A nonempty set.. of residues modulo.. is said to be balanced if for each....., there is a.. with 0 1. Thus, the sequence simplifies to.... = 2.. + 1 3 with.. odd. Finally, we prove that if 2.. + 1 3 is prime, then.. ...
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A nonempty set.. of residues modulo.. is said to be balanced if for each....., there is a.. with 0 <.. =.. /2 such that.. +/-.. mod.. both lie in... We denote the minimum cardinality of a balanced set modulo.. by..(..). Minimal size balanced sets are needed for a winning strategy in the Vector game which was introduced together with balanced sets. In this paper, we describe a polynomial algorithm for constructing a minimal size balanced set modulo.., when.. is from two special classes of primes called lucky primes. We prove that lucky primes are all primes among the sequence.... = 2..+3 + (-1).. 3. Then we prove that the numbers.... = 2..+3 + (-1).. 3 are never prime when.. is odd and.. > 1. Thus, the sequence simplifies to.... = 2.. + 1 3 with.. odd. Finally, we prove that if 2.. + 1 3 is prime, then.. must be a prime.
The problem of locating service facilities with respect to a given set of demands on a graph is considered. The objective function to be minimized is equal to the maximum difference in the distance from a demand point...
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A type of capacity expansion problem on networks is considered. A network N (V, E, C over-bar ) is given and the capacity vector C over-bar is increased so that the capacity of the maximum capacity tree can be increas...
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A type of capacity expansion problem on networks is considered. A network N (V, E, C over-bar ) is given and the capacity vector C over-bar is increased so that the capacity of the maximum capacity tree can be increased to the maximum extent while the total cost for the increment of capacity is within a given budget limit, assuming that the cost of increment of capacity on each edge of network includes both fixed cost and increasing cost. A strong polynomial algorithm is proposed to solve the problem.
In parallel-batching machine scheduling, all jobs in a batch start and complete at the same time, and the processing time of the batch is the maximum processing time of any job in it. For the unbounded parallel-batchi...
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In parallel-batching machine scheduling, all jobs in a batch start and complete at the same time, and the processing time of the batch is the maximum processing time of any job in it. For the unbounded parallel-batching machine scheduling problem of minimizing the maximum lateness, denoted 1|p-batch|L_(max), a dynamic programming algorithm with time complexity O(n^2) is well known in the ***, this algorithm is improved to be an O(n log n) algorithm. In this note, we present another O(n log n) algorithm with simplifications on data structure and implementation details.
In this paper we consider a kind of the inverse shortest paths problem in which we want to adjust the given weights of arcs as less as possible so that a set of given paths become shortest ones from their sources to s...
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In this paper we consider a kind of the inverse shortest paths problem in which we want to adjust the given weights of arcs as less as possible so that a set of given paths become shortest ones from their sources to sinks, under an additional requirement that the weights of all arcs initiated from the same node must be adjusted by the same amount. We first formulate this problem into an LP model, them prove it can be solved in strongly polynomial time.
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