The property of spatial mixing and strong spatial mixing in spin systems has been of interest because of its implications on uniqueness of Gibbs measures on infinite graphs and efficient approximation of counting problems that are otherwise known to be #P hard. In the context of coloring, strong spatial mixing has been established for regular trees in [GS11] when q ≥ α * ∆ + 1 where q the number of colors, ∆ is the degree and α * = 1.763.. is the unique solution to xe −1/x = 1. It has also been established in [GMP05] for bounded degree lattice graphs whenever q ≥ α * ∆ − β for some constant β, where ∆ is the maximum vertex degree of the graph. The latter uses a technique based on recursively constructed coupling of Markov chains whereas the former is based on establishing decay of correlations on the tree. We establish strong spatial mixing of list colorings on arbitrary bounded degree triangle-free graphs whenever the size of the list of each vertex v is at least α∆(v)+ β where ∆(v) is the degree of vertex v and α > α * and β is a constant that only depends on α. We do this by proving the decay of correlations via recursive contraction of the distance between the marginals measured with respect to a suitably chosen error function.

We propose a deterministic algorithm for approximately counting the number of list colorings of a graph. Under the assumption that the graph is triangle free, the size of every list is at least α∆, where α is an arbitrary constant bigger than α * * = 2.8432 . . ., and ∆ is the maximum degree of the graph, we obtain the following results. For the case when the size of the each list is a large constant, we show the existence of a deterministic FPTAS for computing the total number of list colorings. The same deterministic algorithm has complexity 2 O(log 2 n) , without any assumptions on the sizes of the lists, where n is the instance size. We further extend our method to a discrete Markov random field (MRF) model. Under certain assumptions relating the size of the alphabet, the degree of the graph and the interacting potential we again construct a deterministic FPTAS for computing the partition function of a MRF.Our results are not based on the most powerful existing counting technique -rapidly mixing Markov chain method. Rather we build upon concepts from statistical physics, in particular, the decay of correlation phenomena and its implication for the uniqueness of Gibbs measures in infinite graphs. This approach was proposed in two recent papers [?] and [?]. The principle insight of this approach is that the correlation decay property can be established with respect to certain computation tree, as opposed to the conventional correlation decay property with respect to graph theoretic neighborhoods of a given node. This allows truncation of computation at a logarithmic depth in order to obtain polynomial accuracy in polynomial time.

We propose a new method for the problems of computing free energy and surface pressure for various statistical mechanics models on a lattice Z d . Our method is based on representing the free energy and surface pressure in terms of certain marginal probabilities in a suitably modified sublattice of Z d . Then recent deterministic algorithms for computing marginal probabilities are used to obtain numerical estimates of the quantities of interest. The method works under the assumption of Strong Spatial Mixing (SSP), which is a form of a correlation decay.We illustrate our method for the hard-core and monomer-dimer models, and improve several earlier estimates. For example we show that the exponent of the monomer-dimer coverings of Z 3 belongs to the interval [0.78595, 0.78599], improving best previously known estimate of (approximately) [0.7850, 0.7862] obtained in [FP05], [FKLM]. Moreover, we show that given a target additive error > 0, the computational effort of our method for these two models is (1/ ) O(1) both for free energy and surface pressure. In contrast, prior methods, such as transfer matrix method, require exp`(1/ ) O(1)´c omputation effort.

. (2012). Improved bounds for speed scaling in devices obeying the cube-root rule. Theory of Computing, 8(9), 209-229. DOI: 10.4086/toc.2012.v008a009 General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.• Users may download and print one copy of any publication from the public portal for the purpose of private study or research.• You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal ? Take down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Abstract: Speed scaling is a power management technology that involves dynamically changing the speed of a processor. This technology gives rise to dual-objective scheduling problems, where the operating system both wants to conserve energy and optimize some Quality of Service (QoS) measure of the resulting schedule. In the most investigated speed scaling problem in the literature, the QoS constraint is deadline feasibility, and the objective is to minimize the energy used. The standard assumption is that the processor power is of the form s α where s is the processor speed, and α > 1 is some constant; α ≈ 3 for CMOS based processors. In this paper we introduce and analyze a natural class of speed scaling algorithms that we call qOA. The algorithm qOA sets the speed of the processor to be q times the speed that the optimal offline algorithm would run the jobs in the current state. When α = 3, we show that qOA is 6.7-competitive, improving upon the previous best guarantee of 27 achieved by the algorithm Optimal Available (OA). We also give almost matching upper and lower bounds for qOA for general α. Finally, we give the first non-trivial lower bound, namely e α−1 /α, on the competitive ratio of a general deterministic online algorithm for this problem.

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