An Approximate Distribution for the Maximum Order Complexity

In this paper we give an approximate probability distribution for the maximum order complexity of a random binary sequence. This enables the development of statistical tests based on maximum order complexity for the testing of a binary sequence generator. These tests are analogous to those based on linear complexity.     

A complexity measure for families of binary sequences

In earlier papers finite pseudorandom binary sequences were studied, quantitative measures of pseudorandomness of them were introduced and studied, and large families of “good” pseudorandom sequences were constructed. In certain applications (cryptography) it is not enough to know that a family of “good” pseudorandom binary sequences is large, it is a more important property if it has a “rich”, “complex” structure. Correspondingly, the notion of “f-complexity” of a family of binary sequences is introduced. It is shown that the family of “good” pseudorandom binary sequences constructed earlier is also of high f-complexity. Finally, the cardinality of the smallest family achieving a prescibed f-complexity and multiplicity is estimated. This revised version was published online in August 2006 with corrections to the Cover Date.     

Implementation of parallel adder using combinatorial logic architecture

The algorithm and optical implementation of full parallel adder using combina-torial logic architecture have been suggested in this paper.The hybrid full parallel adder,which can perform multiple bits addition simultaneously,is implemented by utilizing VLSItechnology and acoustooptic Bragg cells.The algorithm has the advantages of high parallelismand easy optical implementation.The full parallel adder finishes n-bit addition within the du-ration T_d if the pipeline architecture is employed,whose hardware complexity is O(n~2).     

The computational complexity of pattern formation

The computational complexity of diffusion-limited aggregation and fluid invasion in porous media is studied. The time requirements on an idealized parallel computer for simulating the patterns formed by these models are investigated. It is shown that these growth models are P-complete. These results provide strong evidence that such pattern formation processes are inherently sequential and cannot be simulated efficiently in parallel.     

On the worst case complexity of potential reduction algorithms for linear programming

There are several classes of interior point algorithms that solve linear programming problems in O(√nL) iterations. Among them, several potential reduction algorithms combine both theoretical (O(√nL) iterations) and practical efficiency as they allow the flexibility of line searches in the potential function, and thus can lead to practical implementations. It is a significant open question whether interior point algorithms can lead to better complexity bounds. In the present paper we give some negative answers to this question for the class of potential reduction algorithms. We show that, even if we allow line searches in the potential function, and even for problems that have network structure, the bound O(√nL) is tight for several potential reduction algorithms, i.e., there is a class of examples with network structure, in which the algorithms need at least Θ(√nL) iterations to find an optimal solution. The research of the author was partially supported by a Presidential Young Investigator Award DDM-9158118 with matching funds from Draper Laboratory.     

Maximum algebraic connectivity augmentation is NP-hard

The algebraic connectivity of a graph, which is the second-smallest eigenvalue of the Laplacian of the graph, is a measure of connectivity. We show that the problem of adding a specified number of edges to an input graph to maximize the algebraic connectivity of the augmented graph is NP-hard.     

On the strong p-Helly property

The notion of strong p-Helly hypergraphs was introduced by Golumbic and Jamison in 1985 [M.C. Golumbic, R.E. Jamison, The edge intersection graphs of paths in a tree, J. Combin. Theory Ser. B 38 (1985) 8-22]. Independently, other authors [A. Bretto, S. Ubéda, J. ?erovnik, A polynomial algorithm for the strong Helly property. Inform. Process. Lett. 81 (2002) 55-57, E. Prisner, Hereditary clique-Helly graphs, J. Combin. Math. Combin. Comput. 14 (1993) 216-220, W.D. Wallis, Guo-Hui Zhang, On maximal clique irreducible graphs. J. Combin. Math. Combin. Comput. 8 (1990) 187-193.] have also considered the strong Helly property in other contexts. In this paper, we characterize strong p-Helly hypergraphs. This characterization leads to an algorithm for recognizing such hypergraphs, which terminates within polynomial time whenever p is fixed. In contrast, we show that the recognition problem is co-NP-complete, for arbitrary p. Further, we apply the concept of strong p-Helly hypergraphs to the cliques of a graph, leading to the class of strong p-clique-Helly graphs. For p=2, this class is equivalent to that of hereditary clique-Helly graphs [E. Prisner, Hereditary clique-Helly graphs, J. Combin. Math. Combin. Comput. 14 (1993) 216-220]. We describe a characterization for this class and obtain an algorithm for recognizing such graphs. Again, the algorithm has polynomial-time complexity for p fixed, and we show the corresponding recognition problem to be NP-hard, for arbitrary p.     

Number of alternatives and efficiency in different types of top-management decisions

What are the causes of the efficiency of complex strategic decisions? To answer this question, the impact of information searching, alternative designing, and complexity of a decision problem on its decision quality are analyzed in a longitudinal study of 83 top-management decisions, made the by executive board of a medium-sized firm (1380 employees). Decision quality is negatively influenced by the complexity of the decision problem. Alternative designing has a strong positive impact on decision quality. Information search shows no significant relationship to decision quality. The results indicate that designing of alternatives is an important instrument to counter the challenges of complex strategic decision-problems. However, one should not simply maximize the number of alternatives. There seems to be a very small optimal number beyond which decision quality will decrease. Besides, alternative designing has to be coordinated with other problem-solving activities, namely goal formation, process organization, and information searching.     

Local minima for indefinite quadratic knapsack problems

We consider the complexity of finding a local minimum for the nonconvex Quadratic Knapsack Problem. Global minimization for this example of quadratic programming is NP-hard. Moré and Vavasis have investigated the complexity of local minimization for the strictly concave case of QKP; here we extend their algorithm to the general indefinite case. Our main result is an algorithm that computes a local minimum in O(n(logn)²) steps. Our approach involves eliminating all but one of the convex variables through parametrization, yielding a nondifferentiable problem. We use a technique from computational geometry to address the nondifferentiable problem.Supported in part by the Applied Mathematical Sciences subprogram of the Office of Energy Research, Department of Energy, under contract W-31-109-Eng-38, in part by a Fannie and John Hertz Foundation graduate fellowship, and in part by Department of Energy grant DE-FG02-86ER25013.A000.     

Condorcet Winners for Public Goods

In this work, we consider a public facility allocation problem decided through a voting process under the majority rule. A location of the public facility is a majority rule winner if there is no other location in the network where more than half of the voters would have been closer to than the majority rule winner. We develop fast algorithms for interesting cases with nice combinatorial structures. We show that the computing problem and the decision problem in the general case, where the number of public facilities is more than one and is considered part of the input size, are all NP-hard. Finally, we discuss majority rule decision making for related models.