The Easiest Hard Problem: Number Partitioning

Abstract

The number partitioning problem (NPP) is defined easily: Given a list a1, a2, . . . , an of positive integers, find a partition, that is, a subset A {1,..., n}, minimizing the discrepancy Number partitioning is of considerable importance, both practically and theoretically. Its practical applications range from multiprocessor scheduling and the minimization of VLSI circuit size and delay [102, 504], to public key cryptography [387], to choosing up sides in a ball game [237]. Number partitioning is also one of Garey and Johnson's six basic NP-hard problems that lie at the heart of the theory of NP-completeness [191, 388], and is in fact the only one of these problems that actually deals with numbers. Hence, it is often chosen as a base for NP-completeness proofs of other problems involving numbers, such as bin packing, multiprocessor scheduling [38], quadratic programming, and knapsack problems. The computational complexity of the NPP depends on the type of input numbers {a1, a2 , . . . , an}. Consider the case where the values of aj are positive integers bounded by a constant A. Then the discrepancy E can take on at most nA different values, so the size of the search space is O(nA) instead of O(2n) and it is straightforward to devise an algorithm that explores this reduced space in time polynomial in nA [191]. Of course, such an algorithm does not prove P = NP: a concise encoding of an instance requires O(nlog2 A) bits, and A is not bounded by any power of Iog2 A. This feature of the NPP is called pseudopoly nomiality. The NP-hardness of the NPP becomes apparent when input numbers are of a size exponentially large in n or, after division by the maximal input number, of exponentially high precision. To study typical properties of the NPP, the input numbers are often taken to be independent, identically distributed random variables. Under this probabilistic assumption, the minimal discrepancy E0 is a stochastic quantity.