An Algebraic Model for Combinatorial Problems

• Authors: Richard E. Stearns;Harry B. Hunt, III
• Venue: SIAM Journal on Computing
• Year: 1996

A new algebraic model, called the generalized satisfiability problem (GSP) model, is introduced for representing and solving combinatorial problems. The GSP model is an alternative to the common method in the literature of representing such problems as language-recognition problems. In the GSP model, a problem instance is represented by a set of variables together with a set of terms, and the computational objective is to find a certain sum of products of terms over a commutative semiring. The model is general enough to express all the standard problems about sets of clauses and generalized clauses, all nonserial optimization problems, and all $\{0,1\}$-linear programming problems. The model can also describe many graph problems, often in a very direct structure-preserving way. Two important properties of the model are the following: \begin{enumerate} \item In the GSP model, one can naturally discuss the structure of individual problem instances. The structure of a GSP instance is displayed in a "structure tree." The smaller the "weighted depth" or "channelwidth" of the structure tree for a GSP instance, the faster the instance can be solved by any one of several generic algorithms. \item The GSP model extends easily so as to apply to hierarchically specified problems and enables solutions to instances of such problems to be found directly from the specification rather than from the (often exponentially) larger specified object.