Dependence of ground state energy of classical n-vector spins on n

We study the ground state energy E_G(n) of N classical n-vector spins with the hamiltonian H = - \sum_{i>j} J_ij S_i.S_j where S_i and S_j are n-vectors and the coupling constants J_ij are arbitrary. We prove that E_G(n) is independent of n for all n > n_{max}(N) = floor((sqrt(8N+1)-1) / 2) . We show that this bound is the best possible. We also derive an upper bound for E_G(m) in terms of E_G(n), for m<n. We obtain an upper bound on the frustration in the system, as measured by F(n), which is defined to be (\sum_{i>j} |J_ij| + E_G(n)) / (\sum_{i>j} |J_ij|). We describe a procedure for constructing a set of J_ij's such that an arbitrary given state, {S_i}, is the ground state.Comment: 6 pages, 2 figures, submitted to Physical Review

Maximizing Algebraic Connectivity of Constrained Graphs in Adversarial Environments

This paper aims to maximize algebraic connectivity of networks via topology design under the presence of constraints and an adversary. We are concerned with three problems. First, we formulate the concave maximization topology design problem of adding edges to an initial graph, which introduces a nonconvex binary decision variable, in addition to subjugation to general convex constraints on the feasible edge set. Unlike previous methods, our method is justifiably not greedy and capable of accommodating these additional constraints. We also study a scenario in which a coordinator must selectively protect edges of the network from a chance of failure due to a physical disturbance or adversarial attack. The coordinator needs to strategically respond to the adversary's action without presupposed knowledge of the adversary's feasible attack actions. We propose three heuristic algorithms for the coordinator to accomplish the objective and identify worst-case preventive solutions. Each algorithm is shown to be effective in simulation and we provide some discussion on their compared performance.Comment: 8 pages, submitted to European Control Conference 201

Polynomiality for Bin Packing with a Constant Number of Item Types

We consider the bin packing problem with d different item sizes s_i and item multiplicities a_i, where all numbers are given in binary encoding. This problem formulation is also known as the 1-dimensional cutting stock problem. In this work, we provide an algorithm which, for constant d, solves bin packing in polynomial time. This was an open problem for all d >= 3. In fact, for constant d our algorithm solves the following problem in polynomial time: given two d-dimensional polytopes P and Q, find the smallest number of integer points in P whose sum lies in Q. Our approach also applies to high multiplicity scheduling problems in which the number of copies of each job type is given in binary encoding and each type comes with certain parameters such as release dates, processing times and deadlines. We show that a variety of high multiplicity scheduling problems can be solved in polynomial time if the number of job types is constant

Symmetric Submodular Function Minimization Under Hereditary Family Constraints

We present an efficient algorithm to find non-empty minimizers of a symmetric submodular function over any family of sets closed under inclusion. This for example includes families defined by a cardinality constraint, a knapsack constraint, a matroid independence constraint, or any combination of such constraints. Our algorithm make $O(n^3)$ oracle calls to the submodular function where $n$ is the cardinality of the ground set. In contrast, the problem of minimizing a general submodular function under a cardinality constraint is known to be inapproximable within $o(\sqrt{n/\log n})$ (Svitkina and Fleischer [2008]). The algorithm is similar to an algorithm of Nagamochi and Ibaraki [1998] to find all nontrivial inclusionwise minimal minimizers of a symmetric submodular function over a set of cardinality $n$ using $O(n^3)$ oracle calls. Their procedure in turn is based on Queyranne's algorithm [1998] to minimize a symmetric submodularComment: 13 pages, Submitted to SODA 201

Community Detection in Hypergraphs, Spiked Tensor Models, and Sum-of-Squares

We study the problem of community detection in hypergraphs under a stochastic block model. Similarly to how the stochastic block model in graphs suggests studying spiked random matrices, our model motivates investigating statistical and computational limits of exact recovery in a certain spiked tensor model. In contrast with the matrix case, the spiked model naturally arising from community detection in hypergraphs is different from the one arising in the so-called tensor Principal Component Analysis model. We investigate the effectiveness of algorithms in the Sum-of-Squares hierarchy on these models. Interestingly, our results suggest that these two apparently similar models exhibit significantly different computational to statistical gaps.Comment: In proceedings of 2017 International Conference on Sampling Theory and Applications (SampTA

A consideration of the nature and purpose of mental health social work

Purpose: This paper aims to provide an analysis of the mental health social work role, its contribution to social inclusion, and its ability to translate this into practice. Design/methodology/approach: The paper considers national policy, research and theory to consider the nature of social work and the mental health system. Findings: While social work is ideally placed to challenge the biomedical model and promote social inclusion, organisational and other failings would appear to seriously undermine its ability to do so. Originality/value: The paper considers some important issues facing social work and mental health, and raises points for thought and discussion Keywords: social work, social care, mental health, social inclusion, personalisation, recovery Classification: Conceptual paper / Viewpoin