Quantifying Intractability and the Complexity of Heuristics

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Search and optimization problems are central to all areas of computer scienceand engineering. Finding the optimal layout for a VLSI circuit or the lowest energy configuration of a crystal are both examples of optimization problems.While such problems are believed to be intractable, requiring exponential time to solve the worst-case instances, many heuristic methods have been observed to be relatively successful on instances that arise in different applications.This project addresses questions concerning the quantitative measures of the intractability of search and optimization problems, as opposed to qualitative notionssuch as NP-completeness. The following are some of the questions addressed in this project:1. Which instances of optimization problems are the most intractable ones?2. Exactly how difficult are these problems? 3. What are good heuristic methods for solving optimization problems ? When and how well do they work? 4. Are specific non-complete problems such as factoring also intractable? 5. How much does randomness help in solving problems? 6. Are hard problems suitable for cryptographic applications?If so, what levels of security do they provide these applications? Unconditional answers to these questions first require solving the P=NP problem. However, this project will use two approaches to find the most likely answers to these questions. The first approach is to provide proofs resolving these issues underplausible complexity assumptions. The second approach is to examine restricted but powerful classes of algorithms that include the most successful heuristics for the problemsunder study. This approach will include attempts to both explain the success of such heuristics and to show limitations that can be used as a guide for the likely inherentcomplexity of the problems.