On the complexity of trial and error

Abstract

Motivated by certain applications from physics, biochemistry, economics, and computer science in which the objects under investigation are unknown or not directly accessible because of various limitations, we propose a trial-and-error model to examine search problems in which inputs are unknown. More specifically, we consider constraint satisfaction problems i Ci, where the constraints Ci are hidden, and the goal is to find a solution satisfying all constraints. We can adaptively propose a candidate solution (i.e., trial), and there is a verification oracle that either confirms that it is a valid solution, or returns the index i of a violated constraint (i.e., error), with the exact content of Ci still hidden.We studied the time and trial complexities of a number of natural CSPs, summarized as follows. On one hand, despite the seemingly very little information provided by the oracle, efficient algorithms do exist for Nash, Core, Stable Matching, and SAT problems, whose unknown-input versions are shown to be as hard as the corresponding known-input versions up to a factor of polynomial. The techniques employed vary considerably, including, e.g., order theory and the ellipsoid method with a strong separation oracle.On the other hand, there are problems whose complexities are substantially increased in the unknown-input model. In particular, no time-efficient algorithms exist for Graph Isomorphism and Group Isomorphism (unless PH collapses or P = NP). The proofs use quite nonstandard reductions, in which an efficient simulator is carefully designed to simulate a desirable but computationally unaffordable oracle.Our model investigates the value of input information, and our results demonstrate that the lack of input information can introduce various levels of extra difficulty. The model accommodates a wide range of combinatorial and algebraic structures, and exhibits intimate connections with (and hopefully can also serve as a useful supplement to) certain existing learning and complexity theories.

On the complexity of trial and error for constraint satisfaction problems

“…In [BCZ13], Bei, Chen and Zhang proposed a trial and error model to study algorithmic problems when some input information is lacking. As argued in their paper, the lack of input information can happen when we have only limited knowledge of, and access to the problem.…”

“…As mentioned, in [BCZ13] the authors focused on the hidden versions of some specific constraint satisfaction problems (H-CSPs), whose instances could only be accessed via a revealing oracle. An algorithm in this setting interacts with this revealing oracle to get information about the input instance.…”

“…In this paper, following [BCZ13], we initiate a systematic study of the constraint satisfaction problems in the trial and error model. To achieve this, we first adopt a formal framework for CSPs, and based on this framework we define three types of revealing oracles.…”

“…To achieve this, we first adopt a formal framework for CSPs, and based on this framework we define three types of revealing oracles. This framework also helps to clarify and enrich the model of [BCZ13]. Let us make a quick remark that, our CSP model has a couple of features that may not be quite standard.…”

“…This in principle transfers the study of hidden CSPs to the study of CSPs in the normal setting. We also apply transfer theorems to get results for concrete CSPs, including some problems considered in [BCZ13], for which we usually get much shorter and easier proofs.…”