In the paper, we study the Maximum Satisfiability and the Partial Maximum Satisfiability problems. Using Gallai–Edmonds decomposition, we significantly improve the upper bound for the Maximum Satisfiability problem parameterized above maximum matching in the variable-clause graph. Our algorithm operates with a runtime of O*(2.83^k'), a substantial improvement compared to the previous approach requiring O*(4^k' ), where k' denotes the relevant parameter. Moreover, this result immediately implies O*(1.14977^m) and O*(1.27895^m) time algorithms for the (n, 3)-MaxSAT and (n, 4)-MaxSAT where m is the overall number of clauses. These upper bounds improve prior-known upper bounds equal to O*(1.1554^m) and O*(1.2872^m). We also adapt the algorithm so that it can handle instances of Partial Maximum Satisfiability without losing performance in some cases. Note that this is somewhat surprising, as the existence of even one hard clause can significantly increase the hardness of a problem.

Computing homotopy groups of spheres has long been a fundamental objective in algebraic topology. Various theoretical and algorithmic approaches have been developed to tackle this problem. In this paper we take a step towards the goal of comprehending the group-theoretic structure of the generators of these homotopy groups by leveraging the power of machine learning. Specifically, in the simplicial group setting of Wu's formula, we reformulate the problem of generating simplicial cycles as a problem of sampling from the intersection of algorithmic datasets related to Dyck languages. We present and evaluate language modelling approaches that employ multi-label information for input sequences, along with the necessary group-theoretic toolkit and non-neural baselines.

Heterogeneity, e.g., due to different types of layers or multiple sub-models, poses key challenges in analyzing the generalization behavior of several modern architectures. For instance, descriptors based on Persistent Homology (PH) are being increasingly integrated into Graph Neural Networks (GNNs) to augment them with rich topological features; however, the generalization of such PH schemes remains unexplored. We introduce a novel _compositional_ PAC-Bayes framework that provides a general recipe to analyze a broad spectrum of models including those with heterogeneous layers. Specifically, we provide the first data-dependent generalization bounds for a widely adopted PH vectorization scheme (that subsumes persistence landscapes, images, and silhouettes) as well as PH-augmented GNNs. Using our framework, we also obtain bounds for GNNs and neural nets with ease. Our bounds also inform the design of novel regularizers. Empirical evaluations on several standard real-world datasets demonstrate that our theoretical bounds highly correlate with empirical generalization performance, leading to improved classifier design via our regularizers. Overall, this work bridges a crucial gap in the theoretical understanding of PH methods and general heterogeneous models, paving the way for the design of better models for (graph) representation learning. Our code is available at https://github.com/Aalto-QuML/Compositional-PAC-Bayes.

The Maximum Satisfiability (MAXSAT) problem is an optimization version of the Satisfiability problem (SAT) in which one is given a CNF formula with n variables and needs to find the maximum number of simultaneously satisfiable clauses. Recent works achieved significant progress in proving new upper bounds on the worst-case computational complexity of MAXSAT. All these works reduce general MAXSAT to a special case of MAXSAT where each variable appears a small number of times. So, it is important to design fast algorithms for (n,k)-MAXSAT to construct an efficient exact algorithm for MAXSAT. (n,k)-MAXSAT is a special case of MAXSAT where each variable appears at most k times in the input formula. For the (n,3)-MAXSAT problem, we design a O*(1.1749^n) algorithm improving on the previous record running time of O*(1.191^n). For the (n,4)-MAXSAT problem, we construct a O*(1.3803^n) algorithm improving on the previous best running time of O*(1.4254^n). Using the results, we develop a O*(1.0911^L) algorithm for the MAXSAT where L is a length of the input formula which improves previous algorithm with O*(1.0927^L) running time.