Open Questions in the Theory of Semifeasible Computation

TL;DR

This paper reviews open problems in the theory of semifeasible algorithms, discussing their background, known partial results, and highlighting key questions for future research in computational complexity.

Contribution

It provides a comprehensive overview of open questions in semifeasible computation, summarizing existing knowledge and identifying gaps for future investigation.

Findings

Identifies key open problems in semifeasible algorithms.

Summarizes partial results related to these open questions.

Highlights the importance of understanding semifeasible computation in complexity theory.

Abstract

The study of semifeasible algorithms was initiated by Selman's work a quarter of century ago [Sel79,Sel81,Sel82]. Informally put, this research stream studies the power of those sets L for which there is a deterministic (or in some cases, the function may belong to one of various nondeterministic function classes) polynomial-time function f such that when at least one of x and y belongs to L, then f(x,y) \in L \cap \{x,y\}. The intuition here is that it is saying: ``Regarding membership in L, if you put a gun to my head and forced me to bet on one of x or y as belonging to L, my money would be on f(x,y).'' In this article, we present a number of open problems from the theory of semifeasible algorithms. For each we present its background and review what partial results, if any, are known.