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Abstract A Boolean Finite Synchronous Dynamical System (BFDS, for short) consists of a finite number of objects that each maintains a boolean state, where after individually receiving state assignments, the objects update their state with respect to object-specific time-independent boolean functions synchronously in discrete time steps. The present paper studies the computational complexity of determining, given a boolean finite synchronous dynamical system, a configuration, which is a boolean vector representing the states of the objects, and a positive integer t , whether there exists another configuration from which the given configuration can be reached in t steps. It was previously shown that this problem, which we call the t -Predecessor Problem, is NP-complete even for t = 1 if the update function of an object is either the conjunction of arbitrary fan-in or the disjunction of arbitrary fan-in. This paper studies the computational complexity of the t -Predecessor Problem for a variety of sets of permissible update functions as well as for polynomially bounded t. It also studies the t -Garden-Of-Eden Problem, a variant of the t -Predecessor Problem that asks whether a configuration has a t -predecessor, which itself has no predecessor. The paper obtains complexity theoretical characterizations of all but one of these problems. [ABSTRACT FROM AUTHOR]

Let B be a set of b blue points and R be a set of r red points in the plane. In this paper we study the problem of finding rectangles that contain the maximum number of blue points without containing any red points, known as the maximum-box problem . First we study this problem for axis-aligned rectangles, and propose an exact worst-case optimal O ( r 2 + r b + b log ⁡ b ) time algorithm using O ( r + b ) space to find all maximum boxes. We also provide a 2-approximation algorithm running in O ( ( r + b ) log ⁡ ( r + b ) ) time and using O ( r + b ) space to find a single maximum box in the axis-aligned case. Then we generalize the exact algorithm for the axis-aligned case to find all arbitrarily oriented maximum boxes leading to a worst-case optimal O ( ( r + b ) 2 ( r + log ⁡ b ) ) time algorithm using O ( ( r + b ) 2 ) space to solve the problem. We conclude the paper by discussing time and space trade-offs. Our results improve the previously best known solutions to the maximum-box problem. [ABSTRACT FROM AUTHOR]

In this paper, we introduce obnoxious effects into obnoxious facility location games on a line, where each agent i has a private location x i on a closed interval [ 0 , 1 ] and one facility will be built on a location y in the interval according to the bids of all the agents. In addition, there are two thresholds d 1 and d 2 in the utility function of each agent, where 0 ≤ d 1 ≤ d 2 ≤ 1 . Denote d ( y , x i ) = | y − x i | to be the distance between agent i and the facility on the location y . The utility function of agent i is 0 if d ( y , x i ) is at most d 1 ; 1 if d ( y , x i ) is at least d 2 ; otherwise a linear increasing function between 0 and 1. Each agent attempts to get the largest utility while the social welfare is to maximize the sum of all the agents' utilities. The classic obnoxious facility game [4,11] is a special case of our problem when d 1 = 0 and d 2 = 1 . In this work, we first study the hardness of approximate mechanism design on this generalized problem, which states that our problem cannot admit any deterministic strategy-proof mechanism with bounded approximation ratio if d 1 ≥ 1 2 . Then we limit the thresholds to some ranges, both deterministic and randomized strategy-proof mechanisms are studied, and the approximation ratios vary with the specific values of d 1 and d 2 . [ABSTRACT FROM AUTHOR]

The Small Exponent Test (SET) for exponentiation is an essential batch-verification technique that is widely applied. In this paper, we propose a simplified SET that can securely batch-verify n instances with only n − 1 randomizing exponents. We show that the structure of the proposed batch test is compact in the sense that it works with a minimal number of randomizing exponents for the SET. Thus, our test offers various advantages. Overall, compared to the original SET, the proposed simplified SET is more efficient for any sized batch instance. In particular, unlike the SET, our proposal performs well even when the size of a batch instance is small, e.g., n = 1 , 2 , 3 , and 4. This feature can be also used to significantly reduce pairing computations in a signature scheme where several pairing equations are verified. In addition, our test can be combined easily and generically with existing batch techniques such as the use of sparse exponents, the bucket test for large batch sizes, or an automated tool to generate a batch algorithm. Finally, with our simplified test, an efficient identification algorithm can be constructed to discover incorrect instances in a batch. [ABSTRACT FROM AUTHOR]

Abstract: A mixed dominating set of a simple graph is a subset such that every vertex or edge not in is adjacent or incident to at least one vertex or edge in . The mixed domination problem is to determine a minimum mixed dominating set of . This paper studies mixed domination in graphs from an algorithmic point of view. In particular, a linear-time labeling algorithm for the mixed domination problem in cacti is presented. In addition, we fix an incomplete proof of the NP-completeness of the mixed domination problem in split graphs Zhao et al. (2011) [23]. Finally, we establish a primal–dual algorithm for the mixed domination problem in trees. [Copyright &y& Elsevier]

Abstract: The consensus (string) problem is finding a representative string, called a consensus, of a given set of strings. In this paper we deal with consensus problems considering both distance sum and radius, where the distance sum is the sum of (Hamming) distances from the strings in to the consensus and the radius is the longest (Hamming) distance from the strings in to the consensus. Although there have been results considering either distance sum or radius, there have been no results considering both, to the best of our knowledge. We present the first algorithms for two consensus problems considering both distance sum and radius for three strings: one problem is to find an optimal consensus minimizing both distance sum and radius. The other problem is to find a bounded consensus such that the distance sum is at most and the radius is at most for given constants and . Our algorithms are based on characterization of the lower bounds of distance sum and radius, and thus they solve the problems efficiently. Both algorithms run in linear time. [Copyright &y& Elsevier]

Abstract: Let and be positive integers, . Define to be the minimum positive value of where are positive integers no larger than . Define to be . It is important to find tight bounds for and , in connection to the sum-of-square-roots problem, a famous open problem in computational geometry. The current best lower bound and upper bound are far apart. In this paper, we prove an upper bound of for , which is better than the best known result whenever for some constant . In particular, our result implies an algorithm subexponential in (i.e. with time complexity ) to compare two sums of square roots of integers of value . We then present an algorithm to find exactly in time and in space. As an example, we are able to compute exactly in a few hours on a single PC. The numerical data indicate that the root separation bound is very far away from the true value of . [Copyright &y& Elsevier]

Abstract: There has been great interest in deciding whether a combinatorial structure satisfies some property, or in estimating the value of some numerical function associated with this combinatorial structure, by considering only a randomly chosen substructure of sufficiently large, but constant size. These problems are called property testing and parameter testing, where a property or parameter is said to be testable if it can be estimated accurately in this way. The algorithmic appeal is evident, as, conditional on sampling, this leads to reliable constant-time randomized estimators. Our paper addresses property testing and parameter testing for permutations in a subpermutation perspective; more precisely, we investigate permutation properties and parameters that can be well approximated based on a randomly chosen subpermutation of much smaller size. In this context, we use a theory of convergence of permutation sequences developed by the present authors [C. Hoppen, Y. Kohayakawa, C.G. Moreira, R.M. Sampaio, Limits of permutation sequences through permutation regularity, Manuscript, 2010, 34pp.] to characterize testable permutation parameters along the lines of the work of Borgs et al. [C. Borgs, J. Chayes, L. Lovász, V.T. Sós, B. Szegedy, K. Vesztergombi, Graph limits and parameter testing, in: STOC’06: Proceedings of the 38th Annual ACM Symposium on Theory of Computing, ACM, New York, 2006, pp. 261–270.] in the case of graphs. Moreover, we obtain a permutation result in the direction of a famous result of Alon and Shapira [N. Alon, A. Shapira, A characterization of the (natural) graph properties testable with one-sided error, SIAM J. Comput. 37 (6) (2008) 1703–1727.] stating that every hereditary graph property is testable. [Copyright &y& Elsevier]

Abstract: A numerical elimination method is presented in this paper for floating-point computation in polynomial algebra. The method is designed to calculate one or more polynomials in an elimination ideal by a sequence of matrix rank/kernel computation. The method is reliable in numerical computation with verifiable stability and a sensitivity measurement. Computational experiment shows that the method possesses significant advantages over classical resultant computation in numerical stability and in producing eliminant polynomials with lower degrees and fewer extraneous factors. The elimination algorithm combined with an approximate GCD finder appears to be effective in solving polynomial systems for positive dimensional solutions. [Copyright &y& Elsevier]

Abstract: The problem of monitoring an electric power system by placing as few measurement devices in the system as possible is closely related to the well-known domination problem in graphs. In 2002, Haynes et al. considered the graph theoretical representation of this problem as a variation of the domination problem. They defined a set S to be a power dominating set of a graph if every vertex and every edge in the system is monitored by the set S (following a set of rules for power system monitoring). The power domination number of a graph G is the minimum cardinality of a power dominating set of G. This problem was proved NP-complete even when restricted to bipartite graphs and chordal graphs. In this paper, we present a linear time algorithm for solving the power domination problem in block graphs. As an application of the algorithm, we establish a sharp upper bound for power domination number in block graphs and characterize the extremal graphs. [Copyright &y& Elsevier]

Abstract: Double-loop networks have been widely studied as an architecture for local area networks. It is well known that the minimum distance diagram of a double-loop network yields an L-shape. Given a positive integer , it is desirable to find a double-loop network with its diameter being the minimum among all double-loop networks with nodes. Since the diameter of a double-loop network can be easily computed from its L-shape, one method is to start with a desirable L-shape and then find a double-loop network to realize it. This is a problem discussed by many authors [F. Aguiló, M.A. Fiol, An efficient algorithm to find optimal double loop networks, Discrete Math. 138 (1995) 15–29, R.C. Chan, C.Y. Chen, Z.X. Hong, A simple algorithm to find the steps of double-loop networks, Discrete Appl. Math. 121 (2002) 61–72, C.Y. Chen, F.K. Hwang, The minimum distance diagram of double-loop networks, IEEE Trans. Comput. 49 (2000) 977–979, P. Esqué, F. Aguiló, M.A. Fiol, Double commutative-step diagraphs with minimum diameters, Discrete Math. 114 (1993) 147–157] and it has been open for a long time whether this problem can be solved in time. In this paper, we will provide a simple and efficient -time algorithm for solving this problem. [Copyright &y& Elsevier]

Abstract: Given a bipartite graph , a 2-layered drawing consists of placing nodes in the first node set V on a straight line and placing nodes in the second node set W on a parallel line . For a given ordering of nodes in W on , the one-sided crossing minimization problem asks to find an ordering of nodes in V on so that the number of arc crossings is minimized. A well-known lower bound LB on the minimum number of crossings is obtained by summing up over all node pairs , where denotes the number of crossings generated by arcs incident to u and when u precedes in an ordering. In this paper, we prove that there always exists a solution whose crossing number is at most if the minimum degree of a node in V is at least 5. [Copyright &y& Elsevier]

To establish a multicast connection in a wavelength routed optical network, two steps are needed under the multi-tree model. One is to construct a set of light-trees rooted at the source node such that in each of them at most a specified number of destination nodes are allowed to receive the data and every destination node must be designated in one of them to receive the data. The other is to assign a wavelength to each of the produced light-trees in such a way that two light-trees must be assigned two distinct wavelengths if they use a common link. In this paper we mainly study how to construct a multicast routing of minimal cost under the multi-tree model in optical networks, where the routing cost is total costs of the produced light-trees. We propose a 4-approximation algorithm for this NP-hard problem. [Copyright &y& Elsevier]