Parallel vs. parametric complexity

Abstract

In [5] (also see [4]) we defined a natural and realistic model of parallel computation called the PRAM model without bit operations. It is like the usual PRAM model, the main difference being that no bit operations are provided. It encompasses virtually all known parallel algorithms for (weighted) combinatorial optimization and algebraic problems. In this model we proved that, for some large enough constant b, the mincost-flow for graphs with n vertices cannot be solved deterministically (or with randomization) in √n-/b (expected) time using 2√n/b processors; this is so even if we restrict every cost and capacity to be an integer (nonnegative if it is a capacity) of bit length at most an for some large enough constant a. A similar lower bound is also proved for the max flow problem. It follows that these problems cannot be solved in this model deterministically (or with randomization) in μ (N ^c) (expected) time with 2μ(N ^c) processors, where c is an appropriate positive constant and N is the total bitlength of the input. Since these problems were known to be P-complete, this provides concrete support for the belief that P-completeness implies high parallel complexity, and for the P ≠ NC conjecture itself.

These lower bounds actually follow from a general lower bound which roughly states that if the so-called parametric complexity of a problem is high then it is hard to parallelize in the PRAM model without (or with limited) bit operations. In this lecture I shall explore this relationship between parametric and parallel complexities in detail-it may open a way for investigating parallel complexities of many weighted optimization problems. Main motivation behind the lower bounds for the mincost flow and max flow problems-which are P-complete-was to prove unconditionally a weaker implication of the P ≠ NC conjecture in model that is restricted but realistic. But now one can ask if similar lower bounds can also be proved for other weighted optimization problems whose parallel complexities are open because they are neither known to be P-complete nor have fast parallel algorithms. One prime example is the minimum-weight perfect matching problem. If the edge-weights are in unary, the problem has fast parallel algorithms [2, 6]. In general, its parallel complexity is open. We conjecture that the parametric complexity of this problem is 2μ(N ^c) for some small enough positive ε. It would then follow from the general lower bound that it cannot be solved in the PRAM model without bit operations in o(N ^ε′ time using 2o(Nε′) processors, where N is the input bitlength and ε′ is a small enough positive constant. Several other problems may be investigated in this way: e.g. matroid intersection problems [3], matroid parity problems [3], construction of blocking flows [1], and several problems in computational geometry, robot motion planning, and so forth.