Interdependence of Methods and Representations in Design of Software for Combinatorial Optimization

Practical algorithmicmethods for combinatorial optimization prob lems cannot be considered in isolation from the representations that people use when communicating these problems to computer systems Di erent represen tations de ne di erent problem classes for which distinct types of methods are appropriate Conversely di erent methods have di erent ranges of applicabil ity which have motivated a diverse variety of representations This strong interdependence of method and representation in combinatorial optimization is quite the opposite of what one nds in linear or continuous nonlinear programming where a single standard form permits communication between numerous methods and representations that have been independently developed The most evident consequence of this interdependence has been a tendency for di erent research groups to address combinatorial optimization in separate ways through di erent preferred combinations of representations and methods A related consequence has been a lack of good general purpose combinatorial optimization compter packages We address in this paper the possibility that contrasting representations and their associated methods can in fact be brought together to considerable ad vantage To this end we survey three representations popularly applied in combinatorial optimization algebraic modeling languages constraint logic pro gramming languages and netforms or network diagrams We rst describe the kinds of optimizationmethods and systems most commonly associated with these alternatives Each pair of representations is then considered to show how each has been advantageous and how its advantages have begun to in uence or ought to in uence the design of the other Our current research projects are described in conjunction with several of these comparisons Practical algorithmicmethods for combinatorial optimization problems cannot be considered in isolation from the representations that people use when communicat ing these problems to computer systems Di erent representations de ne di erent problem classes for which distinct types of methods are appropriate Conversely di erent methods have di erent ranges of applicability which have motivated a diverse variety of representations This strong interdependence of method and representation in combinatorial op timization is quite the opposite of what one nds in linear or continuous nonlinear programming where a single standard form permits communication between nu merous methods and representations that have been independently developed One consequence has been the tendency of separate research groups such as the AI and OR communities to address optimization in separate ways through di er ent preferred combinations of representations and methods Another consequence is a lack of good general purpose combinatorial optimization packages as one can see by examining the many ads for other optimization systems in a typical issue of OR MS Today Which kinds of representation for combinatorial optimization are most deserving of study In principle any su ciently rich programming environment can be re garded as a general and powerful modeling system The programming language may be entirely general C Fortran or specialized for mathematical modeling Matlab Mathematica Maple or specialized for optimization as in the case of decades old packages such as OMNI and modern C libraries such as ILOG Solver The drawbacks of programming to describe optimization models are well known however so called matrix generation programs are hard to debug to maintain and to document Our focus in this paper is on higher level representations that allow optimization models to be described non procedurally in terms familiar to human modelers The continuing success of such representations and of modeling systems based on them testi es to their value in optimization We survey in particular three representations popularly applied in combinatorial optimization algebraic modeling languages constraint logic programming languages and netforms or network diagrams We rst describe the kinds of optimization methods and systems most commonly associated with these alternatives Each pair of representations is then considered to show how each has been advantageous and how its advantages have begun to in uence or ought to in uence the design of the other Our current research projects are described in conjunction with several of these comparisons Algebraic modeling languages We take it for granted that virtually any problem of optimizing a linear function of given decision variables subject to linear equations and inequalities in the vari ables can be solved by any of several general purpose algorithms This remarkable property has permitted the development of comprehensive modeling languages for the support of linear programming The idea of a modeling language is to describe a linear program in a form that is natural for people to work with yet that can be translated by a computer system to forms that are required by optimizing algorithms Subscripted data and variables are concisely and symbolically de ned by the language as collections of components indexed over sets explicit data values are provided separately in text or or database tables The objective and constraint equations and inequalities may be speci ed through the use of algebraic expressions AIMMS AMPL GAMS LINGO MGG MPL SML through a description of the activities associated with variables AMPL again MathPro or through a description of the block structure of the constraint matrix MIMI PAM Our particular concern in this paper is with algebraic modeling languages which have exhibited the greatest potential for extension beyond linear programming Al gebraic languages are based on familiar mathematical terminology for functions and comparisons but with modi cations to use the ASCII character set and to permit unambiguous interpretation by a computer system Thus in AMPL whose design is one current focus of our research one may for example write minimize P i I P j J PT t cijtxijt