The Power of Linear Programming for Valued CSPs

A class of valued constraint satisfaction problems (VCSPs) is characterised by a valued constraint language, a fixed set of cost functions on a finite domain. An instance of the problem is specified by a sum of cost functions from the language with the goal to minimise the sum. This framework includes and generalises well-studied constraint satisfaction problems (CSPs) and maximum constraint satisfaction problems (Max-CSPs). Our main result is a precise algebraic characterisation of valued constraint languages whose instances can be solved exactly by the basic linear programming relaxation. Using this result, we obtain tractability of several novel and previously widely-open classes of VCSPs, including problems over valued constraint languages that are: (1) submodular on arbitrary lattices; (2) bisubmodular (also known as k-submodular) on arbitrary finite domains; (3) weakly (and hence strongly) tree-submodular on arbitrary trees.Comment: Corrected a few typo

Polynomial combinatorial algorithms for skew-bisubmodular function minimization

Huber et al. (SIAM J Comput 43:1064–1084, 2014) introduced a concept of skew bisubmodularity, as a generalization of bisubmodularity, in their complexity dichotomy theorem for valued constraint satisfaction problems over the three-value domain, and Huber and Krokhin (SIAM J Discrete Math 28:1828–1837, 2014) showed the oracle tractability of minimization of skew-bisubmodular functions. Fujishige et al. (Discrete Optim 12:1–9, 2014) also showed a min–max theorem that characterizes the skew-bisubmodular function minimization, but devising a combinatorial polynomial algorithm for skew-bisubmodular function minimization was left open. In the present paper we give first combinatorial (weakly and strongly) polynomial algorithms for skew-bisubmodular function minimization

The power of linear programming for general-valued CSPs

Let $D$, called the domain, be a fixed finite set and let $\Gamma$, called the valued constraint language, be a fixed set of functions of the form $f:D^m\to\mathbb{Q}\cup\{\infty\}$, where different functions might have different arity $m$. We study the valued constraint satisfaction problem parametrised by $\Gamma$, denoted by VCSP$(\Gamma)$. These are minimisation problems given by $n$ variables and the objective function given by a sum of functions from $\Gamma$, each depending on a subset of the $n$ variables. Finite-valued constraint languages contain functions that take on only rational values and not infinite values. Our main result is a precise algebraic characterisation of valued constraint languages whose instances can be solved exactly by the basic linear programming relaxation (BLP). For a valued constraint language $\Gamma$, BLP is a decision procedure for $\Gamma$ if and only if $\Gamma$ admits a symmetric fractional polymorphism of every arity. For a finite-valued constraint language $\Gamma$, BLP is a decision procedure if and only if $\Gamma$ admits a symmetric fractional polymorphism of some arity, or equivalently, if $\Gamma$ admits a symmetric fractional polymorphism of arity 2. Using these results, we obtain tractability of several novel classes of problems, including problems over valued constraint languages that are: (1) submodular on arbitrary lattices; (2) $k$-submodular on arbitrary finite domains; (3) weakly (and hence strongly) tree-submodular on arbitrary trees.Comment: A full version of a FOCS'12 paper by the last two authors (arXiv:1204.1079) and an ICALP'13 paper by the first author (arXiv:1207.7213) to appear in SIAM Journal on Computing (SICOMP

Maximizing supermodular functions on product lattices, with application to maximum constraint satisfaction

Recently, a strong link has been discovered between supermodularity on lattices and tractability of optimization problems known as maximum constraint satisfaction problems. This paper strengthens this link. We study the problem of maximizing a supermodular function which is defined on a product of $n$ copies of a fixed finite lattice and given by an oracle. We exhibit a large class of finite lattices for which this problem can be solved in oracle-polynomial time in $n$. We also obtain new large classes of tractable maximum constraint satisfaction problems

Dynamic Pricing Strategy for Maximizing Cloud Revenue

The unexpected growth, flexibility and dynamism of information technology (IT) over the last decade has radically altered the civilization lifestyle and this boom continues as yet. Many nations have been competing to be forefront of this technological revolution, quite embracing the opportunities created by the advancements in this field in order to boost economy growth and to increase the accomplishments of everyday’s life. Cloud computing is one of the most promising achievement of these advancements. However, it faces many challenges and barriers like any new industry. Managing and maximizing such a very complex system business revenue is of paramount importance. The wealth of the cloud protfolio comes from the proceeds of three main services: Infrastructure as a service (IaaS), Software as a service (SaaS), and Platform as a service (PaaS). The Infrastructure as a Service (IaaS) cloud industry that relies on leasing virtual machines (VMs) has a significant portion of business values. Therefore many enterprises show frantic effort to capture the largest portion through the introducing of many different pricing models to satisfy not merely customers’ demands but essentially providers’ requirements. Indeed, one of the most challenging requirements is finding the dynamic equilibrium between two conflicting phenomena: underutilization and surging congestion. Spot instance has been presented as an elegant solution to overcome these situations aiming to gain more profits. However, previous studies on recent spot pricing schemes reveal an artificial pricing policy that does not comply with the dynamic nature of these phenomena. In this thesis, we investigate dynamic pricing of stagnant resources so as to maximize cloud revenue. To achieve this task, we reveal the necessities and objectives that underlie the importance of adopting cloud providers to dynamic price model, analyze adopted dynamic pricing strategy for real cloud enterprises and create dynamic pricing model which could be a strategic pricing model for IaaS cloud providers to increase the marginal profit and also to overcome technical barriers simultaneously. First, we formulate the maximum expected reward under discrete finite-horizon Markovian decisions and characterize model properties under optimum controlling conditions. The initial approach manages one class but multiple fares of virtual machines. For this purpose, the proposed approach leverages Markov decision processes, a number of properties under optimum controlling conditions that characterize a model’s behaviour, and approximate stochastic dynamic programming using linear programming to create a practical model. Second, our seminal work directs us to explore the most sensitive factors that drive price dynamism and to mitigate the high dimensionality of such a large-scale problem through conducting column generation. More specifically we employ a decomposition approach. Third, we observe that most previous work tackled one class of virtual machines merely. Therefore, we extend our study to cover multiple classes of virtual machines. Intuitively, dynamic price of multiple classes model is much more efficient from one side but practically is more challenging from another side. Consequently, our approach of dynamic pricing can scale up or down the price efficiently and effectively according to stagnant resources and load threshold aims to maximize the IaaS cloud revenue