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Abstract: "We investigate here NP optimization problems from a logically definability standpoint. We show that the class of optimization problems whose optimum is definable using first-order formulae coincides with the class of polynomially bounded NP optimization problems on finite structures. After this, we analyze the relative expressive power of various classes of optimization problems that arise in this framework. Some of our results show that logical definability has different implications for NP maximization problems than it has for NP minimization problems, in terms of both expressive power and approximation properties."
Abstract: "Extending a well known property of polynomially bounded NP optimization problems, we observe that, by attaching weights to tuples over the domain of the input, all NP optimization problems admit a certain logical characterization. We show that any NP optimization problem can be stated as a problem in which the constraint conditions can be expressed by a II2 first-order formula and this is the best possible result. We further analyze the weighted analogue of all syntactically defined classes of optimization problems that are known to have good approximation properties: MAX NP, MAX SNP, MAX SNP([pi]), MIN F[pi]1 and MIN F[pi]2(1). All these classes continue to have the same approximation properties in the case of positive weights. Using reductions from multiprover interactive systems, we show that if NP [not subset of] DTIME[2[superscript log[superscript O(1)] n]], the approximation properties of the above classes devaluates considerably when negative weights are also allowed (with the exception of MIN F+II1, where only a weaker deterioration could be proven). It follows that the general weighted versions of MAX 2SAT, Set Cover, Priority Ordering and of some other closely related natural problems are not approximable in quasipolynomial time within a factor of 2[superscript log [superscript [mu]] n] for some [mu]> 0 (which depends on the problem), unless NP [subset of] DTIME[2[superscript log [superscript O(1)] n]]. Under the same hypothesis, we show that the maximization variant of Set Partition is also not superpolylog approximable. A stronger result is proven for the minimization variant of Set Partition: if P [not equal to] NP, then MIN Set Partition cannot be approximated in polynomial time within a factor of n[supersript O(1)]."
Abstract: "In this paper we introduce a new approach to the logical definability of NP optimization problems by focusing on the expressibility of feasible solutions. We show that in this framework first-order sentences capture exactly all polynomially bounded optimization problems. We also show that, assuming P [not equal to] NP, it is an undecidable problem to tell if a given first-order sentence defines an approximable optimization problem. We then isolate a syntactically defined class of NP minimization problems that contains the MIN SET COVER problem and has the property that every problem in it has a logarithmic approximation algorithm. We conclude by giving a machine-independent characterization of the NP [equals?] co-NP problem in terms of logical expressibility of the MAX CLIQUE problem."
Logic and Complexity looks at basic logic as it is used in Computer Science, and provides students with a logical approach to Complexity theory. With plenty of exercises, this book presents classical notions of mathematical logic, such as decidability, completeness and incompleteness, as well as new ideas brought by complexity theory such as NP-completeness, randomness and approximations, providing a better understanding for efficient algorithmic solutions to problems. Divided into three parts, it covers: - Model Theory and Recursive Functions - introducing the basic model theory of propositional, 1st order, inductive definitions and 2nd order logic. Recursive functions, Turing computability and decidability are also examined. - Descriptive Complexity - looking at the relationship between definitions of problems, queries, properties of programs and their computational complexity. - Approximation - explaining how some optimization problems and counting problems can be approximated according to their logical form. Logic is important in Computer Science, particularly for verification problems and database query languages such as SQL. Students and researchers in this field will find this book of great interest.
The themes of the 1997 conference are new theoretical and practical accomplishments in logic programming, new research directions where ideas originating from logic programming can play a fundamental role, and relations between logic programming and other fields of computer science. The annual International Logic Programming Symposium, traditionally held in North America, is one of the main international conferences sponsored by the Association of Logic Programming. The themes of the 1997 conference are new theoretical and practical accomplishments in logic programming, new research directions where ideas originating from logic programming can play a fundamental role, and relations between logic programming and other fields of computer science. Topics include theoretical foundations, constraints, concurrency and parallelism, deductive databases, language design and implementation, nonmonotonic reasoning, and logic programming and the Internet.
The adiabatic quantum computation (AQC) is based on the adiabatic theorem to approximate solutions of the Schrödinger equation. The design of an AQC algorithm involves the construction of a Hamiltonian that describes the behavior of the quantum system. This Hamiltonian is expressed as a linear interpolation of an initial Hamiltonian whose ground state is easy to compute, and a final Hamiltonian whose ground state corresponds to the solution of a given combinatorial optimization problem. The adiabatic theorem asserts that if the time evolution of a quantum system described by a Hamiltonian is large enough, then the system remains close to its ground state. An AQC algorithm uses the adiabatic theorem to approximate the ground state of the final Hamiltonian that corresponds to the solution of the given optimization problem. In this book, we investigate the computational simulation of AQC algorithms applied to the MAX-SAT problem. A symbolic analysis of the AQC solution is given in order to understand the involved computational complexity of AQC algorithms. This approach can be extended to other combinatorial optimization problems and can be used for the classical simulation of an AQC algorithm where a Hamiltonian problem is constructed. This construction requires the computation of a sparse matrix of dimension 2n × 2n, by means of tensor products, where n is the dimension of the quantum system. Also, a general scheme to design AQC algorithms is proposed, based on a natural correspondence between optimization Boolean variables and quantum bits. Combinatorial graph problems are in correspondence with pseudo-Boolean maps that are reduced in polynomial time to quadratic maps. Finally, the relation among NP-hard problems is investigated, as well as its logical representability, and is applied to the design of AQC algorithms. It is shown that every monadic second-order logic (MSOL) expression has associated pseudo-Boolean maps that can be obtained by expanding the given expression, and also can be reduced to quadratic forms. Table of Contents: Preface / Acknowledgments / Introduction / Approximability of NP-hard Problems / Adiabatic Quantum Computing / Efficient Hamiltonian Construction / AQC for Pseudo-Boolean Optimization / A General Strategy to Solve NP-Hard Problems / Conclusions / Bibliography / Authors' Biographies
This book constitutes the refereed proceedings of the 30th International Symposium on Mathematical Foundations of Computer Science, MFCS 2005, held in Gdansk, Poland in August/September 2005. The 62 revised full papers presented together with full papers or abstracts of 7 invited talks were carefully reviewed and selected from 137 submissions. All current aspects in theoretical computer science are addressed, ranging from quantum computing, approximation, automata, circuits, scheduling, games, languages, discrete mathematics, combinatorial optimization, graph theory, networking, algorithms, and complexity to programming theory, formal methods, and mathematical logic.
This book constitutes the refereed proceedings of the Third International Workshop on Parameterized and Exact Computation, IWPEC 2008, held in Victoria, Canada, in May 2008 - co-located with the 40th ACM Symposium on Theory of Computing, STOC 2008. The 17 revised full papers presented together with 3 invited lectures were carefully reviewed and selected from 32 submissions. The topics addressed cover research in all aspects of parameterized and exact computation and complexity, including but not limited to new techniques for the design and analysis of parameterized and exact algorithms, parameterized complexity theory, relationship between parameterized complexity and traditional complexity classifications, applications of parameterized computation, implementation and experiments, high-performance computing and fixed-parameter tractability.
This book is a state-of-the-art introduction into both algorithmic techniques for fixed-parameter tractability and the structural theory of parameterized complexity classes. It presents detailed proofs of recent advanced results that have not appeared in book form before and replaces the earlier publication "Parameterized Complexity" by Downey and Fellows as the definitive book on this subject. The book will interest computer scientists, mathematicians and graduate students engaged with algorithms and problem complexity.