9,508 research outputs found
Frameworks for logically classifying polynomial-time optimisation problems.
We show that a logical framework, based around a fragment of existential second-order logic formerly proposed by others so as to capture the class of polynomially-bounded P-optimisation problems, cannot hope to do so, under the assumption that P ≠ NP. We do this by exhibiting polynomially-bounded maximisation and minimisation problems that can be expressed in the framework but whose decision versions are NP-complete. We propose an alternative logical framework, based around inflationary fixed-point logic, and show that we can capture the above classes of optimisation problems. We use the inductive depth of an inflationary fixed-point as a means to describe the objective functions of the instances of our optimisation problems
Applications of Finite Model Theory: Optimisation Problems, Hybrid Modal Logics and Games.
There exists an interesting relationships between two seemingly distinct fields: logic from the field of Model Theory, which deals with the truth of statements about discrete structures; and Computational Complexity, which deals with the classification of problems by how much of a particular computer resource is required in order to compute a solution. This relationship is known as Descriptive Complexity and it is the primary application of the tools from Model Theory when they are restricted to the finite; this restriction is commonly called Finite Model Theory.
In this thesis, we investigate the extension of the results of Descriptive Complexity from classes of decision problems to classes of optimisation problems. When dealing with decision problems the natural mapping from true and false in logic to yes and no instances of a problem is used but when dealing with optimisation problems, other features of a logic need to be used. We investigate what these features are and provide results in the form of logical frameworks that can be used for describing optimisation problems in particular classes, building on the existing research into this area.
Another application of Finite Model Theory that this thesis investigates is the relative expressiveness of various fragments of an extension of modal logic called hybrid modal logic. This is achieved through taking the Ehrenfeucht-Fraïssé game from Model Theory and modifying it so that it can be applied to hybrid modal logic. Then, by developing winning strategies for the players in the game, results are obtained that show strict hierarchies of expressiveness for fragments of hybrid modal logic that are generated by varying the quantifier depth and the number of proposition and nominal symbols available
Complexity of Discrete Energy Minimization Problems
Discrete energy minimization is widely-used in computer vision and machine
learning for problems such as MAP inference in graphical models. The problem,
in general, is notoriously intractable, and finding the global optimal solution
is known to be NP-hard. However, is it possible to approximate this problem
with a reasonable ratio bound on the solution quality in polynomial time? We
show in this paper that the answer is no. Specifically, we show that general
energy minimization, even in the 2-label pairwise case, and planar energy
minimization with three or more labels are exp-APX-complete. This finding rules
out the existence of any approximation algorithm with a sub-exponential
approximation ratio in the input size for these two problems, including
constant factor approximations. Moreover, we collect and review the
computational complexity of several subclass problems and arrange them on a
complexity scale consisting of three major complexity classes -- PO, APX, and
exp-APX, corresponding to problems that are solvable, approximable, and
inapproximable in polynomial time. Problems in the first two complexity classes
can serve as alternative tractable formulations to the inapproximable ones.
This paper can help vision researchers to select an appropriate model for an
application or guide them in designing new algorithms.Comment: ECCV'16 accepte
The Complexity of Rooted Phylogeny Problems
Several computational problems in phylogenetic reconstruction can be
formulated as restrictions of the following general problem: given a formula in
conjunctive normal form where the literals are rooted triples, is there a
rooted binary tree that satisfies the formula? If the formulas do not contain
disjunctions, the problem becomes the famous rooted triple consistency problem,
which can be solved in polynomial time by an algorithm of Aho, Sagiv,
Szymanski, and Ullman. If the clauses in the formulas are restricted to
disjunctions of negated triples, Ng, Steel, and Wormald showed that the problem
remains NP-complete. We systematically study the computational complexity of
the problem for all such restrictions of the clauses in the input formula. For
certain restricted disjunctions of triples we present an algorithm that has
sub-quadratic running time and is asymptotically as fast as the fastest known
algorithm for the rooted triple consistency problem. We also show that any
restriction of the general rooted phylogeny problem that does not fall into our
tractable class is NP-complete, using known results about the complexity of
Boolean constraint satisfaction problems. Finally, we present a pebble game
argument that shows that the rooted triple consistency problem (and also all
generalizations studied in this paper) cannot be solved by Datalog
Constraint Satisfaction Problems over Numeric Domains
We present a survey of complexity results for constraint satisfaction problems (CSPs) over the integers, the rationals, the reals, and the complex numbers. Examples of such problems are feasibility of linear programs, integer linear programming, the max-atoms problem, Hilbert\u27s tenth problem, and many more. Our particular focus is to identify those CSPs that can be solved in polynomial time, and to distinguish them from CSPs that are NP-hard. A very helpful tool for obtaining complexity classifications in this context is the concept of a polymorphism from universal algebra
Distance Constraint Satisfaction Problems
We study the complexity of constraint satisfaction problems for templates
that are first-order definable in , the integers with
the successor relation. Assuming a widely believed conjecture from finite
domain constraint satisfaction (we require the tractability conjecture by
Bulatov, Jeavons and Krokhin in the special case of transitive finite
templates), we provide a full classification for the case that Gamma is locally
finite (i.e., the Gaifman graph of has finite degree). We show that
one of the following is true: The structure Gamma is homomorphically equivalent
to a structure with a d-modular maximum or minimum polymorphism and
can be solved in polynomial time, or is
homomorphically equivalent to a finite transitive structure, or
is NP-complete.Comment: 35 pages, 2 figure
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