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Nonmonotonicity of phase transitions in a loss network with controls
 Ann. Appl. Probab
, 2006
"... We consider a symmetric tree loss network that supports singlelink (unicast) and multilink (multicast) calls to nearest neighbors and has capacity C on each link. The network operates a control so that the number of multicast calls centered at any node cannot exceed CV and the number of unicast cal ..."
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We consider a symmetric tree loss network that supports singlelink (unicast) and multilink (multicast) calls to nearest neighbors and has capacity C on each link. The network operates a control so that the number of multicast calls centered at any node cannot exceed CV and the number of unicast calls at a link cannot exceed CE, where CE, CV ≤ C. We show that uniqueness of Gibbs measures on the infinite tree is equivalent to the convergence of certain recursions of a related map. For the case CV = 1 and CE = C, we precisely characterize the phase transition surface and show that the phase transition is always nonmonotone in the arrival rate of the multicast calls. This model is an example of a system with hard constraints that has weights attached to both the edges and nodes of the network and can be viewed as a generalization of the hard core model that arises in statistical mechanics and combinatorics. Some of the results obtained also hold for more general models than just the loss network. The proofs rely on a combination of techniques from probability theory and dynamical systems. 1. Introduction. In Section 1.1
1 Local Energy Statistics in Directed Polymers.
, 2007
"... Abstract Recently, Bauke and Mertens conjectured that the local statistics of energies in random spin systems with discrete spin space should, in most circumstances, be the same as in the random energy model. We show that this conjecture holds true as well for directed polymers in random environment ..."
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Abstract Recently, Bauke and Mertens conjectured that the local statistics of energies in random spin systems with discrete spin space should, in most circumstances, be the same as in the random energy model. We show that this conjecture holds true as well for directed polymers in random environment. We also show that, under certain conditions, this conjecture holds for directed polymers even if energy levels that grow moderately with the volume of the system are considered.
Latticebased Algorithms for Number Partitioning in the Hard Phase
"... The number partitioning problem (NPP) is to divide n numbers a1,..., an into two disjoint subsets such that the difference between the two subset sums – the discrepancy, ∆, is minimized. In the balanced version of NPP (BalNPP), the subsets must have the same cardinality. With ajs chosen uniformly fr ..."
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The number partitioning problem (NPP) is to divide n numbers a1,..., an into two disjoint subsets such that the difference between the two subset sums – the discrepancy, ∆, is minimized. In the balanced version of NPP (BalNPP), the subsets must have the same cardinality. With ajs chosen uniformly from [1, R], R> 2n gives the hard phase, when there are no equal partitions (i.e., ∆ = 0) with high probability (whp). In this phase, the minimum partition is also unique whp. Most current methods struggle in the hard phase, as they often perform exhaustive enumeration of all partitions to find the optimum. We propose reductions of NPP and BalNPP in the hard phase to the closest vector problem (CVP). We can solve the original problems by making polynomial numbers of calls to a CVP oracle. In practice, we implement a heuristic which applies basis reduction (BR) to several CVP instances (less than 2n in most cases). This method finds nearoptimal solutions without proof of optimality to NPP problems with reasonably large dimensions – up to n = 75. second, we propose a truncated NPP algorithm, which finds approximate minimum discrepancies for instances on which the BR approach is not effective. In place of the original instance, we solve a modified instance with āj = ⌊aj/T ⌉ for some T ≤ R. We show that the expected optimal discrepancy of the original problem given by the truncated solution, E (∆ ∗ T), is not much different from the expected optimal discrepancy: E (∆ ∗ T) ≤ E (∆∗) + nT/2. This algorithm can be used to find good quality partitions within a short time for problems of sizes up to n = 100. Third, we propose a direct mixed integer programming (MIP) model for NPP and BalNPP. We then solve a latticebased reformulation of the original MIP using standard branchandcut methods. Assuming it terminates, the MIP model is guaranteed to find the optimum partition. 1