Combinatorial Decision Problems Research Articles

The Referenced Vertex Ordering Problem: Theory, Applications, and Solution Methods

We introduce the referenced vertex ordering problem (REVORDER) as a combinatorial decision problem generalizing several vertex ordering problems that already appeared in the scientific literature under different guises. In other words, REVORDER is a generic problem with several possible extensions corresponding to various real-life applications. Previous works show that REVORDER has a polynomial complexity, whereas its optimization version, denoted in our work as MIN REVORDER, is NP-hard. We give a survey of methods and algorithms that can be applied to the solution of MIN REVORDER, and we develop a new enumeration scheme for its solution. Our theoretical analysis of this scheme yields several pruning techniques aimed at the reduction of the number of enumeration nodes. We then discuss how upper and lower bounds can be computed during the enumeration to design a branch-and-bound algorithm. Finally, we validate the branch-and-bound by conducting a large set of computational experiments on instances coming from various real-life applications. Our results highlight that the newly introduced pruning techniques allow for major reductions of computational cost, with a constant solution quality. In result, our branch-and-bound outperforms other existing solution methods: among 180 instances with 60 vertices, it solves 179 instances to optimality whereas the best existing method is only able to solve 109 of them. Moreover, our tests show that our algorithm can solve medium-scale instances including up to 500 vertices, which opens the perspective of handling new real-life problems. Our implementation of the branch-and-bound algorithm, together with all instances we have used, is publicly available on GitLab.

Joint Access Point Assignment and Power Allocation in Multi-Tier Hybrid RF/VLC HetNets

This paper investigates the joint problem of access point (AP) assignment and power allocation (PA) in a three-tier hybrid radio frequency/visible light communication (VLC) heterogeneous network (HetNet). The main goal is to maximize the HetNet’s sum-rate under practical constraints such as APs’ power budgets and users’ quality-of-service (QoS) requirements, while maintaining an acceptable level of illumination in the VLC system. When this design problem is formulated mathematically, it turns out to be a combinatorial decision problem that involves non-linear constraints, and hence is NP hard. To efficiently obtain good quality solutions for the formulated problem, a reformulation into the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">college admission model</i> is first performed. Then, a distributed and low-complexity algorithm based on matching theory and an efficient heuristic PA scheme are proposed to obtain a good quality suboptimal solution for the joint problem. Simulation results highlight the robustness of the proposed solution and its significant gain in network sum-rate as compared to different benchmark schemes. The effect of various system parameters such as the minimum QoS and maximum illumination requirements on the performance of the proposed solution is studied. Finally, the theoretical analysis of convergence, stability, and complexity of the proposed technique is performed.

Easy-hard phase transition in parameter estimation for optical waveguides

The determination of the parameters of cylindrical optical waveguides, e.g. the diameters vec {d}=(d_1,ldots ,d_r) of r layers of (semi-) transparent optical fibres, can be executed by inverse evaluation of the scattering intensities that emerge under monochromatic illumination. The inverse problem can be solved by optimising the mismatch R(vec {d}) between the measured and simulated scattering patterns. The global optimum corresponds to the correct parameter values. The mismatch R(vec {d}) can be seen as an energy landscape as a function of the diameters. In this work, we study the structure of the energy landscape for different values of the complex refractive indices vec {n}, for r=1 and r=2 layers. We find that for both values of r, depending on the values of vec {n}, two very different types of energy landscapes exist, respectively. One type is dominated by one global minimum and the other type exhibits a multitude of local minima. From an algorithmic viewpoint, this corresponds to easy and hard phases, respectively. Our results indicate that the two phases are separated by sharp phase-transition lines and that the shape of these lines can be described by one “critical” exponent b, which depends slightly on r. Interestingly, the same exponent also describes the dependence of the number of local minima on the diameters. Thus, our findings are comparable to previous theoretical studies on easy-hard transitions in basic combinatorial optimisation or decision problems like Travelling Salesperson and Satisfiability. To our knowledge our results are the first indicating the existence of easy-hard transitions for a real-world optimisation problem of technological relevance.

A Recursive Scenario Decomposition Algorithm for Combinatorial Multistage Stochastic Optimisation Problems

Stochastic programming is concerned with decision making under uncertainty, seeking an optimal policy with respect to a set of possible future scenarios. This paper looks at multistage decision problems where the uncertainty is revealed over time. First, decisions are made with respect to all possible future scenarios. Secondly, after observing the random variables, a set of scenario specific decisions is taken. Our goal is to develop algorithms that can be used as a back-end solver for high-level modeling languages. In this paper we propose a scenario decomposition method to solve multistage stochastic combinatorial decision problems recursively. Our approach is applicable to general problem structures, utilizes standard solving technology and is highly parallelizable. We provide experimental results to show how it efficiently solves benchmarks with hundreds of scenarios.

Genetic algorithm for supply chain modelling: basic concepts and applications

As a subfield of artificial intelligence, genetic algorithm (GA) was introduced in the 1970s to tackle various types (both continuous and discrete) of combinatorial decision problems facing many business enterprises. These problems include routine but complex managerial challenges associated with supply chain activities of sourcing, making, selling, and delivering goods and services. With the emergence of supply chain principles in today's business world, GA has increased its role in improving managerial decision-making processes and subsequently enhancing supply chain efficiency by avoiding the sub-optimisation of problem solutions. Despite its application potentials, we have seen the limited use of GA for supply chain management. To make the best use of GA for supply chain management, this paper introduces the theoretical underpinning of GA and then explains how effectively it works for solving difficult supply chain problems. In so doing, this paper reviews the past record of success in GA applications to supply chain fields and then identifies the most promising areas of supply chain management in which to apply GA.

Multi-variable distributed backtracking with sessions

The Constraint Satisfaction Problem (CSP) formalism is used to represent many combinatorial decision problems instances simply and efficiently. However, many such problems cannot be solved on a single, centralized computer for various reasons (e.g., their excessive size or privacy). The Distributed CSP (DisCSP) extends the CSP model to allow such combinatorial decision problems to be modelled and handled. In this paper, we propose a complete DisCSP-solving algorithm, called Distributed Backtracking with Sessions (DBS), which can solve DisCSP so that each agent encapsulates a whole "complex" problem with many variables and constraints. We prove that the algorithm is sound and complete, and generates promising experimental results.

Phase transition in the countdown problem

We present a combinatorial decision problem, inspired by the celebrated quiz show called Countdown, that involves the computation of a given target number T from a set of k randomly chosen integers along with a set of arithmetic operations. We find that the probability of winning the game evidences a threshold phenomenon that can be understood in the terms of an algorithmic phase transition as a function of the set size k. Numerical simulations show that such probability sharply transitions from zero to one at some critical value of the control parameter, hence separating the algorithm's parameter space in different phases. We also find that the system is maximally efficient close to the critical point. We derive analytical expressions that match the numerical results for finite size and permit us to extrapolate the behavior in the thermodynamic limit.

A rolling horizon approach for disruption management of railway rolling stock

This paper deals with real-time disruption management of rolling stock in passenger railway transportation. We describe a generic framework for dealing with disruptions of railway rolling stock schedules. The framework is presented as an online combinatorial decision problem, where the uncertainty of a disruption is modeled by a sequence of information updates. To decompose the problem and to reduce the computation time, we propose a rolling horizon approach: rolling stock decisions are only considered if they are within a certain time horizon from the time of rescheduling. The schedules are then revised as time progresses and new information becomes available. We extend an existing model for rolling stock scheduling to the specific requirements of the real-time situation, and we apply it in the rolling horizon framework. We perform computational tests on instances constructed from real-life cases of Netherlands Railways (NS), the main operator of passenger trains in the Netherlands. We explore the consequences of different settings of the approach for the trade-off between solution quality and computation time.

Stochastic Constraint Programming: A Scenario-Based Approach

To model combinatorial decision problems involving uncertainty and probability, we introduce scenario based stochastic constraint programming. Stochastic constraint programs contain both decision variables, which we can set, and stochastic variables, which follow a discrete probability distribution. We provide a semantics for stochastic constraint programs based on scenario trees. Using this semantics, we can compile stochastic constraint programs down into conventional (non-stochastic) constraint programs. This allows us to exploit the full power of existing constraint solvers. We have implemented this framework for decision making under uncertainty in stochastic OPL, a language which is based on the OPL constraint modelling language [Hentenryck et al., 1999]. To illustrate the potential of this framework, we model a wide range of problems in areas as diverse as portfolio diversification, agricultural planning and production/inventory management.

Convex hulls, oracles, and homology

This paper presents a new algorithm for the convex hull problem, which is based on a reduction to a combinatorial decision problem CompletenessC, which in turn can be solved by a simplicial homology computation. Like other convex hull algorithms, our algorithm is polynomial (in the size of input plus output) for simplicial or simple input. We show that the “no”-case of CompletenessC has a certificate that can be checked in polynomial time (if integrity of the input is guaranteed).

Complexity-theoretic models of phase transitions in search problems

In recent years, numerous studies have observed that many hard combinatorial decision problems exhibit behaviour described as a ‘ phase-transition’. This is the phenomenon whereby typical instances of a problem display a dramatic shift in certain characteristics as some parameter of the instances is varied. Such characteristics include the likelihood of an instance having a solution and the time taken by a search algorithm. The apparent pervasiveness of phase-transitions in hard combinatorial search problems has led to contrasting claims being advanced concerning to what extent all NP-complete problems exhibit phase-transitions. The established importance of exploiting phase-transition effects in the design of search heuristics provides a strong motivation for assessing how valid such claims may be. In this paper we argue that questions concerning the generality of phase-transition phenomena are, at present, ill-defined. In order to address this difficulty, we propose and examine rigorous complexity-theoretic models of the statement ‘the decision problem D has a phase-transition’. Within these models it is proved that for certain ‘natural’ definitions contrasting results about phase-transition behaviour can be proved.

Haar spectra-based entropy approach to quasi-minimisation of FBDDs

An information theoretic approach, to exploit the additional degree of freedom associated with don&apos;t cares of incompletely specified Boolean functions, is applied to quasi-minimisation of free binary decision diagrams (FBDDs). The concept of entropy and equivocation is formulated through paired Haar spectra of incompletely specified Boolean functions. The likelihood metric expressed in terms of selected spectral coefficients is used to simplify the process of don&apos;t care allocation. The approach is general and can be extended to other combinatorial decision problems.

Probabilistic analysis of combinatorial algorithms: A bibliography with selected annotations

Probabilistic methods in evaluation of performance efficiency of combinatorial optimization algorithms are of continuously growing interest, and rapidly increasing effort of researchers in this field is observed. The present paper is a bibliography which contains 70 references dealing with probabilistic evaluation of time complexity and performance accuracy of deterministic algorithms for combinatorial decision and optimization problems. Some entries of the bibliography, mainly those having appreared in journals and nonperiodical issues of limited distribution are shortly annotated (18 references). Basic notions and definitions facilitating better understanding and plain presentation of different results are given.

The one-one equivalence of some general combinatorial decision problems.

The one-one equivalence of some general combinatorial decision problems.

Combinatorial systems with axiom.

Introduction:* The purpose of this paper is to investigate the general decision problems associated with a number of combinatorial systems with axiom. In particular, we shall show the many-one equivalence of the general halting problem for Turing machines, the general decision problem for Thue systems with axiom, the general decision problem for semi-Thue systems with axiom, and the general decision problem for Post normal systems with axiom. This, combined with a recent result of Overbeek [5], shows that every recursively enumerable (r.e.) many-one degree (of unsolvability) is represented by each of these general problems for systems with axiom. Finally, this latter result is proven to be best possible in that it does not hold for every r.e. one-one degree. Historical Background: Semi-Thue systems, Thue systems and Post normal systems were defined by Post as proper subsets of canonical forms. Decision problems associated with these systems have been studied by various authors, e.g., [l], [2], [3], [4], [6], [7], and [8]. In particular, W. E. Singletary [7] has combined results of his own and those of others in such a way as to provide an effective proof of the (r.e.) equivalence of the general decision problems which are of concern to us here. The stronger results to be proven here were announced in [4] and form part of an extensive study into the equivalence of general combinatorial decision problems. Preliminary Definitions: A semi-Thue system T is a pair (Σ,R) where Σ is a finite alphabet and R is a finite set of productions of the form a —» β, for a and β words over Σ. T is said to be a Thue system if a -* β belongs to R implies β —> a is also in R. For any arbitrary pair of words W, W r over Σ, we say that Wr is an immediate successor of W in T, denoted T(W, W) if W = P a Q, W = P β Q, P and Q are words over Σ, and a — β is inR.

A note on disjunctive form tautologies

Cook [1] and Karp [2] have shown that a large class of combinatorial decision problems can be solved in time bounded by a polynomial in the size of the problem iff there is a polynomial time procedure for deciding whether a disjunctive formula with at most three literals per disjunct in the propositional calculus is a tautology.

A Dynamic Programming Approach to a Bidder Selection Problem

Abstract This paper deals with a set of combinatorial decision problems that arise in large-scale governmental or military procurement operations. After stating the problems, they are then mathematically formulated as linear integer programming problems. However, because a specified sensitivity analysis is required, a dynamic programming approach is shown to give all solutions needed after one pass through the DP tableaus. A model has been implemented and is currently being used by a large buying activity of the Department of the Army.

Combinatorial systems I. Cylindrical problems

The reduction of one combinatorial decision problem PI to another P2 can be achieved by constructing an effective map r : Pa --~ P2 such that each problem p in /'1 is reducible to the corresponding problem r in P2 (el. [4]). Judicious choice of ~b can sometimes ensure that p ~,~ r If, in addition, all the problems in P2, are cylinders [2] then p 41 r so that the reduction of P1 to P2 is best possible. This is because any cylinder has maximum one-one degree within its many-one degree--two cylinders which are many-one equivalent are actually one-one equivalent. The primary aim of this paper is, therefore, to identify some combinatorial problems which are always cylinders. Our investigation is effected by considering those functions on the natural numbers whose values are finite sets of natural numbers--system functions--which possess certain formal properties in common with those arising naturally from G6del numbering of many kinds of combinatorial system. This approach also enables us to construct system functions with noncylindrical problems--a matter reserved for a later paper. Many aspects of combinatorial systems (partial propositional calculi, Markov algorithms semi-Thue systems, etc.) are comprehended in Tarski's definition of deductive systems [5]. Accordingly we first expound a general method of recognizing cylindrical combinatorial problems of deductive systems and then apply it to system functions. For an explanation of the concepts of recursive-function theory as used in this paper the reader is referred to [2]; the notion of a cylinder, which is central to our discussion, and its elementary properties are presented on p. 89. The idea of using system functions in the theory of combinatorial systems is due to W. E. Singletary. The points in the ensuing development at which I am indebted to him are too numerous to mention.

The many-one equivalence of some general combinatorial decision problems

The many-one equivalence of some general combinatorial decision problems

The equivalence of some general combinatorial decision problems

The equivalence of some general combinatorial decision problems