On parallel versus sequential approximation

In this paper we deal with the class NCX of NP Optimization problems that are approximable within constant ratio in NC. This class is the parallel counterpart of the class APX. Our main motivation here is to reduce the study of sequential and parallel approximability to the same framework. To this aim, we first introduce a new kind of NC-reduction that preserves the relative error of the approximate solutions and show that the class NCX has {em complete} problems under this reducibility. An important subset of NCX is the class MAXSNP, we show that MAXSNP-complete problems have a threshold on the parallel approximation ratio that is, there are positive constants $epsilon_1$, $epsilon_2$ such that although the problem can be approximated in P within $epsilon_1$ it cannot be approximated in NC within epsilon_2$, unless P=NC. This result is attained by showing that the problem of approximating the value obtained through a non-oblivious local search algorithm is P-complete, for some values of the approximation ratio. Finally, we show that approximating through non-oblivious local search is in average NC.Postprint (published version

CS 740: Algorithms, Complexity and the Theory of Computability

The objective of this course is to use the formal algorithmic system provided by Turing machines as a tool to analyze the complexity of decision and optimization problems and the algorithms that solve them. The topics to be covered include • the definition of the time and space complexity of a deterministic algorithm• the classes of deterministic polynomial and non-polynomial time languages• the complexity of nondeterministic algorithms• the P=NP question (relationship between solvability by deterministic and nondeterministic polynomial time algorithms)• the implications of a solution to the P=NP question• NP completeness and examples of NP complete problems• classes of NP complete problems• techniques for approximate solutions of NP complete problem

An Approximate Algorithm Combining P Systems and Ant Colony Optimization for Traveling Salesman Problems

This paper proposes an approximate optimization algorithm combining P systems with ant colony optimization, called ACOPS, to solve traveling salesman prob- lems, which are well-known and extensively studied NP-complete combinatorial optimization problems. ACOPS uses the pheromone model and pheromone update rules defined by ant colony optimization algorithms, and the hierarchical membrane structure and transformation/communication rules of P systems. First, the parameter setting of the ACOPS is discussed. Second, extensive experiments and statistical analysis are investigated. It is shown that the ACOPS is superior to Nishida's algorithms and its counterpart ant colony optimization algorithms, in terms of the quality of solutions and the number of function evaluations

Discriminating Codes in Geometric Setups

We study geometric variations of the discriminating code problem. In the \emph{discrete version} of the problem, a finite set of points $P$ and a finite set of objects $S$ are given in $\mathbb{R}^d$. The objective is to choose a subset $S^* \subseteq S$ of minimum cardinality such that for each point $p_i \in P$, the subset $S_i^* \subseteq S^*$ covering $p_i$ satisfies $S_i^*\neq \emptyset$, and each pair $p_i,p_j \in P$, $i \neq j$, we have $S_i^* \neq S_j^*$. In the \emph{continuous version} of the problem, the solution set $S^*$ can be chosen freely among a (potentially infinite) class of allowed geometric objects. In the 1-dimensional case ($d=1$), the points in $P$ are placed on a horizontal line $L$, and the objects in $S$ are finite-length line segments aligned with $L$ (called intervals). We show that the discrete version of this problem is NP-complete. This is somewhat surprising as the continuous version is known to be polynomial-time solvable. Still, for the 1-dimensional discrete version, we design a polynomial-time $2$-approximation algorithm. We also design a PTAS for both discrete and continuous versions in one dimension, for the restriction where the intervals are all required to have the same length. We then study the 2-dimensional case ($d=2$) for axis-parallel unit square objects. We show that both continuous and discrete versions are NP-complete, and design polynomial-time approximation algorithms that produce $(16\cdot OPT+1)$-approximate and $(64\cdot OPT+1)$-approximate solutions respectively, using rounding of suitably defined integer linear programming problems. We show that the identifying code problem for axis-parallel unit square intersection graphs (in $d=2$) can be solved in the same manner as for the discrete version of the discriminating code problem for unit square objects

Complexity and heuristics in ruled-based algorithmic music composition

Successful algorithmic music composition requires the efficient creation of works that reflect human preferences. In examining this key issue, we make two main contributions in this dissertation: analysis of the computational complexity of algorithmic music composition, and methods to produce music that approximates a commendable human effort. We use species counterpoint as our compositional model, wherein a set of stylistic and grammatical rules governs the search for suitable countermelodies to match a given melody. Our analysis of the complexity of rule-based music composition considers four different types of computational problems: decision, enumeration, number, and optimization. For restricted versions of the decision problem, we devise a polynomial algorithm by constructing a non-deterministic finite state transducer. This transducer can also solve corresponding restricted versions of the enumeration and number problems. The general forms of the four types of problems, however, are respectively NP-complete, #P-complete, NP-complete in the strong sense, and NP-equivalent. We prove this by first reducing from the well known Three-Dimensional Matching problem to the music composition decision problem, and then by reducing among the music problems themselves. In order to compose music both correct and human-like, we formulate new “artistry” rules to supplement traditional rules of musical style and grammar. We also propose the fuzzy application of these artistry rules, to complement the crisp application of the traditional rules. We then suggest two methods to model human preferences: (1) distinguish an expert’s compositions from alternative compositions by determining rule weights; (2) train an artificial neural network to reflect an expert’s musical preferences through the latter’s evaluations of a set of compositions. We were able to approximate that elusive factor of human preference with better than 75% accuracy. To solve the optimization problem, we adapt two different search algorithms: best-first search with branch-and-bound pruning (for m ≥ 1 optimal solutions), and a genetic algorithm (for m ≥ 1 near-optimal solutions). Through these algorithms, we test the techniques of rule weightings and of trained neural networks as evaluation functions. Our adaptation of the genetic algorithm produced optimal countermelodies in execution time favorably comparable to that taken by the best-first algorithm

Sampling-based Approximation Algorithms for Multi-stage Stochastic Optimization

Stochastic optimization problems provide a means to model uncertainty in the input data where the uncertainty is modeled by a probability distribution over the possible realizations of the data. We consider a broad class of these problems, called {it multi-stage stochastic programming problems with recourse}, where the uncertainty evolves through a series of stages and one take decisions in each stage in response to the new information learned. These problems are often computationally quite difficult with even very specialized (sub)problems being #P-complete. We obtain the first fully polynomial randomized approximation scheme (FPRAS) for a broad class of multi-stage stochastic linear programming problems with any constant number of stages, without placing any restrictions on the underlying probability distribution or on the cost structure of the input. For any fixed $k$, for a rich class of $k$-stage stochastic linear programs (LPs), we show that, for any probability distribution, for any $epsilon>0$, one can compute, with high probability, a solution with expected cost at most $(1+e)$ times the optimal expected cost, in time polynomial in the input size, $frac{1}{epsilon}$, and a parameter $lambda$ that is an upper bound on the cost-inflation over successive stages. Moreover, the algorithm analyzed is a simple and intuitive algorithm that is often used in practice, the {it sample average approximation} (SAA) method. In this method, one draws certain samples from the underlying distribution, constructs an approximate distribution from these samples, and solves the stochastic problem given by this approximate distribution. This is the first result establishing that the SAA method yields near-optimal solutions for (a class of) multi-stage programs with a polynomial number of samples. As a corollary of this FPRAS, by adapting a generic rounding technique of Shmoys and Swamy, we also obtain the first approximation algorithms for the analogous class of multi-stage stochastic integer programs, which includes the multi-stage versions of the set cover, vertex cover, multicut on trees, facility location, and multicommodity flow problems

Numerical Methods for Non-divergence Form Second Order Linear Elliptic Partial Differential Equations and Discontinuous Ritz Methods for Problems from the Calculus of Variations

This dissertation consists of three integral parts. Part one studies discontinuous Galerkin approximations of a class of non-divergence form second order linear elliptic PDEs whose coefficients are only continuous. An interior penalty discontinuous Galerkin (IP-DG) method is developed for this class of PDEs. A complete analysis of the proposed IP-DG method is carried out, which includes proving the stability and error estimate in a discrete W2;p-norm [W^2,p-norm]. Part one also studies the convergence of the vanishing moment method for this class of PDEs. The vanishing moment method refers to a PDE technique for approximating these PDEs by a family of fourth order PDEs. Detailed proofs of uniform H1 [H^1] and H2 [H^2]-stability estimates for the approximate solutions and their convergence are presented. Part two studies finite element approximations of a class of calculus of variations problems which exhibit so-called Lavrentiev gap phenomenon (LGP), whose solutions often contain singularities. The LGP incapacitates all standard numerical methods, especially the finite element method, as they fail to produce a correct approximate solution. To overcome the difficulty, an enhanced finite element method based on a truncation technique is developed in this part of the dissertation. The proposed enhanced finite element method is shown to numerically converge on several benchmark problems with the LGP. Part three of the dissertation develops a discontinuous Galerkin numerical framework for general calculus of variations problems, which is called the discontinuous Ritz (DR) methodology and can be regarded as the counterpart of the discontinuous Galerkin (DG) methodology for PDEs. Conceptually, it approximates the admissible space by the DG spaces which consist of totally discontinuous piecewise polynomials and approximates the underlying energy functional by discrete energy functionals defined on the DG spaces. The main idea here is to construct the desired discrete energy functional by using the newly developed DG finite element calculus theory, which only requires replacing the gradient operator in the energy functional by the corresponding DG finite element discrete gradient and adding the standard interior penalty terms. It is shown that for a certain class of functionals the proposed DR method does indeed converge to the true solution

Analytic approximation of solutions of parabolic partial differential equations with variable coefficients

A complete family of solutions for the one-dimensional reaction-diffusion equation $$u_{xx}(x,t)-q(x)u(x,t) = u_t(x,t)$$ with a coefficient $q$ depending on $x$ is constructed. The solutions represent the images of the heat polynomials under the action of a transmutation operator. Their use allows one to obtain an explicit solution of the noncharacteristic Cauchy problem for the considered equation with sufficiently regular Cauchy data as well as to solve numerically initial boundary value problems. In the paper the Dirichlet boundary conditions are considered however the proposed method can be easily extended onto other standard boundary conditions. The proposed numerical method is shown to reveal good accuracy.Comment: 8 pages, 1 figure. Minor updates to the tex