On the Complexity of Trial and Error

TL;DR

This paper introduces a trial-and-error model for constraint satisfaction problems, analyzing how limited information impacts the complexity of solving various computational problems, with some remaining efficiently solvable and others becoming intractable.

Contribution

It provides a formal framework for understanding the impact of limited information on problem complexity and establishes bounds for several key problems within this model.

Findings

Efficient algorithms exist for Nash, Core, Stable Matching, and SAT problems despite limited information.

Unknown-input versions of these problems are as hard as their known-input counterparts, up to polynomial factors.

Certain problems like Graph Isomorphism become significantly harder, with no efficient algorithms under standard assumptions.

Abstract

Motivated by certain applications from physics, biochemistry, economics, and computer science, in which the objects under investigation are not accessible because of various limitations, we propose a trial-and-error model to examine algorithmic issues in such situations. Given a search problem with a hidden input, we are asked to find a valid solution, to find which we can propose candidate solutions (trials), and use observed violations (errors), to prepare future proposals. In accordance with our motivating applications, we consider the fairly broad class of constraint satisfaction problems, and assume that errors are signaled by a verification oracle in the format of the index of a violated constraint (with the content of the constraint still hidden). Our discoveries are summarized as follows. On one hand, despite the seemingly very little information provided by the verification oracle, efficient algorithms do exist for a number of important problems. For the Nash, Core, Stable Matching, and SAT problems, the unknown-input versions are as hard as the corresponding known-input versions, up to a factor of polynomial. We further give almost tight bounds on the latter two problems' trial complexities. On the other hand, there are problems whose complexities are substantially increased in the unknown-input model. In particular, no time-efficient algorithms exist (under standard hardness assumptions) for Graph Isomorphism and Group Isomorphism problems. The tools used to achieve these results include order theory, strong ellipsoid method, and some non-standard reductions. Our model investigates the value of information, and our results demonstrate that the lack of input information can introduce various levels of extra difficulty. The model exhibits intimate connections with (and we hope can also serve as a useful supplement to) certain existing learning and complexity theories.