A Circuit-Based Proof of Toda′s Theorem

We present a simple proof of Toda′s result (Toda (1989), in "Proceedings, 30th Annual IEEE Symposium on Foundations of Computer Science," pp. 514-519), which states that circled plus P is hard for the Polynomial Hierarchy under randomized reductions. Our approach is circuit-based in the sense that we start with uniform circuit definitions of the Polynomial Hierarchy and apply the Valiant-Vazirani lemma on these circuits (Valiant and Vazirani (1986), Thoeret. Comput. Sci.47, 85-93).

A note on resolving infeasibility in linear programs by constraint relaxation

We study the problem of finding a set of constraints of minimum cardinality which when relaxed in an infeasible linear program, make it feasible. We show the problem is NP-hard even when the constraint matrix is totally unimodular and prove polynomial-time solvability when the constraint matrix and the right-hand-side together form a totally unimodular matrix.

Minimizing total completion time on a batch processing machine with job families

We consider the problem of minimizing the total completion time on a single batch processing machine. The set of jobs to be scheduled can be partitioned into a number of families, where all jobs in the same family have the same processing time. The machine can process at most B jobs simultaneously as a batch, and the processing time of a batch is equal to the processing time of the longest job in the batch. We analyze that properties of an optimal schedule and develop a dynamic programming algorithm of polynomial time complexity when the number of job families is fixed. The research is motivated by the problem of scheduling burn-in ovens in the semiconductor industry

Some new results regarding spikes and a heuristic for spike construction

his paper addresses the problem of minimizing the number of columns with superdiagonal nonzeroes (viz., spiked columns) in a square, nonsingular linear system of equations which is to be solved by Gaussian elimination. The exact focus is on a class of min-spike heuristics in which the rows and columns of the coefficient matrix are first permuted to block lower-triangular form. Subsequently, the number of spiked columns in each irreducible block and their heights above the diagonal are minimized heuristically. We show that ifevery column in an irreducible block has exactly two nonzeroes, i.e., is a doubleton, then there is exactly one spiked column. Further, if there is at least one non-doubleton column, there isalways an optimal permutation of rows and columns under whichnone of the doubleton columns are spiked. An analysis of a few benchmark linear programs suggests that singleton and doubleton columns can abound in practice. Hence, it appears that the results of this paper can be practically useful. In the rest of the paper, we develop a polynomial-time min-spike heuristic based on the above results and on a graph-theoretic interpretation of doubleton columns.

The Reproducing Kernel DMS-FEM: 3D Shape Functions and Applications to Linear Solid Mechanics

We propose a family of 3D versions of a smooth finite element method (Sunilkumar and Roy 2010), wherein the globally smooth shape functions are derivable through the condition of polynomial reproduction with the tetrahedral B-splines (DMS-splines) or tensor-product forms of triangular B-splines and ID NURBS bases acting as the kernel functions. While the domain decomposition is accomplished through tetrahedral or triangular prism elements, an additional requirement here is an appropriate generation of knotclouds around the element vertices or corners. The possibility of sensitive dependence of numerical solutions to the placements of knotclouds is largely arrested by enforcing the condition of polynomial reproduction whilst deriving the shape functions. Nevertheless, given the higher complexity in forming the knotclouds for tetrahedral elements especially when higher demand is placed on the order of continuity of the shape functions across inter-element boundaries, we presently emphasize an exploration of the triangular prism based formulation in the context of several benchmark problems of interest in linear solid mechanics. In the absence of a more rigorous study on the convergence analyses, the numerical exercise, reported herein, helps establish the method as one of remarkable accuracy and robust performance against numerical ill-conditioning (such as locking of different kinds) vis-a-vis the conventional FEM.

Boxicity and Poset Dimension

Let G be a simple, undirected, finite graph with vertex set V(G) and edge set E(C). A k-dimensional box is a Cartesian product of closed intervals a(1), b(1)] x a(2), b(2)] x ... x a(k), b(k)]. The boxicity of G, box(G) is the minimum integer k such that G can be represented as the intersection graph of k-dimensional boxes, i.e. each vertex is mapped to a k-dimensional box and two vertices are adjacent in G if and only if their corresponding boxes intersect. Let P = (S, P) be a poset where S is the ground set and P is a reflexive, anti-symmetric and transitive binary relation on S. The dimension of P, dim(P) is the minimum integer l such that P can be expressed as the intersection of t total orders. Let G(P) be the underlying comparability graph of P. It is a well-known fact that posets with the same underlying comparability graph have the same dimension. The first result of this paper links the dimension of a poset to the boxicity of its underlying comparability graph. In particular, we show that for any poset P, box(G(P))/(chi(G(P)) - 1) <= dim(P) <= 2box(G(P)), where chi(G(P)) is the chromatic number of G(P) and chi(G(P)) not equal 1. The second result of the paper relates the boxicity of a graph G with a natural partial order associated with its extended double cover, denoted as G(c). Let P-c be the natural height-2 poset associated with G(c) by making A the set of minimal elements and B the set of maximal elements. We show that box(G)/2 <= dim(P-c) <= 2box(G) + 4. These results have some immediate and significant consequences. The upper bound dim(P) <= 2box(G(P)) allows us to derive hitherto unknown upper bounds for poset dimension. In the other direction, using the already known bounds for partial order dimension we get the following: (I) The boxicity of any graph with maximum degree Delta is O(Delta log(2) Delta) which is an improvement over the best known upper bound of Delta(2) + 2. (2) There exist graphs with boxicity Omega(Delta log Delta). This disproves a conjecture that the boxicity of a graph is O(Delta). (3) There exists no polynomial-time algorithm to approximate the boxicity of a bipartite graph on n vertices with a factor of O(n(0.5-epsilon)) for any epsilon > 0, unless NP=ZPP.

New Approximation Algorithms for Minimum Cycle Bases of Graphs

We consider the problem of computing an approximate minimum cycle basis of an undirected non-negative edge-weighted graph G with m edges and n vertices; the extension to directed graphs is also discussed. In this problem, a {0,1} incidence vector is associated with each cycle and the vector space over F-2 generated by these vectors is the cycle space of G. A set of cycles is called a cycle basis of G if it forms a basis for its cycle space. A cycle basis where the sum of the weights of the cycles is minimum is called a minimum cycle basis of G. Cycle bases of low weight are useful in a number of contexts, e.g. the analysis of electrical networks, structural engineering, chemistry, and surface reconstruction. Although in most such applications any cycle basis can be used, a low weight cycle basis often translates to better performance and/or numerical stability. Despite the fact that the problem can be solved exactly in polynomial time, we design approximation algorithms since the performance of the exact algorithms may be too expensive for some practical applications. We present two new algorithms to compute an approximate minimum cycle basis. For any integer k >= 1, we give (2k - 1)-approximation algorithms with expected running time O(kmn(1+2/k) + mn((1+1/k)(omega-1))) and deterministic running time O(n(3+2/k) ), respectively. Here omega is the best exponent of matrix multiplication. It is presently known that omega < 2.376. Both algorithms are o(m(omega)) for dense graphs. This is the first time that any algorithm which computes sparse cycle bases with a guarantee drops below the Theta(m(omega) ) bound. We also present a 2-approximation algorithm with expected running time O(M-omega root n log n), a linear time 2-approximation algorithm for planar graphs and an O(n(3)) time 2.42-approximation algorithm for the complete Euclidean graph in the plane.

Can one electrochemically measure the statistical morphology of a rough electrode

We have developed a theory for an electrochemical way of measuring the statistical properties of a nonfractally rough electrode. We obtained the expression for the current transient on a rough electrode which shows three times regions: short and long time limits and the transition region between them. The expressions for these time ranges are exploited to extract morphological information about the surface roughness. In the short and long time regimes, we extract information regarding various morphological features like the roughness factor, average roughness, curvature, correlation length, dimensionality of roughness, and polynomial approximation for the correlation function. The formulas for the surface structure factors (the measure of surface roughness) of rough surfaces in terms of measured reversible and diffusion-limited current transients are also obtained. Finally, we explore the feasibility of making such measurements.

Popular matchings with variable item copies

We consider the problem of matching people to items, where each person ranks a subset of items in an order of preference, possibly involving ties. There are several notions of optimality about how to best match a person to an item; in particular, popularity is a natural and appealing notion of optimality. A matching M* is popular if there is no matching M such that the number of people who prefer M to M* exceeds the number who prefer M* to M. However, popular matchings do not always provide an answer to the problem of determining an optimal matching since there are simple instances that do not admit popular matchings. This motivates the following extension of the popular matchings problem: Given a graph G = (A U 3, E) where A is the set of people and 2 is the set of items, and a list < c(1),...., c(vertical bar B vertical bar)> denoting upper bounds on the number of copies of each item, does there exist < x(1),...., x(vertical bar B vertical bar)> such that for each i, having x(i) copies of the i-th item, where 1 <= xi <= c(i), enables the resulting graph to admit a popular matching? In this paper we show that the above problem is NP-hard. We show that the problem is NP-hard even when each c(i) is 1 or 2. We show a polynomial time algorithm for a variant of the above problem where the total increase in copies is bounded by an integer k. (C) 2011 Elsevier B.V. All rights reserved.

Multiple UAV Coalitions for a Search and Prosecute Mission

Unmanned aerial vehicles (UAVs) have the potential to carry resources in support of search and prosecute operations. Often to completely prosecute a target, UAVs may have to simultaneously attack the target with various resources with different capacities. However, the UAVs are capable of carrying only limited resources in small quantities, hence, a group of UAVs (coalition) needs to be assigned that satisfies the target resource requirement. The assigned coalition must be such that it minimizes the target prosecution delay and the size of the coalition. The problem of forming coalitions is computationally intensive due to the combinatorial nature of the problem, but for real-time applications computationally cheap solutions are required. In this paper, we propose decentralized sub-optimal (polynomial time) and decentralized optimal coalition formation algorithms that generate coalitions for a single target with low computational complexity. We compare the performance of the proposed algorithms to that of a global optimal solution for which we need to solve a centralized combinatorial optimization problem. This problem is computationally intensive because the solution has to (a) provide a coalition for each target, (b) design a sequence in which targets need to be prosecuted, and (c) take into account reduction of UAV resources with usage. To solve this problem we use the Particle Swarm Optimization (PSO) technique. Through simulations, we study the performance of the proposed algorithms in terms of mission performance, complexity of the algorithms and the time taken to form the coalition. The simulation results show that the solution provided by the proposed algorithms is close to the global optimal solution and requires far less computational resources.

On the performance of strain smoothing for quadratic and enriched finite element approximations (XFEM/GFEM/PUFEM)

By using the strain smoothing technique proposed by Chen et al. (Comput. Mech. 2000; 25: 137-156) for meshless methods in the context of the finite element method (FEM), Liu et al. (Comput. Mech. 2007; 39(6): 859-877) developed the Smoothed FEM (SFEM). Although the SFEM is not yet well understood mathematically, numerical experiments point to potentially useful features of this particularly simple modification of the FEM. To date, the SFEM has only been investigated for bilinear and Wachspress approximations and is limited to linear reproducing conditions. The goal of this paper is to extend the strain smoothing to higher order elements and to investigate numerically in which condition strain smoothing is beneficial to accuracy and convergence of enriched finite element approximations. We focus on three widely used enrichment schemes, namely: (a) weak discontinuities; (b) strong discontinuities; (c) near-tip linear elastic fracture mechanics functions. The main conclusion is that strain smoothing in enriched approximation is only beneficial when the enrichment functions are polynomial (cases (a) and (b)), but that non-polynomial enrichment of type (c) lead to inferior methods compared to the standard enriched FEM (e.g. XFEM). Copyright (C) 2011 John Wiley & Sons, Ltd.

Nonlinear static and dynamic damped response of a composite rectangular plate resting on a two-parameter elastic foundation

Nonlinear static and dynamic response analyses of a clamped. rectangular composite plate resting on a two-parameter elastic foundation have been studied using von Karman's relations. Incorporating the material damping, the governing coupled, nonlinear partial differential equations are obtained for the plate under step pressure pulse load excitation. These equations have been solved by a one-term solution and by applying Galerkin's technique to the deflection equation. This yields an ordinary nonlinear differential equation in time. The nonlinear static solution is obtained by neglecting the time-dependent variables. Thc nonlinear dynamic damped response is obtained by applying the ultraspherical polynomial approximation (UPA) technique. The influences of foundation modulus, shear modulus, orthotropy, etc. upon the nonlinear static and dynamic responses have been presented.

Hierarchies of circuit classes that are closed under complement

We examine three hierarchies of circuit classes and show they are closed under complementation. (1) The class of languages recognized by a family of polynomial size skew circuits with width O(w), are closed under complement. (2) The class of languages recognized by family of polynomial size circuits with width O(w) and polynomial tree-size, are closed under complement. (3) The class of languages recognized by a family of polynomial size, O(log(n)) depth, bounded AND fan-in with OR fan-in f (f⩾log(n)) circuits are closed under complement. These improve upon the results of (i) Immerman (1988) and Szelepcsenyi (1988), who show that 𝒩L𝒪𝒢 is closed under complementation, and (ii) Borodin et al. (1989), who show that L𝒪𝒢𝒞ℱL is closed under complement

Fast computation of Legendre and Zernike moments

This paper presents recursive algorithms for fast computation of Legendre and Zernike moments of a grey-level image intensity distribution. For a binary image, a contour integration method is developed for the evaluation of Legendre moments using only the boundary information. A method for recursive calculation of Zernike polynomial coefficients is also given. A square-to-circular image transformation scheme is introduced to minimize the computation involved in Zernike moment functions. The recursive formulae can also be used in inverse moment transforms to reconstruct the original image from moments. The mathematical framework of the algorithms is given in detail, and illustrated with binary and grey-level images.

Photochemical intramolecular gamma-hydrogen abstraction in a ketone and an alkene and the role of tunneling of hydrogen

Intramolecular gamma-hydrogen abstraction reactions were examined in pentane-2-one and 2-methyl-1-pentene in their lowest triplet states using the AM1 semi-empirical molecular orbital method with the complete geometry optimization in the unrestricted Hartree-Fock frame. The results reveal that the oxygen atom of the carbonyl group and the end carbon atom of the olefinic bond acquire high free valence and spin density indices in their respective lowest triplet states, leading to abstraction of hydrogen from the gamma-position relative to the carbonyl and olefinic bonds. The theoretical energy profiles fit with a polynomial and the probability of tunneling of hydrogen was estimated by the WKB (Wentzel, Kramer and Brillouin) method. The results, after thermal averaging of the rate constants, reveal that tunneling of hydrogen is significant at room temperature.