Random-Order Enumeration for Self-Reducible NP-Problems

TL;DR

This paper introduces new algorithms for uniformly random enumeration of solutions in self-reducible NP problems, improving efficiency and delay guarantees over traditional uniform generators, with parallelization strategies for further enhancement.

Contribution

The paper develops novel random-order enumeration algorithms with polynomial delay for various classes of self-reducible NP problems, surpassing the efficiency of existing uniform generator approaches.

Findings

Polynomial delay enumeration algorithms for NP problems in different hierarchies.

A Las Vegas algorithm with expected polynomial delay for problems with FPTAS.

A parallelization method achieving near-optimal enumeration delay.

Abstract

In plenty of data analysis tasks, a basic and time-consuming process is to produce a large number of solutions and feed them into downstream processing. Various enumeration algorithms have been developed for this purpose. An enumeration algorithm produces all solutions of a problem instance without repetition. To be a statistically meaningful representation of the solution space, solutions are required to be enumerated in uniformly random order. This paper studies a set of self-reducible NP-problems in three hierarchies, where the problems are polynomially countable ($Sr_{NP}^{FP}$), admit FPTAS ($Sr_{NP}^{FPTAS}$), and admit FPRAS ($Sr_{NP}^{FPRAS}$), respectively. The trivial algorithm based on a (almost) uniform generator is in fact inefficient. We provide a new insight that the (almost) uniform generator is not the end of the story. More efficient algorithmic frameworks are proposed to enumerate solutions in uniformly random order for problems in these three hierarchies. (1) For problems in $Sr_{NP}^{FP}$, we show a random-order enumeration algorithm with polynomial delay (PDREnum); (2) For problems in $Sr_{NP}^{FPTAS}$, we show a Las Vegas random-order enumeration algorithm with expected polynomial delay (PDLVREnum); (3) For problems in $Sr_{NP}^{FPRAS}$, we devise a fully polynomial delay Atlantic City random-order enumeration algorithm with expected delay polynomial in the input size and the given error probability $\delta$ (FPACREnum), which has a probability of at least $1-\delta$ becoming a Las Vegas random-order enumeration algorithm. Finally, to further improve the efficiency of the random-order enumeration algorithms, based on the master/slave paradigm, we present a parallelization with 1.5-optimal enumeration delay and running time, along with the theoretical analysis.