Improving the efficiency of AC matching and unification

This report consists of three independent parts that each study important steps in the matching and unification process for AC theories. In the first part we consider the problem of AC matching where there is just a single (variadic) AC symbol and no free function symbols in the pattern and subject. We show that even this restricted problem is NP-complete. We give some search methods and empirical results. In the second part we consider the full AC matching problem where there is no restriction on AC and free functions symbols allowed in the pattern and subject. Our approach is to build a hierarchy of bipartite graph matching problems which encodes all the possible solutions of subproblems. Certain sets of solutions to the graph problems are then used to construct simplified AC systems which are solved by a constrained search. In the final part we focus on one of the computationally intensive steps in current AC unification algorithms : the extraction of potential unifiers from a diophantine basis. We show that certain sub-problems are NP-complete and derive a new search algorithm which is shown to be at worst equivalent to the best published algorithm and which is potentially much better

Complexity of Manipulation, Bribery, and Campaign Management in Bucklin and Fallback Voting

A central theme in computational social choice is to study the extent to which voting systems computationally resist manipulative attacks seeking to influence the outcome of elections, such as manipulation (i.e., strategic voting), control, and bribery. Bucklin and fallback voting are among the voting systems with the broadest resistance (i.e., NP-hardness) to control attacks. However, only little is known about their behavior regarding manipulation and bribery attacks. We comprehensively investigate the computational resistance of Bucklin and fallback voting for many of the common manipulation and bribery scenarios; we also complement our discussion by considering several campaign management problems for Bucklin and fallback.Comment: 28 page

Lemmings is PSPACE-complete

Lemmings is a computer puzzle game developed by DMA Design and published by Psygnosis in 1991, in which the player has to guide a tribe of lemming creatures to safety through a hazardous landscape, by assigning them specific skills that modify their behavior in different ways. In this paper we study the optimization problem of saving the highest number of lemmings in a given landscape with a given number of available skills. We prove that the game is PSPACE-complete, even if there is only one lemming to save, and only Builder and Basher skills are available. We thereby settle an open problem posed by Cormode in 2004, and again by Forisek in 2010. However we also prove that, if we restrict the game to levels in which the available Builder skills are only polynomially many (and there is any number of other skills), then the game is solvable in NP. Similarly, if the available Basher, Miner, and Digger skills are polynomially many, the game is solvable in NP. Furthermore, we show that saving the maximum number of lemmings is APX-hard, even when only one type of skill is available, whatever this skill is. This contrasts with the membership in P of the decision problem restricted to levels with no "deadly areas" (such as water or traps) and only Climber and Floater skills, as previously established by Cormode.Comment: 26 pages, 11 figure