Maximum bounded 3-dimensional matching is MAX SNP-complete Viggo Kann Royal Institute of Technology, Stockholm viggo@nada.kth.se Abstract We prove that maximum three dimensional matching is a MAX SNP-hard problem. If the number of occurrences of elements in triples is bounded by a constant the problem is MAX SNP-complete. As corollaries we prove that maximum covering by 3-sets and maximum covering of a graph by triangles are MAX SNP-hard. The problems are MAX SNP-complete if the number of occurrences of the elements and the degree of the nodes respectively are bounded by a constant. Keywords Approximation, combinatorial problems, computational complexity, MAX SNP-complete problems. Introduction Identifying which combinatorial problems are easy to solve and which are hard is an important and challenging task. One of the most accepted ways to prove that a problem is hard is to prove it NP-complete. If an optimisation problem is NP-complete we are almost certain that it cannot be solved optimally in polynomial time. In practice though, it is often sufficient to find an approximative solution, which is near the optimal solution, and in many cases this can be done quite fast. For example the TSP (Travelling Salesman Problem) with triangular inequality is NP-complete, but in polynomial time it can be solved approximately within a factor 3 2 , i.e. one can find a trip of length at most 3 2 times the shortest trip possible [Ch]. Another even more striking example is the bin-packing problem, which is NP-complete but can be approximated within every constant in polynomial time [KaKa]. Such a scheme for approximating within every constant is called a PTAS (Polynomial Time Approximation Scheme). In [PaYa] Papadimitriouand Yannakakis defined (syntactically) a complexity class, called MAX SNP, together with a concept of reduction, called L-reduction, which preserves approximability with constants. All problems in MAX SNP can be approximated within a constant, because MAX SNP is closed under L-reduction, but many are not known to admit a PTAS. Several problems were shown to be complete in MAX SNP, for example maximum 3-satisfiability and maximum independent set in a graph with bounded degree. All attempts to construct a PTAS for a MAX SNP-complete problem have failed and hence it seems reasonable to conjecture that no such scheme exists, in particular since if one problem had a PTAS then every problem in MAX SNP would admit a PTAS. The general feeling is that some NP-complete problems may be harder than others to approximate in polynomial time. An important task is to classify the classical NP-complete problems according to approximability. As seen above we know that some of them have a PTAS, some are in MAX SNP and some seem to be even harder to approximate, for example unbounded maximum independent set [BeSch]. InthispaperweshowthattheoptimisationversionoftheNP-complete3-dimensionalmatchingproblem is MAX SNP-hard. This is shown by a reduction from the problem of maximum bounded 3-satisfiability, which is known to be MAX SNP-complete [PaYa]. By reducing from 3-dimensional matching we also show that the optimisation versions of two related problems, covering by 3-sets and node covering by triangles, are MAX SNP-hard. 1