pseudo-Boolean Optimization Research Articles

Classes of submodular constraints expressible by graph cuts

Submodular constraints play an important role both in theory and practice of valued constraint satisfaction problems (VCSPs). It has previously been shown, using results from the theory of combinatorial optimisation, that instances of VCSPs with submodular constraints can be minimised in polynomial time. However, the general algorithm is of order O(n 6) and hence rather impractical. In this paper, by using results from the theory of pseudo-Boolean optimisation, we identify several broad classes of submodular constraints over a Boolean domain which are expressible using binary submodular constraints, and hence can be minimised in cubic time. Furthermore, we describe how our results translate to certain optimisation problems arising in computer vision.

The expressive power of binary submodular functions

We investigate whether all Boolean submodular functions can be decomposed into a sum of binary submodular functions over a possibly larger set of variables. This question has been considered in several different contexts in computer science, including computer vision, artificial intelligence, and pseudo-Boolean optimisation. Using a connection between the expressive power of valued constraints and certain algebraic properties of functions, we answer this question negatively. Our results have several corollaries. First, we characterise precisely which submodular polynomials of arity 4 can be expressed by binary submodular polynomials. Next, we identify a novel class of submodular functions of arbitrary arities which can be expressed by binary submodular functions, and therefore minimised efficiently using a so-called expressibility reduction to the Min-Cut problem. More importantly, our results imply limitations on this kind of reduction and establish, for the first time, that it cannot be used in general to minimise arbitrary submodular functions. Finally, we refute a conjecture of Promislow and Young on the structure of the extreme rays of the cone of Boolean submodular functions.

The impact of parametrization in memetic evolutionary algorithms

Memetic (evolutionary) algorithms integrate local search into the search process of evolutionary algorithms. As computational resources have to be spread adequately among local and evolutionary search, one has to care about when to apply local search and how much computational effort to devote to local search. Often local search is called with a fixed frequency and run for a fixed number of iterations, the local search depth. There is empirical evidence that these parameters have a significant impact on performance, but a theoretical understanding as well as concrete design guidelines are missing. We initiate the rigorous theoretical analysis of memetic algorithms. To this end, we consider a simple memetic algorithm for pseudo-Boolean optimization that captures basic working principles of memetic algorithms—the interplay of genetic operators like mutation and selection with local search. We present function classes where even small changes of the parametrization have a strong impact on performance. For almost every reasonable parameter setting we construct a function that, with high probability, can be optimized in polynomial time. However, changing the local search depth by a small additive term in any direction yields a superpolynomial optimization time, with high probability. For another class of functions altering the local search frequency by a factor of 2 even yields exponential optimization times. Our results show exemplarily that parametrizing memetic evolutionary algorithms can be extremely hard. Moreover, this work yields insights into the dynamic behavior of memetic algorithms and contributes to a theoretical foundation of hybrid metaheuristics.

A UNIFYING FRAMEWORK FOR GENERALIZED CONSTRAINT ACQUISITION

When a practical problem can be modeled as a constraint satisfaction problem (CSP), which is a set of constraints that need to be satisfied, it can be solved using many constraint programming techniques. In many practical applications, while users can recognize examples of where a CSP should be satisfied or violated, they cannot articulate the specification of the CSP itself. In these situations, it can be helpful if the computer can take an active role in learning the CSP from examples of its solutions and non-solutions. This is called constraint acquisition. This paper introduces a framework for constraint acquisition in which one can uniformly define and formulate constraint acquisition problems of different types as optimization problems. The difference between constraint acquisition problems within the framework is not only in the type of constraints that need to be acquired but also in the learning objective. The generic framework can be instantiated to obtain specific formulations for acquiring classical, fuzzy, weighted or probabilistic constraints. The paper shows as an example how recent techniques for acquiring classical constraints can be directly obtained from the framework. Specifically, the formulation obtained from the framework to acquire classical CSPs with the minimum number of violated examples is equivalent to a simple pseudo-boolean optimization problem, thus being efficiently solvable by using many available optimization tools. The paper also reports empirical results on constraint acquisition methods to show the utility of the framework.

FPGA implementation of a stochastic neural network for monotonic pseudo-Boolean optimization

In this paper a FPGA implementation of a novel neural stochastic model for solving constrained NP-hard problems is proposed and developed. The model exploits pseudo-Boolean functions both to express the constraints and to define the cost function, interpreted as energy of a neural network. A wide variety of NP-hard problems falls in the class of problems that can be solved by this model, particularly those having a quadratic pseudo-Boolean penalty function. The proposed hardware implementation provides high computation speed by exploiting parallelism, as the neuron update and the constraint violation check can be performed in parallel over the whole network. The neural system has been tested on random and benchmark graphs, showing good performance with respect to the same heuristic for the same problems. Furthermore, the computational speed of the FPGA implementation has been measured and compared to software implementation. The developed architecture featured dramatically faster computation, with respect to the software implementation, even adopting a low-cost FPGA chip.

Weighted stability number of graphs and weighted satisfiability: The two facets of pseudo-Boolean optimization

We exhibit links between pseudo-Boolean optimization, graph theory and logic. We show the equivalence of maximizing a pseudo-Boolean function and finding a maximum weight stable set; symmetrically minimizing a pseudo-Boolean function is shown to be equivalent to solving a weighted satisfiability problem.

Joint routing and topology formation in multihop UWB networks

This paper addresses the throughput optimization problem in multihop ultra-wideband (UWB) networks by jointly considering network topology formation and routing. Given a spatial distribution of UWB devices and traffic requirement, we want to form piconets and select paths to maximize the network throughput. Although there have been several works considering the problem of selecting paths to achieve the optimal throughput in multihop wireless networks, to the best of our knowledge, none of them takes the topology formation into the consideration. In this paper, we use Boolean matrices to model role assignment in UWB networks and formulate the throughput optimization problem as a nonlinear programming (NLP) problem. Since the throughput optimization problem is NP-hard, we give an upper bound of the optimal throughput by relaxing some constraints and using pseudo-Boolean optimization to linearize the NLP. We prove that the solution of the upper bound is at most three times of the optimal throughput. Based on the topology formed by solving the upper bound, we formulate a lower bound of the optimal throughput as a linear programming problem and use column generation to solve the lower bound. Numerical results show that the lower bound is very close to the upper bound. Simulation results demonstrate the effectiveness of the scheme.

On Using Cutting Planes in Pseudo-Boolean Optimization

Cutting planes are a well-known, widely used, and very eectiv e technique for Integer Linear Programming (ILP). However, cutting plane techniques are seldom used in PseudoBoolean Optimization (PBO) algorithms. This paper addresses the utilization of Gomory mixed-integer and clique cuts, in Satisabilit y-based algorithms for PBO, and shows how these cuts can be used for computing lower bounds and for learning new constraints. A side result of learning new constraints is that the utilization of cutting planes enables nonchronological backtracking. Besides cutting planes, the paper also shows that the utilization of search restarts in PBO can be eectiv e in practice, allowing the computation of tighter lower bounds each time the search restarts. The more aggressive lower bounds result from the constraints learned due to the utilization of cutting planes. Experimental results show that the integration of cutting planes and search restarts in a SAT-based algorithm for PBO yields a competitive new solution for PBO.

Pseudo-Boolean optimization

This survey examines the state of the art of a variety of problems related to pseudo-Boolean optimization, i.e. to the optimization of set functions represented by closed algebraic expressions. The main parts of the survey examine general pseudo-Boolean optimization, the specially important case of quadratic pseudo-Boolean optimization (to which every pseudo-Boolean optimization can be reduced), several other important special classes, and approximation algorithms.

Synthesis of Narrowing Enquiries

A canonical model of decision making, with a disjunctive restriction [1] and the concept of a Pareto set Π(A) which describes the uncertainty region [3], is used as a basis for developing a method for active narrowing enquiries for solving canonical pseudoboolean optimization problems.

Optimal cell flipping to minimize channel density in VLSI design and pseudo-Boolean optimization

Cell flipping in VLSI design is an operation in which some of the cells are replaced with their “mirror images” with respect to a vertical axis, while keeping them in the same slot. After the placement of all the cells, one can apply cell flipping in order to further decrease the total area, approximating this objective by minimizing total wire length, channel width, etc. However, finding an optimal set of cells to be flipped is usually a difficult problem. In this paper we show that cell flipping can be efficiently applied to minimize channel density in the standard cell technology. We show that an optimal flipping pattern can be found in O(p( n c ) c) time, where n, p and c denote the number of nets, pins and channels, respectively. Moreover, in the one channel case (i.e. when c = 1) the cell flipping problem can be solved in O( p log n) time. For the multi-channel case we present both an exact enumeration scheme and a mixed-integer program that generates an approximate solution very quickly. We present computational results on examples up to 139 channels and 65000 cells.

Inference Duality as a Basis for Sensitivity Analysis

The constraint programming community has recently begun to address certain types of optimization problems. These problems tend to be discrete or to have discrete elements. Although sensitivity analysis is well developed for continuous problems, progress in this area for discrete problems has been limited. This paper proposes a general approach to sensitivity analysis that applies to both continuous and discrete problems. In the continuous case, particularly in linear programming, sensitivity analysis can be obtained by solving a dual problem. One way to broaden this result is to generalize the classical idea of a dual to that of an ’’inference dual,‘‘ which can be defined for any optimization problem. To solve the inference dual is to obtain a proof of the optimal value of the problem. Sensitivity analysis can be interpreted as an analysis of the role of each constraint in this proof. This paper shows that traditional sensitivity analysis for linear programming is a special case of this approach. It also illustrates how the approach can work out in a discrete problem by applying it to 0-1 linear programming (linear pseudo-boolean optimization).

On the structure of all minimum cuts in a network and applications

This paper presents a characterization of all minimum cuts, separating a source from a sink in a network. A binary relation is associated with any maximum flow in this network, and minimum cuts are identified with closures for this relation. As a consequence, finding all minimum cuts reduces to a straightforward enumeration. Applications of this results arise in sensitivity and parametric analyses of networks, the vertex packing and maximum closure problems, in unconstrained pseudo-boolean optimization and project selection, as well as in other areas of application of minimum cuts.

An Approximation Technique for Pseudo-Boolean Maximization Problems

Abstract A partitioning procedure is proposed for obtaining approximate solutions to pseudo-Boolean optimization problems (e.g., zero-one polynomial programs). Such a procedure is useful not only in its ability to provide a bound value but further as a means for initializing optimum seeking methods, such as implicit enumeration techniques, employed to insure optimality.