Minimal winning coalitions and orders of criticality

In this paper, we analyze the order of criticality in simple games, under the light of minimal winning coalitions. The order of criticality of a player in a simple game is based on the minimal number of other players that have to leave so that the player in question becomes pivotal. We show that this definition can be formulated referring to the cardinality of the minimal blocking coalitions or minimal hitting sets for the family of minimal winning coalitions; moreover, the blocking coalitions are related to the winning coalitions of the dual game. Finally, we propose to rank all the players lexicographically accounting the number of coalitions for which they are critical of each order, and we characterize this ranking using four independent axioms.

Sequence-specific minimizers via polar sets

Minimizers are efficient methods to sample k-mers from genomic sequences that unconditionally preserve sufficiently long matches between sequences. Well-established methods to construct efficient minimizers focus on sampling fewer k-mers on a random sequence and use universal hitting sets (sets of k-mers that appear frequently enough) to upper bound the sketch size. In contrast, the problem of sequence-specific minimizers, which is to construct efficient minimizers to sample fewer k-mers on a specific sequence such as the reference genome, is less studied. Currently, the theoretical understanding of this problem is lacking, and existing methods do not specialize well to sketch specific sequences. We propose the concept of polar sets, complementary to the existing idea of universal hitting sets. Polar sets are k-mer sets that are spread out enough on the reference, and provably specialize well to specific sequences. Link energy measures how well spread out a polar set is, and with it, the sketch size can be bounded from above and below in a theoretically sound way. This allows for direct optimization of sketch size. We propose efficient heuristics to construct polar sets, and via experiments on the human reference genome, show their practical superiority in designing efficient sequence-specific minimizers. A reference implementation and code for analyses under an open-source license are at https://github.com/kingsford-group/polarset.

On the Practical Performance of Minimal Hitting Set Algorithms from a Diagnostic Perspective

Minimal hitting sets (MHSs) meliorate our reasoning in many applications, including AI planning, CNF/DNF conversion, and program debugging. When following Reiter’s ”theory of diagnosis from first principles”, minimal hitting sets are also essential to the diagnosis problem, since diagnoses can be characterized as the minimal hitting sets of conflicts in thebehavior of a faulty system. While the large amount of application options led to the advent of a variety of corresponding MHS algorithms, for diagnostic purposes we still lack a comparative evaluation assessing performance characteristics. In this paper, we thus empirically evaluate a set of complete algorithms relevant for diagnostic purposes in synthetic andreal-world scenarios. We consider in our experimental evaluation also how cardinality constraints on the solution space, as often established in practice for diagnostic purposes, influence performance in terms of run-time and memory usage.