DOCSLIB.ORG

AND/OR Branch-And-Bound Search for Pure 0/1 Integer Linear Programming Problems

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
AND/OR Branch-And-Bound Search for Pure 0/1 Integer Linear Programming Problems AND/OR Branch-and-Bound Search for Pure 0/1 Integer Linear Programming Problems Radu Marinescu and Rina Dechter School of Information and Computer Science University of California, Irvine, CA 92697-3425 {radum,dechter}@ics.uci.edu Abstract. AND/OR search spaces have recently been introduced as a unify- ing paradigm for advanced algorithmic schemes for graphical models. The main virtue of this representation is its sensitivity to the structure of the model, which can translate into exponential time savings for search algorithms. In this paper we extend the recently introduced AND/OR Branch-and-Bound algorithm (AOBB) [1] for solving pure 0/1 Integer Linear Programs [2]. Since the variable selection can have a dramatic impact on search performance, we introduce a new dynamic AND/OR Branch-and-Bound algorithm able to accommodate variable ordering heuristics. The effectiveness of the dynamic AND/OR approach is demonstrated on a variety of benchmarks for pure 0/1 integer programming, including instances from the MIPLIB library, real-world combinatorial auctions and random unca- pacitated warehouse location problems. 1 Introduction A constraint optimization problem is the minimization/maximization of an objective function subject to a set of constraints on the possible values of a set of independent decision variables. An important class of constraint optimization problems are the Inte- ger Linear Programming problems (ILP) [2] where the objective is to optimize a linear function of integer-valued variables, subject to a set of linear equality or inequality con- straints defined on subsets of variables. The classical approach to solving ILPs is the branch-and-bound method [3] which maintains the best solution found so far, while discarding partial solutions which cannot improve on the best. The AND/OR search space for graphical models [4] is a newly introduced frame- work for search that is sensitive to the independencies in the model, often resulting in exponentially reduced complexities. It is based on a pseudo-tree that captures indepen- dencies in the graphical model, resulting in a search tree exponential in the depth of the pseudo-tree, rather than in the number of variables. The AND/OR Branch-and-Bound algorithm (AOBB) is a new search method that explores the AND/OR search tree for solving optimization tasks in graphical models [1]. In this paper we present an extension of the algorithm for solving optimization problems from the class of pure 0/1 Integer Linear Programs [2]. A pure 0/1 integer linear program is a linear program where all the decision variables are restricted to be either 0 or 1 at the optimal solution. Since variable selection can have a dramatic impact on search performance [2], we introduce a dynamic AND/OR Branch-and-Bound search algorithm that combines the AND/OR decomposition principle with variable ordering heuristics. There are two orthogonal approaches to incorporating dynamic orderings into AOBB. The first one im- proves AOBB by applying an independent semantic variable ordering heuristic whenever the partial order dictated by the decomposition principle allows. The second, orthogonal approach gives priority to the semantic variable ordering heuristic and applies problem decomposition as a secondary principle. We demonstrate empirically the efficiency of the dynamic AND/OR Branch-and-Bound approach on several benchmarks for pure 0/1 integer linear programming problems, including test instances from the MIPLIB library, combinatorial auctions simulating radio spectrum allocation and random uncapacitated warehouse location problems. The paper is organized as follows. In Section 2 we present background on con- straint optimization problems and integer linear programming. Section 3 presents the AND/OR search space as well as an efficient heuristic for constructing low depth bal- anced pseudo-trees. In Section 4 we introduce the AND/OR Branch-and-Bound algo- rithm, specialized for solving pure 0/1 integer linear programs. In Section 5 we intro- duce the dynamic AND/OR Branch-and-Bound algorithm. Section 6 shows our empir- ical evaluation and Section 7 concludes. 2 Background 2.1 Constraint Optimization Problems A finite Constraint Optimization Problem (COP) is a four-tuple hX , D, C,zi, where X = {X1, ..., Xn} is a set of variables, D = {D1, ..., Dn} is a set of finite domains, C = {C1, ..., Cm} is a set of constraints on the variables and z is a global cost func- tion (i.e. objective function) to be optimized. The scope of a constraint Ci, denoted scope(Ci) ⊆ X , is the set of arguments of Ci. Constraints can be expressed exten- sionally, through relations, or intentionally, by a mathematical formula (equality or inequality). An optimal solution to a COP is a complete value assignment to all the variables such that every constraint is satisfied and the objective function is minimized or maximized. With every COP instance we can associate a constraint graph G which has a node for each variable and connects any two nodes whose variables appear in the scope of the same constraint. The induced graph of G relative to an ordering d of its variables, denoted G∗(d), is obtained by processing the nodes in reverse order of d. For each node all its earlier neighbors are connected, including neighbors connected by previously added edges. Given a graph and an ordering of its nodes, the width of a node is the number of edges connecting it to nodes lower in the ordering. The induced width of a graph, denoted w∗(d), is the maximum width of nodes in the induced graph. 2.2 Integer Linear Programming A Linear Program (LP) consists of a set of continuous variables and a set of linear con- straints (equalities or inequalities). The goal is to optimize a global linear cost function subject to the constraints. One of the standard forms of a linear program is: min{c>x | Ax ≤ b, x ≥ 0} (1) where c ∈ Rn, b ∈ Rm, A ∈ Rm×n and x ∈ Rn. Here c represents the cost vector and x is the vector of decision variables. The vector b and the matrix A define the m linear constraints. Linear programs are usually solved by Dantzig’s SIMPLEX method [5]. A Mixed Integer Linear Programming (MILP) problem is a linear program where some of the decision variables are constrained to have only integer values at the optimal solution. An important special case is a decision variable xi that is integer with 0 ≤ xi ≤ 1. This forces xi to be either 0 or 1 at the solution. Variables like xi are called 0/1 or binary integer variables. Subsequently, a MILP problem with binary integer variables is also called a 0/1 Mixed Integer Linear Programming problem. A pure 0/1 Integer Linear Programming problem is a MILP where all the decision variables are binary. Pure 0/1 ILPs can formulate many practical problems such as capital budgeting [6], cargo loading [7], processor allocation in distributed systems [8] or combinatorial auctions [9, 10]. Clearly, any pure 0/1 integer linear program can be viewed as a finite COP instance hX , D, C,zi with linear constraints and a linear objective function. In the remaining z n c X of the paper we will consider a minimization problem defined by = Pi=1 i i subject to m linear constraints C = {C1, ..., Cm}, over n binary decision variables X = {X1, ..., Xn}. 2.3 Branch-and-Bound Search for Constraint Optimization Branch-and-Bound (BB) is a general search method for solving constraint optimization problems [3]. It traverses the search tree defined by the problem, where internal nodes represent partial assignments and leaf nodes denote complete ones, which may or may not be optimal. During the traversal, which is usually depth first, BB maintains un upper bound ub, the cost of the best solution found so far. At each internal node the algorithm computes a lower bound lb on the optimal extension of the current partial assignment. When lb ≥ ub, the current best cost cannot be improved and the algorithm backtracks pruning the subtree below the current node. Otherwise, the algorithm moves forward and tries to instantiate the next variable in the ordering. In the context of pure 0/1 integer linear programs, the lower bound of a subproblem is obtained by solving its linear relaxation (i.e. relaxing the integrality restrictions). In this case the branching process can fail at a particular node for one of the following reasons: (i) the LP solution can be integer; or (ii) the LP problem can be infeasible; or (iii) the lower bound exceeds the upper bound (for more details see [2,3]). 3 AND/OR Search Spaces The classical way to do search is to instantiate variables one at a time, following a static/dynamic variable ordering. In the simplest case, this process defines a search tree (called here OR search tree), whose nodes represent states in the space of partial as- signments. The traditional search space does not capture independencies that appear in = + − + − + minimize : z 7A 3B 2C 5D 6E 8F A subject to : − + ≤ 3 A 12 B C 3 B − 2B + 5C − 3D ≤ −2 2 A + B − 4E ≤ 2 C E A − 3E + F ≤1 A, B,C, D, E, F ∈{}1,0 D F (a) (b) OR A AND 0 1 OR B B AND 0 1 0 1 OR C E C E C E C E AND 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 OR D F F D F F D F F D F AND 1 0 1 0 1 0 1 0 1 0 1 1 0 0 1 0 1 0 1 (c) Fig. 1. The AND/OR search space.
Load more

Recommended publications
  • Adversarial Search
    Adversarial Search In which we examine the problems that arise when we try to plan ahead in a world where other agents are planning against us. Outline 1. Games 2. Optimal Decisions in Games 3. Alpha-Beta Pruning 4. Imperfect, Real-Time Decisions 5. Games that include an Element of Chance 6. State-of-the-Art Game Programs 7. Summary 2 Search Strategies for Games • Difference to general search problems deterministic random – Imperfect Information: opponent not deterministic perfect Checkers, Backgammon, – Time: approximate algorithms information Chess, Go Monopoly incomplete Bridge, Poker, ? information Scrabble • Early fundamental results – Algorithm for perfect game von Neumann (1944) • Our terminology: – Approximation through – deterministic, fully accessible evaluation information Zuse (1945), Shannon (1950) Games 3 Games as Search Problems • Justification: Games are • Games as playground for search problems with an serious research opponent • How can we determine the • Imperfection through actions best next step/action? of opponent: possible results... – Cutting branches („pruning“) • Games hard to solve; – Evaluation functions for exhaustive: approximation of utility – Average branching factor function chess: 35 – ≈ 50 steps per player ➞ 10154 nodes in search tree – But “Only” 1040 allowed positions Games 4 Search Problem • 2-player games • Search problem – Player MAX – Initial state – Player MIN • Board, positions, first player – MAX moves first; players – Successor function then take turns • Lists of (move,state)-pairs – Goal test
    [Show full text]
  • A Lifted Linear Programming Branch-And-Bound Algorithm for Mixed Integer Conic Quadratic Programs Juan Pablo Vielma, Shabbir Ahmed, George L
    A Lifted Linear Programming Branch-and-Bound Algorithm for Mixed Integer Conic Quadratic Programs Juan Pablo Vielma, Shabbir Ahmed, George L. Nemhauser, H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, 765 Ferst Drive NW, Atlanta, GA 30332-0205, USA, {[email protected], [email protected], [email protected]} This paper develops a linear programming based branch-and-bound algorithm for mixed in- teger conic quadratic programs. The algorithm is based on a higher dimensional or lifted polyhedral relaxation of conic quadratic constraints introduced by Ben-Tal and Nemirovski. The algorithm is different from other linear programming based branch-and-bound algo- rithms for mixed integer nonlinear programs in that, it is not based on cuts from gradient inequalities and it sometimes branches on integer feasible solutions. The algorithm is tested on a series of portfolio optimization problems. It is shown that it significantly outperforms commercial and open source solvers based on both linear and nonlinear relaxations. Key words: nonlinear integer programming; branch and bound; portfolio optimization History: February 2007. 1. Introduction This paper deals with the development of an algorithm for the class of mixed integer non- linear programming (MINLP) problems known as mixed integer conic quadratic program- ming problems. This class of problems arises from adding integrality requirements to conic quadratic programming problems (Lobo et al., 1998), and is used to model several applica- tions from engineering and finance. Conic quadratic programming problems are also known as second order cone programming problems, and together with semidefinite and linear pro- gramming (LP) problems are special cases of the more general conic programming problems (Ben-Tal and Nemirovski, 2001a).
    [Show full text]
  • A Branch-And-Price Approach with Milp Formulation to Modularity Density Maximization on Graphs
    A BRANCH-AND-PRICE APPROACH WITH MILP FORMULATION TO MODULARITY DENSITY MAXIMIZATION ON GRAPHS KEISUKE SATO Signalling and Transport Information Technology Division, Railway Technical Research Institute. 2-8-38 Hikari-cho, Kokubunji-shi, Tokyo 185-8540, Japan YOICHI IZUNAGA Information Systems Research Division, The Institute of Behavioral Sciences. 2-9 Ichigayahonmura-cho, Shinjyuku-ku, Tokyo 162-0845, Japan Abstract. For clustering of an undirected graph, this paper presents an exact algorithm for the maximization of modularity density, a more complicated criterion to overcome drawbacks of the well-known modularity. The problem can be interpreted as the set-partitioning problem, which reminds us of its integer linear programming (ILP) formulation. We provide a branch-and-price framework for solving this ILP, or column generation combined with branch-and-bound. Above all, we formulate the column gen- eration subproblem to be solved repeatedly as a simpler mixed integer linear programming (MILP) problem. Acceleration tech- niques called the set-packing relaxation and the multiple-cutting- planes-at-a-time combined with the MILP formulation enable us to optimize the modularity density for famous test instances in- cluding ones with over 100 vertices in around four minutes by a PC. Our solution method is deterministic and the computation time is not affected by any stochastic behavior. For one of them, column generation at the root node of the branch-and-bound tree arXiv:1705.02961v3 [cs.SI] 27 Jun 2017 provides a fractional upper bound solution and our algorithm finds an integral optimal solution after branching. E-mail addresses: (Keisuke Sato) [email protected], (Yoichi Izunaga) [email protected].
    [Show full text]
  • Linear Programming
    Lecture 18 Linear Programming 18.1 Overview In this lecture we describe a very general problem called linear programming that can be used to express a wide variety of different kinds of problems. We can use algorithms for linear program- ming to solve the max-flow problem, solve the min-cost max-flow problem, find minimax-optimal strategies in games, and many other things. We will primarily discuss the setting and how to code up various problems as linear programs At the end, we will briefly describe some of the algorithms for solving linear programming problems. Specific topics include: • The definition of linear programming and simple examples. • Using linear programming to solve max flow and min-cost max flow. • Using linear programming to solve for minimax-optimal strategies in games. • Algorithms for linear programming. 18.2 Introduction In the last two lectures we looked at: — Bipartite matching: given a bipartite graph, find the largest set of edges with no endpoints in common. — Network flow (more general than bipartite matching). — Min-Cost Max-flow (even more general than plain max flow). Today, we’ll look at something even more general that we can solve algorithmically: linear pro- gramming. (Except we won’t necessarily be able to get integer solutions, even when the specifi- cation of the problem is integral). Linear Programming is important because it is so expressive: many, many problems can be coded up as linear programs (LPs). This especially includes problems of allocating resources and business 95 18.3. DEFINITION OF LINEAR PROGRAMMING 96 supply-chain applications. In business schools and Operations Research departments there are entire courses devoted to linear programming.
    [Show full text]
  • Integer Linear Programs
    20 ________________________________________________________________________________________________ Integer Linear Programs Many linear programming problems require certain variables to have whole number, or integer, values. Such a requirement arises naturally when the variables represent enti- ties like packages or people that can not be fractionally divided — at least, not in a mean- ingful way for the situation being modeled. Integer variables also play a role in formulat- ing equation systems that model logical conditions, as we will show later in this chapter. In some situations, the optimization techniques described in previous chapters are suf- ficient to find an integer solution. An integer optimal solution is guaranteed for certain network linear programs, as explained in Section 15.5. Even where there is no guarantee, a linear programming solver may happen to find an integer optimal solution for the par- ticular instances of a model in which you are interested. This happened in the solution of the multicommodity transportation model (Figure 4-1) for the particular data that we specified (Figure 4-2). Even if you do not obtain an integer solution from the solver, chances are good that you’ll get a solution in which most of the variables lie at integer values. Specifically, many solvers are able to return an ‘‘extreme’’ solution in which the number of variables not lying at their bounds is at most the number of constraints. If the bounds are integral, all of the variables at their bounds will have integer values; and if the rest of the data is integral, many of the remaining variables may turn out to be integers, too.
    [Show full text]
  • Artificial Intelligence Spring 2019 Homework 2: Adversarial Search
    Artificial Intelligence Spring 2019 Homework 2: Adversarial Search PROGRAMMING In this assignment, you will create an adversarial search agent to play the 2048-puzzle game. A demo of ​ ​ the game is available here: gabrielecirulli.github.io/2048. ​ ​ I. 2048 As A Two-Player Game ​ II. Choosing a Search Algorithm: Expectiminimax ​ III. Using The Skeleton Code ​ IV. What You Need To Submit ​ V. Important Information ​ VI. Before You Submit ​ I. 2048 As A Two-Player Game 2048 is played on a 4×4 grid with numbered tiles which can slide up, down, left, or right. This game can ​ ​ be modeled as a two player game, in which the computer AI generates a 2- or 4-tile placed randomly on the board, and the player then selects a direction to move the tiles. Note that the tiles move until they either (1) collide with another tile, or (2) collide with the edge of the grid. If two tiles of the same number collide in a move, they merge into a single tile valued at the sum of the two originals. The resulting tile cannot merge with another tile again in the same move. Usually, each role in a two-player games has a similar set of moves to choose from, and similar objectives (e.g. chess). In 2048 however, the player roles are inherently asymmetric, as the Computer AI ​ ​ places tiles and the Player moves them. Adversarial search can still be applied! Using your previous experience with objects, states, nodes, functions, and implicit or explicit search trees, along with our skeleton code, focus on optimizing your player algorithm to solve 2048 as efficiently and consistently ​ ​ as possible.
    [Show full text]
  • Linear Programming Notes X: Integer Programming
    Linear Programming Notes X: Integer Programming 1 Introduction By now you are familiar with the standard linear programming problem. The assumption that choice variables are infinitely divisible (can be any real number) is unrealistic in many settings. When we asked how many chairs and tables should the profit-maximizing carpenter make, it did not make sense to come up with an answer like “three and one half chairs.” Maybe the carpenter is talented enough to make half a chair (using half the resources needed to make the entire chair), but probably she wouldn’t be able to sell half a chair for half the price of a whole chair. So, sometimes it makes sense to add to a problem the additional constraint that some (or all) of the variables must take on integer values. This leads to the basic formulation. Given c = (c1, . , cn), b = (b1, . , bm), A a matrix with m rows and n columns (and entry aij in row i and column j), and I a subset of {1, . , n}, find x = (x1, . , xn) max c · x subject to Ax ≤ b, x ≥ 0, xj is an integer whenever j ∈ I. (1) What is new? The set I and the constraint that xj is an integer when j ∈ I. Everything else is like a standard linear programming problem. I is the set of components of x that must take on integer values. If I is empty, then the integer programming problem is a linear programming problem. If I is not empty but does not include all of {1, . , n}, then sometimes the problem is called a mixed integer programming problem.
    [Show full text]
  • Chapter 11: Basic Linear Programming Concepts
    CHAPTER 11: BASIC LINEAR PROGRAMMING CONCEPTS Linear programming is a mathematical technique for finding optimal solutions to problems that can be expressed using linear equations and inequalities. If a real-world problem can be represented accurately by the mathematical equations of a linear program, the method will find the best solution to the problem. Of course, few complex real-world problems can be expressed perfectly in terms of a set of linear functions. Nevertheless, linear programs can provide reasonably realistic representations of many real-world problems — especially if a little creativity is applied in the mathematical formulation of the problem. The subject of modeling was briefly discussed in the context of regulation. The regulation problems you learned to solve were very simple mathematical representations of reality. This chapter continues this trek down the modeling path. As we progress, the models will become more mathematical — and more complex. The real world is always more complex than a model. Thus, as we try to represent the real world more accurately, the models we build will inevitably become more complex. You should remember the maxim discussed earlier that a model should only be as complex as is necessary in order to represent the real world problem reasonably well. Therefore, the added complexity introduced by using linear programming should be accompanied by some significant gains in our ability to represent the problem and, hence, in the quality of the solutions that can be obtained. You should ask yourself, as you learn more about linear programming, what the benefits of the technique are and whether they outweigh the additional costs.
    [Show full text]
  • Integer Linear Programming
    Introduction Linear Programming Integer Programming Integer Linear Programming Subhas C. Nandy ([email protected]) Advanced Computing and Microelectronics Unit Indian Statistical Institute Kolkata 700108, India. Introduction Linear Programming Integer Programming Organization 1 Introduction 2 Linear Programming 3 Integer Programming Introduction Linear Programming Integer Programming Linear Programming A technique for optimizing a linear objective function, subject to a set of linear equality and linear inequality constraints. Mathematically, maximize c1x1 + c2x2 + ::: + cnxn Subject to: a11x1 + a12x2 + ::: + a1nxn b1 ≤ a21x1 + a22x2 + ::: + a2nxn b2 : ≤ : am1x1 + am2x2 + ::: + amnxn bm ≤ xi 0 for all i = 1; 2;:::; n. ≥ Introduction Linear Programming Integer Programming Linear Programming In matrix notation, maximize C T X Subject to: AX B X ≤0 where C is a n≥ 1 vector |- cost vector, A is a m× n matrix |- coefficient matrix, B is a m × 1 vector |- requirement vector, and X is an n× 1 vector of unknowns. × He developed it during World War II as a way to plan expenditures and returns so as to reduce costs to the army and increase losses incurred by the enemy. The method was kept secret until 1947 when George B. Dantzig published the simplex method and John von Neumann developed the theory of duality as a linear optimization solution. Dantzig's original example was to find the best assignment of 70 people to 70 jobs subject to constraints. The computing power required to test all the permutations to select the best assignment is vast. However, the theory behind linear programming drastically reduces the number of feasible solutions that must be checked for optimality.
    [Show full text]
  • Chapter 7. Linear Programming and Reductions
    Chapter 7 Linear programming and reductions Many of the problems for which we want algorithms are optimization tasks: the shortest path, the cheapest spanning tree, the longest increasing subsequence, and so on. In such cases, we seek a solution that (1) satisfies certain constraints (for instance, the path must use edges of the graph and lead from s to t, the tree must touch all nodes, the subsequence must be increasing); and (2) is the best possible, with respect to some well-defined criterion, among all solutions that satisfy these constraints. Linear programming describes a broad class of optimization tasks in which both the con- straints and the optimization criterion are linear functions. It turns out an enormous number of problems can be expressed in this way. Given the vastness of its topic, this chapter is divided into several parts, which can be read separately subject to the following dependencies. Flows and matchings Introduction to linear programming Duality Games and reductions Simplex 7.1 An introduction to linear programming In a linear programming problem we are given a set of variables, and we want to assign real values to them so as to (1) satisfy a set of linear equations and/or linear inequalities involving these variables and (2) maximize or minimize a given linear objective function. 201 202 Algorithms Figure 7.1 (a) The feasible region for a linear program. (b) Contour lines of the objective function: x1 + 6x2 = c for different values of the profit c. x x (a) 2 (b) 2 400 400 Optimum point Profit = $1900 ¡ ¡ ¡ ¡ ¡ 300 ¢¡¢¡¢¡¢¡¢ 300 ¡ ¡ ¡ ¡ ¡ ¢¡¢¡¢¡¢¡¢ ¡ ¡ ¡ ¡ ¡ ¢¡¢¡¢¡¢¡¢ 200 200 c = 1500 ¡ ¡ ¡ ¡ ¡ ¢¡¢¡¢¡¢¡¢ ¡ ¡ ¡ ¡ ¡ ¢¡¢¡¢¡¢¡¢ c = 1200 ¡ ¡ ¡ ¡ ¡ 100 ¢¡¢¡¢¡¢¡¢ 100 ¡ ¡ ¡ ¡ ¡ ¢¡¢¡¢¡¢¡¢ c = 600 ¡ ¡ ¡ ¡ ¡ ¢¡¢¡¢¡¢¡¢ x1 x1 0 100 200 300 400 0 100 200 300 400 7.1.1 Example: profit maximization A boutique chocolatier has two products: its flagship assortment of triangular chocolates, called Pyramide, and the more decadent and deluxe Pyramide Nuit.
    [Show full text]
  • Linear Programming
    Stanford University | CS261: Optimization Handout 5 Luca Trevisan January 18, 2011 Lecture 5 In which we introduce linear programming. 1 Linear Programming A linear program is an optimization problem in which we have a collection of variables, which can take real values, and we want to find an assignment of values to the variables that satisfies a given collection of linear inequalities and that maximizes or minimizes a given linear function. (The term programming in linear programming, is not used as in computer program- ming, but as in, e.g., tv programming, to mean planning.) For example, the following is a linear program. maximize x1 + x2 subject to x + 2x ≤ 1 1 2 (1) 2x1 + x2 ≤ 1 x1 ≥ 0 x2 ≥ 0 The linear function that we want to optimize (x1 + x2 in the above example) is called the objective function.A feasible solution is an assignment of values to the variables that satisfies the inequalities. The value that the objective function gives 1 to an assignment is called the cost of the assignment. For example, x1 := 3 and 1 2 x2 := 3 is a feasible solution, of cost 3 . Note that if x1; x2 are values that satisfy the inequalities, then, by summing the first two inequalities, we see that 3x1 + 3x2 ≤ 2 that is, 1 2 x + x ≤ 1 2 3 2 1 1 and so no feasible solution has cost higher than 3 , so the solution x1 := 3 , x2 := 3 is optimal. As we will see in the next lecture, this trick of summing inequalities to verify the optimality of a solution is part of the very general theory of duality of linear programming.
    [Show full text]
  • Best-First and Depth-First Minimax Search in Practice
    Best-First and Depth-First Minimax Search in Practice Aske Plaat, Erasmus University, [email protected] Jonathan Schaeffer, University of Alberta, [email protected] Wim Pijls, Erasmus University, [email protected] Arie de Bruin, Erasmus University, [email protected] Erasmus University, University of Alberta, Department of Computer Science, Department of Computing Science, Room H4-31, P.O. Box 1738, 615 General Services Building, 3000 DR Rotterdam, Edmonton, Alberta, The Netherlands Canada T6G 2H1 Abstract Most practitioners use a variant of the Alpha-Beta algorithm, a simple depth-®rst pro- cedure, for searching minimax trees. SSS*, with its best-®rst search strategy, reportedly offers the potential for more ef®cient search. However, the complex formulation of the al- gorithm and its alleged excessive memory requirements preclude its use in practice. For two decades, the search ef®ciency of ªsmartº best-®rst SSS* has cast doubt on the effectiveness of ªdumbº depth-®rst Alpha-Beta. This paper presents a simple framework for calling Alpha-Beta that allows us to create a variety of algorithms, including SSS* and DUAL*. In effect, we formulate a best-®rst algorithm using depth-®rst search. Expressed in this framework SSS* is just a special case of Alpha-Beta, solving all of the perceived drawbacks of the algorithm. In practice, Alpha-Beta variants typically evaluate less nodes than SSS*. A new instance of this framework, MTD(ƒ), out-performs SSS* and NegaScout, the Alpha-Beta variant of choice by practitioners. 1 Introduction Game playing is one of the classic problems of arti®cial intelligence.
    [Show full text]