Zero-sum problems with congruence conditions

For a finite abelian group $G$ and a positive integer $d$, let $\mathsf s_{d \mathbb N} (G)$ denote the smallest integer $\ell \in \mathbb N_0$ such that every sequence $S$ over $G$ of length $|S| \ge \ell$ has a nonempty zero-sum subsequence $T$ of length $|T| \equiv 0 \mod d$. We determine $\mathsf s_{d \mathbb N} (G)$ for all $d\geq 1$ when $G$ has rank at most two and, under mild conditions on $d$, also obtain precise values in the case of $p$-groups. In the same spirit, we obtain new upper bounds for the Erd{\H o}s--Ginzburg--Ziv constant provided that, for the $p$-subgroups $G_p$ of $G$, the Davenport constant $\mathsf D (G_p)$ is bounded above by $2 \exp (G_p)-1$. This generalizes former results for groups of rank two

On the existence of zero-sum subsequences of distinct lengths

In this paper, we obtain a characterization of short normal sequences over a finite Abelian p-group, thus answering positively a conjecture of Gao for a variety of such groups. Our main result is deduced from a theorem of Alon, Friedland and Kalai, originally proved so as to study the existence of regular subgraphs in almost regular graphs. In the special case of elementary p-groups, Gao's conjecture is solved using Alon's Combinatorial Nullstellensatz. To conclude, we show that, assuming every integer satisfies Property B, this conjecture holds in the case of finite Abelian groups of rank two.Comment: 10 pages, to appear in Rocky Mountain Journal of Mathematic

Are There Incongruent Ground States in 2D Edwards-Anderson Spin Glasses?

We present a detailed proof of a previously announced result (C.M. Newman and D.L. Stein, Phys. Rev. Lett. v. 84, pp. 3966--3969 (2000)) supporting the absence of multiple (incongruent) ground state pairs for 2D Edwards-Anderson spin glasses (with zero external field and, e.g., Gaussian couplings): if two ground state pairs (chosen from metastates with, e.g., periodic boundary conditions) on the infinite square lattice are distinct, then the dual bonds where they differ form a single doubly-infinite, positive-density domain wall. It is an open problem to prove that such a situation cannot occur (or else to show --- much less likely in our opinion --- that it indeed does happen) in these models. Our proof involves an analysis of how (infinite-volume) ground states change as (finitely many) couplings vary, which leads us to a notion of zero-temperature excitation metastates, that may be of independent interest.Comment: 18 pages (LaTeX); 1 figure; minor revisions; to appear in Commun. Math. Phy

False-Name Manipulation in Weighted Voting Games is Hard for Probabilistic Polynomial Time

False-name manipulation refers to the question of whether a player in a weighted voting game can increase her power by splitting into several players and distributing her weight among these false identities. Analogously to this splitting problem, the beneficial merging problem asks whether a coalition of players can increase their power in a weighted voting game by merging their weights. Aziz et al. [ABEP11] analyze the problem of whether merging or splitting players in weighted voting games is beneficial in terms of the Shapley-Shubik and the normalized Banzhaf index, and so do Rey and Rothe [RR10] for the probabilistic Banzhaf index. All these results provide merely NP-hardness lower bounds for these problems, leaving the question about their exact complexity open. For the Shapley--Shubik and the probabilistic Banzhaf index, we raise these lower bounds to hardness for PP, "probabilistic polynomial time", and provide matching upper bounds for beneficial merging and, whenever the number of false identities is fixed, also for beneficial splitting, thus resolving previous conjectures in the affirmative. It follows from our results that beneficial merging and splitting for these two power indices cannot be solved in NP, unless the polynomial hierarchy collapses, which is considered highly unlikely