets of weighted polymorphisms and show how the closed sets of this Galois connection can be characterized. These results provide a new approach to studying the complexity of discrete optimization. We use this approach to identify certain maximal tractable subproblems of the general problem and hence derive a complete classification of complexity for the Boolean case
Counting Constraint Satisfaction Problems
This chapter surveys counting Constraint Satisfaction Problems (counting CSPs, or #CSPs) and their computational complexity. It aims to provide an introduction to the main concepts and techniques, and present a representative selection of results and open problems. It does not cover holants, which are the subject of a separate chapter
The Power of Linear Programming for Valued CSPs
A class of valued constraint satisfaction problems (VCSPs) is characterised
by a valued constraint language, a fixed set of cost functions on a finite
domain. An instance of the problem is specified by a sum of cost functions from
the language with the goal to minimise the sum. This framework includes and
generalises well-studied constraint satisfaction problems (CSPs) and maximum
constraint satisfaction problems (Max-CSPs).
Our main result is a precise algebraic characterisation of valued constraint
languages whose instances can be solved exactly by the basic linear programming
relaxation. Using this result, we obtain tractability of several novel and
previously widely-open classes of VCSPs, including problems over valued
constraint languages that are: (1) submodular on arbitrary lattices; (2)
bisubmodular (also known as k-submodular) on arbitrary finite domains; (3)
weakly (and hence strongly) tree-submodular on arbitrary trees.Comment: Corrected a few typo
Computational strategies for dissecting the high-dimensional complexity of adaptive immune repertoires
The adaptive immune system recognizes antigens via an immense array of
antigen-binding antibodies and T-cell receptors, the immune repertoire. The
interrogation of immune repertoires is of high relevance for understanding the
adaptive immune response in disease and infection (e.g., autoimmunity, cancer,
HIV). Adaptive immune receptor repertoire sequencing (AIRR-seq) has driven the
quantitative and molecular-level profiling of immune repertoires thereby
revealing the high-dimensional complexity of the immune receptor sequence
landscape. Several methods for the computational and statistical analysis of
large-scale AIRR-seq data have been developed to resolve immune repertoire
complexity in order to understand the dynamics of adaptive immunity. Here, we
review the current research on (i) diversity, (ii) clustering and network,
(iii) phylogenetic and (iv) machine learning methods applied to dissect,
quantify and compare the architecture, evolution, and specificity of immune
repertoires. We summarize outstanding questions in computational immunology and
propose future directions for systems immunology towards coupling AIRR-seq with
the computational discovery of immunotherapeutics, vaccines, and
immunodiagnostics.Comment: 27 pages, 2 figure
Flow-augmentation III: Complexity dichotomy for Boolean CSPs parameterized by the number of unsatisfied constraints
We study the parameterized problem of satisfying ``almost all'' constraints
of a given formula over a fixed, finite Boolean constraint language
, with or without weights. More precisely, for each finite Boolean
constraint language , we consider the following two problems. In Min
SAT, the input is a formula over and an integer , and
the task is to find an assignment that
satisfies all but at most constraints of , or determine that no such
assignment exists. In Weighted Min SAT), the input additionally
contains a weight function and an integer ,
and the task is to find an assignment such that (1) satisfies
all but at most constraints of , and (2) the total weight of the
violated constraints is at most . We give a complete dichotomy for the
fixed-parameter tractability of these problems: We show that for every Boolean
constraint language , either Weighted Min SAT is FPT; or
Weighted Min SAT is W[1]-hard but Min SAT is FPT; or Min
SAT is W[1]-hard. This generalizes recent work of Kim et al. (SODA
2021) which did not consider weighted problems, and only considered languages
that cannot express implications (as is used to, e.g.,
model digraph cut problems). Our result generalizes and subsumes multiple
previous results, including the FPT algorithms for Weighted Almost 2-SAT,
weighted and unweighted -Chain SAT, and Coupled Min-Cut, as well as
weighted and directed versions of the latter. The main tool used in our
algorithms is the recently developed method of directed flow-augmentation (Kim
et al., STOC 2022)
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
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