1,383 research outputs found
Valued Constraint Satisfaction Problems over Infinite Domains
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 ; the task is to find values for the variables such that the sum of the cost functions is at most .
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
The Combined Basic LP and Affine IP Relaxation for Promise VCSPs on Infinite Domains
Convex relaxations have been instrumental in solvability of constraint satisfaction problems (CSPs), as well as in the three different generalisations of CSPs: valued CSPs, infinite-domain CSPs, and most recently promise CSPs. In this work, we extend an existing tractability result to the three generalisations of CSPs combined: We give a sufficient condition for the combined basic linear programming and affine integer programming relaxation for exact solvability of promise valued CSPs over infinite-domains. This extends a result of Brakensiek and Guruswami [SODA\u2720] for promise (non-valued) CSPs (on finite domains)
The combined basic LP and affine IP relaxation for promise VCSPs on infinite domains
Convex relaxations have been instrumental in solvability of constraint
satisfaction problems (CSPs), as well as in the three different generalisations
of CSPs: valued CSPs, infinite-domain CSPs, and most recently promise CSPs. In
this work, we extend an existing tractability result to the three
generalisations of CSPs combined: We give a sufficient condition for the
combined basic linear programming and affine integer programming relaxation for
exact solvability of promise valued CSPs over infinite-domains. This extends a
result of Brakensiek and Guruswami [SODA'20] for promise (non-valued) CSPs (on
finite domains).Comment: Full version of an MFCS'20 pape
The right to the city to contrast the decay of urban spaces : the architectural upgrade of Corso Garibaldi railway station, in Naples
Deep changes in urban frameworks, challenge contemporary cities, where often,
common spaces formerly central, risk an indissoluble physical degradation, due to
processes of social marginalization. Architects are asked to face the developmental
dynamics for spaces of anonymity, every day more numerous, working on previous
attitudes as aggregation and relation junctions. Moving from the description of the
design criteria that inform the rehabilitation for the ancient area of Corso Garibaldi
in Naples, the paper introduces a critical thinking about the links between space’
performances and attitudes towards inclusivity. Urban and architectural solutions are
privileged means in order to return to citizenship the right to public space, reaffirming
the concept of common space as dwelling, residence of the community, with the
creation of new social ties and the growth of local shared identities.Peer Reviewe
Submodular Functions and Valued Constraint Satisfaction Problems over Infinite Domains
Valued constraint satisfaction problems (VCSPs) are a large class of combinatorial optimisation problems. It is desirable to classify the computational complexity of VCSPs depending on a fixed set of allowed cost functions in the input. Recently, the computational complexity of all VCSPs for finite sets of cost functions over finite domains has been classified in this sense. 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 homogeneous cost functions. We remark that in this paper the infinite domain will always be the set of rational numbers. We show that such VCSPs can be solved in polynomial time when the cost functions are additionally submodular, and that this is indeed a maximally tractable class: adding any cost function that is not submodular leads to an NP-hard VCSP
When the catenary degree agrees with the tame degree in numerical semigroups of embedding dimension three
We characterize numerical semigroups of embedding dimension three having the same catenary and tame degrees.García Sánchez is supported by the projects MTM2010-15595, FQM-343, FQM-5849, and FEDER funds. The contents of this article are part of Viola’s master’s thesis. Part of this work was done while she visited the Univerisidad de Granada under the European Erasmus mobility program
Longest Common Substring and Longest Palindromic Substring in Time
The Longest Common Substring (LCS) and Longest Palindromic Substring (LPS)
are classical problems in computer science, representing fundamental challenges
in string processing. Both problems can be solved in linear time using a
classical model of computation, by means of very similar algorithms, both
relying on the use of suffix trees. Very recently, two sublinear algorithms for
LCS and LPS in the quantum query model have been presented by Le Gall and
Seddighin~\cite{GallS23}, requiring and
queries, respectively. However, while the query
model is fascinating from a theoretical standpoint, its practical applicability
becomes limited when it comes to crafting algorithms meant for actual execution
on real hardware. In this paper we present, for the first time, a
quantum algorithm for both LCS and LPS working
in the circuit model of computation. Our solutions are simpler than previous
ones and can be easily translated into quantum procedures. We also present
actual implementations of the two algorithms as quantum circuits working in
and time,
respectively
Algebraic Approach to Approximation
Following the success of the so-called algebraic approach to the study of
decision constraint satisfaction problems (CSPs), exact optimization of valued
CSPs, and most recently promise CSPs, we propose an algebraic framework for
valued promise CSPs.
To every valued promise CSP we associate an algebraic object, its so-called
valued minion. Our main result shows that the existence of a homomorphism
between the associated valued minions implies a polynomial-time reduction
between the original CSPs. We also show that this general reduction theorem
includes important inapproximability results, for instance, the
inapproximability of almost solvable systems of linear equations beyond the
random assignment threshold
Algebraic approach to approximation
Following the success of the so-called algebraic approach to the study of decision constraint satisfaction problems (CSPs), exact optimization of valued CSPs, and most recently promise CSPs, we propose an algebraic framework for valued promise CSPs. To every valued promise CSP we associate an algebraic object, its so-called valued minion. Our main result shows that the existence of a homomorphism between the associated valued minions implies a polynomial-time reduction between the original CSPs. We also show that this general reduction theorem includes important inapproximability results, for instance, the inapproximability of almost solvable systems of linear equations beyond the random assignment threshold
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