Information Processing Letters 26 (1987/88) 223-230 North-Holland 11 January 1988 A POLYNOMIAL CHARACTERIZATION OF SOME GRAPH PARTITIONING PROBLEMS Claudio ARBIB * Dipartimento di Informatica e Sistemistica, Universita’ degli Studi “La Sapienza”‘, Via Eudossiana 18, I-00184 Roma, Italy Communicated by L. Boasson Received March 1987 Revised June 1987 The Uniform Partition (UP) and the Simple Max Partition (SMP) problems arc NP-complete graph partitioning problems, and polynomial-time algorithms are known for few classes of graphs only. In the present paper, the class of line-graphs is considered and a polynomial algorithm is proposed to solve both problems in this class. When the instance space is extended to digraphs, a characterization is possible which leads to similar results. Keywork Combinatorial problem, computational complexity, graph partitioning, line-graph zyxwvutsrqponmlkjihgfedcbaZYXWV Introduction The Uniform Partition and the Simple Max Partition of a graph (from now on, UP and SMP for short) are unweighed and simplified versions of the general Graph Partitioning problem. In the general formulation, a graph, which can be weighed either on the set of edges or on that of vertices, is to be k-partitioned (i.e., its set of vertices must be divided in k subsets) without violating a set of capacity constraints. The typical objective function is defined as the total weight of the edges removed. Graph Partitioning models are often involved in Network Design (see, ,Sor example, [5,6]). An example is given by the Task Assignment Prob- lem: we are given a multiprocessor system with m parallel processors linked by a single channel. A set of n tasks T is to be partitioned into k subsets Ti, and each subset assigned to a single processor i. Let G = (V, E), with ]V 1 = fz, be a computa- tional weighed graph representing interrelation among processes: any edge between bwo vertices * Present affiliation: Department of Electronic Engineering, Universrty of Rome, Via Orazio Raimondo, 00173 Rome, Italy. (processes) is weighed by the amount of informa- tion required for communication. Assume the m processors to be identical, as- sume tj to be the time required for the completion of process j, and B as an upper bound to the completion time of all processes assigned to the same processor (i.e., a real-time requirement): the solution of the Graph Partitioning problem associ- ated with G clearly yields an assignment of tasks to processors whieh minimizes the channel load and matches the real-time requirement. The most intuitive formulation of Graph Parti- tioning in terms of Binary Programming leads to a nonlinear objective function, whereas other linear formulations are in general rather expensive as for the number of O-1 variables involved and/or the number of constraints. In UP we have k = 2 and no weight either on vertices or on edges: the objective function is the number of edges removed, while the two subsets of vertices are required to have the same cardinal- ity. This constraint is even relaxed in the Simple Max Partition problem: nevertheless, both prob- lems turn out to be NP-complete. As for the solution methods, stochastic search can effectively be applied to solve UP: a local search-based heuristic for UP, proposed in the 0020-0190/88/$3.50 0 1988, Elsevier Science Publishers B.V. (North-Holland) 223