Binary Constraints Research Articles

Continuous-Time Models for Short-Term Scheduling of Multipurpose Batch Plants: A Comparative Study

During the last two decades, the problem of short-term scheduling of multiproduct and multipurpose batch plants has gained increasing attention in the academic, research, and manufacturing communities, predominantly because of the challenges and the high economic incentives. In the last 10 years, numerous formulations have been proposed in the literature based on continuous representations of time. The continuous-time formulations have proliferated because of their established advantages over discrete-time representations and in the quest to reduce the integrality gap and the resulting computational complexities. The various continuous-time models can be broadly classified into three distinct categories: slot-based, global event-based, and unit-specific event-based formulations. In this paper, we compare and evaluate the performance of six such models, based on our implementations using several benchmark example problems from the literature. Two different objective functions, maximization of profit and minimization of makespan, are considered, and the models are assessed with respect to different metrics such as the problem size (in terms of the number of binary variables, continuous variables, and constraints), computational times (on the same computer), and number of nodes needed to reach zero integrality gap. Two additional computational studies with resource constraints such as utility requirements are also considered.

Sufficient Global Optimality Conditions for Non-convex Quadratic Minimization Problems With Box Constraints

In this paper we establish conditions which ensure that a feasible point is a global minimizer of a quadratic minimization problem subject to box constraints or binary constraints. In particular, we show that our conditions provide a complete characterization of global optimality for non-convex weighted least squares minimization problems. We present a new approach which makes use of a global subdifferential. It is formed by a set of functions which are not necessarily linear functions, and it enjoys explicit descriptions for quadratic functions. We also provide numerical examples to illustrate our optimality conditions.

A Hierarchy of Integrable Lattice Soliton Equationsand New Integrable Symplectic Map

Starting from a discrete spectral problem, a hierarchy ofintegrable lattice soliton equations is derived. It is shown thatthe hierarchy is completely integrable in the Liouville sense andpossesses discrete bi-Hamiltonian structure. A new integrablesymplectic map and finite-dimensional integrable systems are givenby nonlinearization method. The binary Bargmann constraint givesrise to a Bäcklund transformation for the resultingintegrable lattice equations. At last, conservation laws of thehierarchy are presented.

The satisfiability threshold for randomly generated binary constraint satisfaction problems

AbstractWe study two natural models of randomly generated constraint satisfaction problems. We determine how quickly the domain size must grow with n to ensure that these models are robust in the sense that they exhibit a non‐trivial threshold of satisfiability, and we determine the asymptotic order of that threshold. We also provide resolution complexity lower bounds for these models. One of our results immediately yields a theorem regarding homomorphisms between two random graphs. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2006

A GRASP algorithm to solve the unicost set covering problem

The set covering problem (SCP) is a well-known combinatorial optimization problem. This paper presents a GRASP algorithm to solve a special SCP case known in the literature as the unicost set covering problem. The algorithm incorporates a local improvement procedure based on the heuristics to solve binary constraint satisfiability problems (SAT). The quality of the proposed algorithm is tested on a set of reference instances, comparing the obtained results with those found in the literature. Our algorithm improves the best-known solutions for many of these instances.

Solving Sparse Random Instances of Max Cut and Max 2-CSP in Linear Expected Time

We show that a maximum cut of a random graph below the giant-component threshold can be found in linear space and linear expected time by a simple algorithm. In fact, the algorithm solves a more general class of problems, namely binary 2-variable constraint satisfaction problems. In addition to Max Cut, such Max 2-CSPs encompass Max Dicut, Max 2-Lin, Max 2-Sat, Max-Ones-2-Sat, maximum independent set, and minimum vertex cover. We show that if a Max 2-CSP instance has an ‘underlying’ graph which is a random graph is a deterministic upper bound on the solution time. Then, analysing the tails of the distribution of this quantity for a component of a random graph yields our result. Towards this end we derive some useful properties of binomial distributions and simple random walks.

Person and number agreement in Dumi

Aissen (1999, 2003) argues that prominence hierarchy effects in morphosyntax are governed by fixed rankings of markedness constraints related to the hierarchies themselves by harmonic alignment (Prince and Smolensky 1993). In this article, I analyze the effects of prominence hierarchies on agreement in Dumi, an endangered Kiranti language spoken in Eastern Nepal (van Driem 1993), and argue that the empirical facts can be captured best if hierarchy effects follow from freely rankable binary preference and markedness constraints, not from the fixed ranking of markedness constraints. The analysis further reveals the relevance of number hierarchies for agreement, which are largely neglected in the literature. Thus Dumi resorts to the hierarchy plural > dual > singular that competes with the parallel effects of the more standard person hierarchy (1st > 2nd > 3rd person) on agreement control and also triggers an as yet undocumented number inverse marking, which indicates if one argument of a verb is more prominent for number than the other, but less prominent for person.

Sensitivity of the Optimum to Perturbations of the Profit or Weight of an Item in the Binary Knapsack Problem

In the binary single constraint Knapsack Problem, denoted KP, we are given a knapsack of fixed capacity c and a set of n items. Each item j, j = 1,...,n, has an associated size or weight wj and a profit pj. The goal is to determine whether or not item j, j = 1,...,n, should be included in the knapsack. The objective is to maximize the total profit without exceeding the capacity c of the knapsack. In this paper, we study the sensitivity of the optimum of the KP to perturbations of either the profit or the weight of an item. We give approximate and exact interval limits for both cases (profit and weight) and propose several polynomial time algorithms able to reach these interval limits. The performance of the proposed algorithms are evaluated on a large number of problem instances.

A generalized MKdV hierarchy, tri-Hamiltonian structure, higher-order binary constrained flows and its integrable couplings system

A subalgebra of higher-dimension loop algebra A 2 ˜ is constructed, from which a new 3 × 3 isospectral problem is designed. By making use of Tu’s scheme, an integrable Hamiltonian hierarchy of equations in the sense of Liouville is obtained, which possesses tri-Hamiltonian structure. As reduction cases of the hierarchy presented in this paper, the generalized MKdV equation is engendered. By establishing binary symmetric constraints, three constrained flows of the hierarchy are presented, which are then reduced to Hamiltonian systems. Finally, an integrable coupling system is obtained by constructing a high-dimension loop algebra.

RCC8 binary constraint network can be consistently extended

The RCC8 constraint language developed by Randell et al. has been popularly adopted by the Qualitative Spatial Reasoning and GIS communities. The recent observation that RCC8 composition table describes only weak composition instead of composition raises questions about Renz and Nebel's maximality results about the computational complexity of reasoning with RCC8. This paper shows that any consistent RCC8 binary constraint network (RCC8 network for short) can be consistently extended. Given Θ, an RCC8 network, and z, a fresh variable, suppose x T y ∈ Θ and T is contained in the weak composition of R and S. This means that we can add two new constraints x R z and z S y to Θ without changing the consistency of the network. The result guarantees the applicability to RCC8 of one key technique, (Theorem 5) of [J. Renz, B. Nebel, On the complexity of qualitative spatial reasoning: A maximal tractable fragment of the Region Connection Calculus. Artificial Intelligence 108 (1999) 69–123], which allows the transfer of tractability of a set of RCC8 relations to its closure under composition, intersection, and converse.

Universal Decentralized Estimation in a Bandwidth Constrained Sensor Network

Consider a situation where a set of distributed sensors and a fusion center wish to cooperate to estimate an unknown parameter over a bounded interval [-U,U]. Each sensor collects one noise-corrupted sample, performs a local estimation, and transmits a message to the fusion center, while the latter combines the received messages to produce a final estimate. This correspondence investigates optimal local estimation and final fusion schemes under the constraint that the communication from each sensor to the fusion center must be a one-bit message. Such a binary message constraint is well motivated by the bandwidth limitation of the communication links, fusion center, and by the limited power budget of local sensors. In the absence of bandwidth constraint and assuming the noises are bounded to the interval [-U,U], additive, independent, but otherwise unknown, the classical estimation theory suggests that a total of O(U2/epsi2) sensors are necessary and sufficient in order for the sensors and the fusion center to jointly estimate the unknown parameter within epsi root mean squared error (MSE). It is shown in this correspondence that the same remains true even with the binary message constraint. Furthermore, the optimal decentralized estimation scheme suggests allocating 1/2 of the sensors to estimate the first bit of the unknown parameter, 1/4 of the sensors to estimate the second bit, and so on

A Genetic Algorithm for Searching Spatial Configurations

Searching spatial configurations is a particular case of maximal constraint satisfaction problems, where constraints expressed by spatial and nonspatial properties guide the search process. In the spatial domain, binary spatial relations are typically used for specifying constraints while searching spatial configurations. Searching configurations is particularly intractable when configurations are derived from a combination of objects, which involves a hard combinatorial problem. This paper presents a genetic algorithm (GA) that combines a direct and an indirect approach to treating binary constraints in genetic operators. A new genetic operator combines randomness and heuristics for guiding the reproduction of new individuals in a population. Individuals are composed of spatial objects whose relationships are indexed by a content measure. This paper describes the GA and presents experimental results that compare the genetic versus a deterministic and a local-search algorithm. These experiments show the convenience of using a GA when the complexity of the queries and databases do no guarantee the tractability of a deterministic strategy.

Relation Algebras and their Application in Temporal and Spatial Reasoning

Qualitative temporal and spatial reasoning is in many cases based on binary relations such as before, after, starts, contains, contact, part of, and others derived from these by relational operators. The calculus of relation algebras is an equational formalism; it tells us which relations must exist, given several basic operations, such as Boolean operations on relations, relational composition and converse. Each equation in the calculus corresponds to a theorem, and, for a situation where there are only finitely many relations, one can construct a composition table which can serve as a look up table for the relations involved. Since the calculus handles relations, no knowledge about the concrete geometrical objects is necessary. In this sense, relational calculus is "pointless". Relation algebras were introduced into temporal reasoning by Allen (1983, Communications of the ACM 26(1), 832--843) and into spatial reasoning by Egenhofer and Sharma (1992, Fifth International Symposium on Spatial Data Handling, Charleston, SC). The calculus of relation algebras is also well suited to handle binary constraints as demonstrated e.g. by Ladkin and Maddux (1994, Journal of the ACM 41(3), 435--469). In the present paper I will give an introduction to relation algebras, and an overview of their role in qualitative temporal and spatial reasoning.

Supermodular functions and the complexity of M AX CSP

In this paper we study the complexity of the maximum constraint satisfaction problem (M AX CSP) over an arbitrary finite domain. An instance of M AX CSP consists of a set of variables and a collection of constraints which are applied to certain specified subsets of these variables; the goal is to find values for the variables which maximize the number of simultaneously satisfied constraints. Using the theory of sub- and supermodular functions on finite lattice-ordered sets, we obtain the first examples of general families of efficiently solvable cases of M AX CSP for arbitrary finite domains. In addition, we provide the first dichotomy result for a special class of non-Boolean M AX CSP, by considering binary constraints given by supermodular functions on a totally ordered set. Finally, we show that the equality constraint over a non-Boolean domain is non-supermodular, and, when combined with some simple unary constraints, gives rise to cases of M AX CSP which are hard even to approximate.

A multiobjective optimization approach to smart growth in land development

In this paper we apply a multiobjective optimization model of Smart Growth to land development. The term Smart Growth is meant to describe development strategies—that do not promote urban sprawl. However, the term is somewhat open to interpretation. The multiobjective aspects arise when considering the conflicting interests of the various stakeholders involved in land development decisions: the government planner, the environmentalist, the conservationist, and the land developer. We present a formulation—employing linear and convex quadratic objective functions subject to polyhedral and binary constraints for the stakeholders. The resulting optimization problems are convex, quadratic mixed integer programs that are NP-complete. We report numerical results with this model for Montgomery County, Maryland, and present them using a geographic information system (GIS).

Factorization of a hierarchy of the lattice soliton equations from a binary Bargmann symmetry constraint

Staring from a discrete spectral problem, a hierarchy of the lattice soliton equations is derived. It is shown that each lattice equation in resulting hierarchy is Liouville integrable discrete Hamiltonian system. The binary nonlinearization of the Lax pairs and the adjoint Lax pairs of the resulting hierarchy is discussed. Each lattice soliton equation in the resulting hierarchy can be factored by an integrable symplectic map and a finite-dimensional integrable system in Liouville sense. Especially, factorization of a discrete Kdv equation is given.

Improved biclustering of microarray data demonstrated through systematic performance tests

A new algorithm is presented for fitting the plaid model, a biclustering method developed for clustering gene expression data. The approach is based on speedy individual differences clustering and uses binary least squares to update the cluster membership parameters, making use of the binary constraints on these parameters and simplifying the other parameter updates. The performance of both algorithms is tested on simulated data sets designed to imitate (normalised) gene expression data, covering a range of biclustering configurations. Empirical distributions for the components of these data sets, including non-systematic error, are derived from a real set of microarray data. A set of two-way quality measures is proposed, based on one-way measures commonly used in information retrieval, to evaluate the quality of a retrieved bicluster with respect to a target bicluster in terms of both genes and samples. By defining a one-to-one correspondence between target biclusters and retrieved biclusters, the performance of each algorithm can be assessed. The results show that, using appropriately selected starting criteria, the proposed algorithm out-performs the original plaid model algorithm across a range of data sets. Furthermore, through the rigorous assessment of the plaid model a benchmark for future evaluation of biclustering methods is established.

An Application of Ramsey Theory to Coding for the Optical Channel

In this paper, we analyze bi-infinite sequences over the alphabet $\{0,1,\ldots,q-1\}$, for an arbitrary $q \geq 2$, that satisfy the q-ary ghost pulse (qGP) constraint. A sequence $\x = {(x_k)}_{k \in \Z} \in \{0,1,\ldots,q-1\}^{\Z}$ satisfies the qGP constraint if for all $k,l,m \in \Z$ such that $x_k$, $x_l$ and $x_m$ are nonzero and equal, $x_{k+l-m}$ is also nonzero. This constraint arises in the context of coding for communication over a fiber optic medium. We show, using techniques from Ramsey theory, that if $\x$ satisfies the qGP constraint, then the set $\supp(\x) = \{l \in \Z:\ x_l \neq 0\}$ is the disjoint union of cosets of some subgroup, $k\Z$, of $\Z$, and a set of zero density. We provide much sharper results in the special cases of $q = 2$ and $q=3$. In the former case, we show that the corresponding binary ghost pulse constraint has zero capacity, and based on our results for the latter case, we conjecture that the capacity of the ternary ghost pulse constraint is also zero.

Robust Parameter Settings for Variation Operators by Measuring the Resampling Ratio: A Study on Binary Constraint Satisfaction Problems

In this article, we try to provide insight into the consequence of mutation and crossover rates when solving binary constraint satisfaction problems. This insight is based on a measurement of the space searched by an evolutionary algorithm. From data empirically acquired we describe the relation between the success ratio and the searched space. This is achieved using the resampling ratio, which is a measure for the amount of points revisited by a search algorithm. This relation is based on combinations of parameter settings for the variation operators. We then show that the resampling ratio is useful for identifying the quality of parameter settings, and provide a range that corresponds to robust parameter settings.

Short-term Scheduling of Non-sequential Multipurpose Batch Processes with Transfer and Set-up Times Under Various Intermediate Storage Policies

In this paper, a mixed integer linear programming (MILP) model for short-term scheduling of non-sequential multipurpose batch processes is presented. The processes consist of a single production line, i.e. one unit per stage. The intermediate storage policies considered are unlimited intermediate storage (UIS), no intermediate storage (NIS), zero-wait (ZW) and finite intermediate storage (FIS) policies. Transfer and sequence dependent setup times are also taken into account under four intermediate storage policies. The objective of this study is to determine the optimal production sequence for minimizing makespan under the different storage policies. The basic idea of the proposed MILP model comes from the fact that the production sequence can be determined by comparing the orders of all pairs of products. In contrast to previously published models, the proposed model has fewer binary and continuous variables and fewer constraints, and can be solved in significantly less computational time. The effectiveness of the proposed formulation is illustrated by several examples, and the numerical results and comparisons are provided.