Refutation of Spectral Graph Theory Conjectures with Monte Carlo Search

Three conjectures in extremal spectral graph theory

We prove three conjectures regarding the maximization of spectral invariants over certain families of graphs. Our most difficult result is that the join of P2 and Pn−2 is the unique graph of maximum spectral radius over all planar graphs. This was conjectured by Boots and Royle in 1991 and independently by Cao and Vince in 1993. Similarly, we prove a conjecture of Cvetković and Rowlinson from 1...

Philosophical conjectures and their refutation.

Sir Karl Popper is well known for explicating science in falsificationist terms, for which his degree of corroboration formalism, C(h,e,b), has become little more than a symbol. For example, de Queiroz and Poe in this issue argue that C(h,e,b) reduces to a single relative (conditional) probability, p(e,hb), the likelihood of evidence e, given both hypothesis h and background knowledge b, and in...

Beautiful conjectures in graph theory

Nested Monte-Carlo Search

Many problems have a huge state space and no good heuristic to order moves so as to guide the search toward the best positions. Random games can be used to score positions and evaluate their interest. Random games can also be improved using random games to choose a move to try at each step of a game. Nested Monte-Carlo Search addresses the problem of guiding the search toward better states when...

Improved Monte-Carlo Search

Monte-Carlo search has been successful in many non-deterministic games, and recently in deterministic games with high branching factor. One of the drawbacks of the current approaches is that even if the iterative process would last for a very long time, the selected move does not necessarily converge to a game-theoretic optimal one. In this paper we introduce a new algorithm, UCT, which extends...