One of the biggest open problems in mathematics may be solved within the next decade, according to a poll of computer scientists. A solution to the so-called P versus NP problem is worth $1 million and could have a profound effect on computing, and perhaps even the entire world. The problem is a question about how long algorithms take to run and whether some hard mathematical problems are actually easy to solve. P and NP both represent groups of mathematical problems, but it isn't known if these groups are actually identical.
In recent years, there has been much interest in phase transitions of combinatorial problems. Phase transitions have been successfully used to analyze combinatorial optimization problems, characterize their typical-case features and locate the hardest problem instances. In this paper, we study phase transitions of the asymmetric Traveling Salesman Problem (ATSP), an NP-hard combinatorial optimization problem that has many real-world applications. Using random instances of up to 1,500 cities in which intercity distances are uniformly distributed, we empirically show that many properties of the problem, including the optimal tour cost and backbone size, experience sharp transitions as the precision of intercity distances increases across a critical value. Our experimental results on the costs of the ATSP tours and assignment problem agree with the theoretical result that the asymptotic cost of assignment problem is pi 2 /6 the number of cities goes to infinity. In addition, we show that the average computational cost of the well-known branch-and-bound subtour elimination algorithm for the problem also exhibits a thrashing behavior, transitioning from easy to difficult as the distance precision increases. These results answer positively an open question regarding the existence of phase transitions in the ATSP, and provide guidance on how difficult ATSP problem instances should be generated.
Matos, Tiago (Universidade de Sao Paulo) | Bergamo, Yannick P. (Universidade de Sao Paulo) | Silva, Valdinei Freire da (Universidade de Sao Paulo) | Cozman, Fabio G. (Universidade de Sao Paulo) | Costa, Anna Helena Reali (Universidade de Sao Paulo)
Suppose an agent builds a policy that satisfactorily solves a decision problem; suppose further that some aspects of this policy are abstracted and used as starting point in a new, different decision problem. How can the agent accrue the benefits of the abstract policy in the new concrete problem? In this paper we propose a framework for simultaneous reinforcement learning, where the abstract policy helps start up the policy for the concrete problem, and both policies are refined through exploration. We report experiments that demonstrate that our framework is effective in speeding up policy construction for practical problems.
However, if it is incorrect, there must be some relationship between the two that allows them to live harmoniously within a single theory. The nature of this relationship is the focus of this article. In passing we note that the theory of weak problem solvers has been well-developed for over a decade; see Kilsson (1971) for example. Some aspects of MYCIN don't fit the problem reduction For example, a THE AI MAGAZINE Summer 1983 25 production whose action part is a conjunction of atomic formulae corresponds to a separate operator for each atomic formula in the conjunction. MYCIN's search strategy effectively applies such operators in a group.
Alvin, Chris (Louisiana State University, Baton Rouge) | Gulwani, Sumit (Microsoft Research) | Majumdar, Rupak (Max Planck Institute for Software Systems) | Mukhopadhyay, Supratik (Louisiana State University, Baton Rouge)
This paper presents a semi-automated methodology for generating geometric proof problems of the kind found in a high-school curriculum. We formalize the notion of a geometry proof problem and describe an algorithm for generating such problems over a user-provided figure. Our experimental results indicate that our problem generation algorithm can effectively generate proof problems in elementary geometry. On a corpus of 110 figures taken from popular geometry textbooks, our system generated an average of about 443 problems per figure in an average time of 4.7 seconds per figure.