Fast algorithms for N-dimensional restrictions of hard problems

Abstract

Let M be a parallel RAM with p processors and arithmetic operations addition and subtraction recognizing L ⊂ Nn in T steps. (Inputs for M are given integer by integer, not bit by bit.) Then L can be recognized by a (sequential) linear search algorithm (LSA) in O(n4(log(n) + T + log(p))) steps. Thus many n-dimensional restrictions of NP-complete problems (binary programming, traveling salesman problem, etc.) and even that of the uniquely optimum traveling salesman problem, which is ΔP2-complete, can be solved in polynomial time by an LSA. This result generalizes the construction of a polynomial LSA for the n-dimensional restriction of the knapsack problem previously shown by the author, and destroys the hope of proving nonpolynomial lower bounds on LSAs for any problem that can be recognized by a PRAM as above with 2poly(n) processors in poly(n) time. © 1988, ACM. All rights reserved.