Winner Robustness via Swap- and Shift-Bribery: Parameterized Counting Complexity and Experiments

We study the parameterized complexity of counting variants of Swap- and Shift-Bribery, focusing on the parameterizations by the number of swaps and the number of voters. Facing several computational hardness results, using sampling we show experimentally that Swap-Bribery offers a new approach to the robustness analysis of elections.

Counting Complexity for Reasoning in Abstract Argumentation

In this paper, we consider counting and projected model counting of extensions in abstract argumentation for various semantics. When asking for projected counts we are interested in counting the number of extensions of a given argumentation framework while multiple extensions that are identical when restricted to the projected arguments count as only one projected extension. We establish classical complexity results and parameterized complexity results when the problems are parameterized by treewidth of the undirected argumentation graph. To obtain upper bounds for counting projected extensions, we introduce novel algorithms that exploit small treewidth of the undirected argumentation graph of the input instance by dynamic programming (DP). Our algorithms run in time double or triple exponential in the treewidth depending on the considered semantics. Finally, we take the exponential time hypothesis (ETH) into account and establish lower bounds of bounded treewidth algorithms for counting extensions and projected extension.

THE THOMPSON–HIGMAN MONOIDS Mk,i: THE ${\mathcal J}$-ORDER, THE ${\mathcal D}$-RELATION, AND THEIR COMPLEXITY

The Thompson–Higman groups Gk,i have a natural generalization to monoids, called Mk,i, and inverse monoids, called Invk,i. We study some structural features of Mk,i and Invk,i and investigate the computational complexity of related decision problems. The main interest of these monoids is their close connection with circuits and circuit complexity. The maximal subgroups of Mk,1 and Invk,1 are isomorphic to the groups Gk,j (1 ≤ j ≤ k - 1); so we rediscover all the Thompson–Higman groups within Mk,1. Deciding the Green relations [Formula: see text] and [Formula: see text] of Mk,1, when the inputs are words over a finite generating set of Mk,1, is in P. When a circuit-like generating set is used for Mk,1 then deciding [Formula: see text] is coDP-complete (where DP is the complexity class consisting of differences of sets in NP). The multiplier search problem for [Formula: see text] is xNPsearch-complete, whereas the multiplier search problems of [Formula: see text] and [Formula: see text] are not in xNPsearch unless NP = coNP. The class of search problems xNPsearch is introduced as a slight generalization of NPsearch. Deciding [Formula: see text] for Mk,1 when the inputs are words over a circuit-like generating set, is ⊕k-1• NP -complete; for any h ≥ 2, ⊕h•NP is a modular counting complexity class, whose verification problems are in NP. Related problems for partial circuits are the image size problem (which is # • NP-complete), and the image size modulo h problem (which is ⊕h•NP-complete). For Invk,1 over a circuit-like generating set, deciding [Formula: see text] is ⊕k-1P-complete. It is interesting that the little known complexity classes coDP and ⊕k-1•NP play a central role in Mk,1.