Based on the framework of disjunctive propositional logic, we first provide a syntactic representation for Scott domains. Precisely, we establish a category of consistent disjunctive sequent calculi with consequence relations, and show it is equivalent to that of Scott domains with Scott-continuous functions. Furthermore, we illustrate the approach to solving recursive domain equations by introducing some standard domain constructions, such as lifting and sums. The subsystems relation on consistent finitary disjunctive sequent calculi makes these domain constructions continuous. Solutions to recursive domain equations are given by constructing the least fixed point of a continuous function.

In the k-means problem with penalties, we are given a data set $${\cal D} \subseteq \mathbb{R}^\ell $$ of n points where each point $$j \in {\cal D}$$ is associated with a penalty cost pj and an integer k. The goal is to choose a set $${\rm{C}}S \subseteq {{\cal R}^\ell }$$ with |CS| ≤ k and a penalized subset $${{\cal D}_p} \subseteq {\cal D}$$ to minimize the sum of the total squared distance from the points in D / Dp to CS and the total penalty cost of points in Dp, namely $$\sum\nolimits_{j \in {\cal D}\backslash {{\cal D}_p}} {d^2}(j,{\rm{C}}S) + \sum\nolimits_{j \in {{\cal D}_p}} {p_j}$$. We employ the primal-dual technique to give a pseudo-polynomial time algorithm with an approximation ratio of (6.357+ε) for the k-means problem with penalties, improving the previous best approximation ratio 19.849+∊ for this problem given by Feng et al. in Proceedings of FAW (2019).

The paper works within the framework of punctual computability, which is focused on eliminating unbounded search from constructions in algebra and infinite combinatorics. We study punctual numberings, that is, uniform computations for families S of primitive recursive functions. The punctual reducibility between numberings is induced by primitive recursive functions. This approach gives rise to upper semilattices of degrees, which are called Rogers pr-semilattices. We show that any infinite, uniformly primitive recursive family S induces an infinite Rogers pr-semilattice R. We prove that the semilattice R does not have minimal elements, and every nontrivial interval inside R contains an infinite antichain. In addition, every non-greatest element from R is a part of an infinite antichain. We show that the $\Sigma_1$ -fragment of the theory Th(R) is decidable.

The concept of a synchronizing word is a very important notion in the theory of finite automata. We consider the associated decision problem to decide if a given DFA possesses a synchronizing word of length at most k, where k is the standard parameter. We show that this problem DFA-SW is equivalent to the problem Monoid Factorization introduced by Cai, Chen, Downey, and Fellows. Apart from the known $\textsf{W}[2]$ -hardness results, we show that these problems belong to $\textsf{A}[2]$ , $\textsf{W}[\textsf{P}],$ and $\textsf{WNL}$ . This indicates that DFA-SW is not complete for any of these classes, and hence, we suggest a new parameterized complexity class $\textsf{W}[\textsf{Sync}]$ as a proper home for these (and more) problems. We present quite a number of problems that belong to $\textsf{W}[\textsf{Sync}]$ or are hard or complete for this new class.

In a strategic-form game, a strategy profile is an equilibrium if no viable coalition of agents (or players) benefits (in the Pareto sense) from jointly changing their strategies. Weaker or stronger equilibrium notions can be defined by considering various restrictions on coalition formation. For instance, in a Nash equilibrium, it is assumed that viable coalitions are singletons, and in a super strong equilibrium, it is assumed that every coalition is viable. Restrictions on coalition formation can be justified by communication limitations, coordination problems, or institutional constraints. In this paper, inspired by social structures in various real-life scenarios, we introduce certain restrictions on coalition formation, and on their basis, we introduce a number of equilibrium notions. As an application, we study our equilibrium notions in resource selection games (RSGs), and we present a complete set of existence and nonexistence results for general RSGs and their important special cases.

Given a plane graph $G=(V,E)$ , a Petrie tour of G is a tour P of G that alternately turns left and right at each step. A Petrie tour partition of G is a collection ${\mathscr P}=\{P_1,\ldots,P_q\}$ of Petrie tours so that each edge of G is in exactly one tour $P_i \in {\mathscr P}$ . A Petrie tour P is called a Petrie cycle if all its vertices are distinct. A Petrie cycle partition of G is a collection ${\mathscr C}=\{C_1,\ldots,C_p\}$ of Petrie cycles so that each vertex of G is in exactly one cycle $C_i \in {\mathscr C}$ . In this paper, we study the properties of 3-regular plane graphs that have Petrie cycle partitions and 4-regular plane multi-graphs that have Petrie tour partitions. Given a 4-regular plane multi-graph $G=(V,E)$ , a 3-regularization of G is a 3-regular plane graph $G_3$ obtained from G by splitting every vertex $v\in V$ into two degree-3 vertices. G is called Petrie partitionable if it has a 3-regularization that has a Petrie cycle partition. The general version of this problem is motivated by a data compression method, tristrip, used in computer graphics. In this paper, we present a simple characterization of Petrie partitionable graphs and show that the problem of determining if G is Petrie partitionable is NP-complete.