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Systems of Boolean equations of low degree arise in a natural way when analyzing block ciphers. The cipher's round functions relate the secret key to auxiliary variables that are introduced by each successive round. In algebraic…
Many of the recent remarkable advances in computer vision and language models can be attributed to the success of transfer learning via the pre-training of large foundation models. However, a theoretical framework which explains this…
We consider a general Kaluza-Klein reduction of a truncated Lovelock theory. We find necessary geometric conditions for the reduction to be consistent. The resulting lower-dimensional theory is a higher derivative scalar-tensor theory,…
The purpose of this work is to develop a version of Forman's discrete Morse theory for simplicial complexes, based on internal strong collapses. Classical discrete Morse theory can be viewed as a generalization of Whitehead's collapses,…
Continuing results from JCDCGGG 2016 and 2017, we solve several new cases of the simple foldability problem -- deciding which crease patterns can be folded flat by a sequence of (some model of) simple folds. We give new efficient algorithms…
The goal of this paper is to prove that several variants of deciding whether a poset can be (weakly) embedded into a small Boolean lattice, or to a few consecutive levels of a Boolean lattice, are NP-complete, answering a question of Griggs…
A fundamental pursuit in complexity theory concerns reducing worst-case problems to average-case problems. There exist complexity classes such as PSPACE that admit worst-case to average-case reductions. However, for many other classes such…
Given a rooted tree and a ranking of its leaves, what is the minimum number of inversions of the leaves that can be attained by ordering the tree? This variation of the problem of counting inversions in arrays originated in mathematical…
We explore the intricate interdependent relationship among counting problems, considered from three frameworks for such problems: Holant Problems, counting CSP and weighted H-colorings. We consider these problems for general complex valued…
We propose developing the theory of consequences of morasses relevant in mathematical applications in the language alternative to the usual one, replacing commonly used structures by families of sets originating with Velleman's neat…
Is reduction always a good scientific strategy? Does it always lead to a gain in information? The very existence of the special sciences above and beyond physics seems to hint no. Previous research has shown that dimension reduction…
The minimization problem for propositional formulas is an important optimization problem in the second level of the polynomial hierarchy. In general, the problem is Sigma-2-complete under Turing reductions, but restricted versions are…
Like termination, confluence is a central property of rewrite systems. Unlike for termination, however, there exists no known complexity hierarchy for confluence. In this paper we investigate whether the decreasing diagrams technique can be…
Jaeger, Vertigan, and Welsh [15] proved a dichotomy for the complexity of evaluating the Tutte polynomial at fixed points: The evaluation is #P-hard almost everywhere, and the remaining points admit polynomial-time algorithms. Dell,…
We show that the recent hierarchy of semidefinite programming relaxations based on non-commutative polynomial optimization and reduced density matrix variational methods exhibits an interesting paradox when applied to the bosonic case: even…
We show $\textsf{EOPL}=\textsf{PLS}\cap\textsf{PPAD}$. Here the class $\textsf{EOPL}$ consists of all total search problems that reduce to the End-of-Potential-Line problem, which was introduced in the works by Hubacek and Yogev (SICOMP…
Tightness is a generalisation of the notion of convexity: a space is tight if and only if it is "as convex as possible", given its topological constraints. For a simplicial complex, deciding tightness has a straightforward exponential time…
We show how to hang a picture by wrapping rope around n nails, making a polynomial number of twists, such that the picture falls whenever any k out of the n nails get removed, and the picture remains hanging when fewer than k nails get…
Ceballos and Pons introduced the $s$-weak order on $s$-decreasing trees, for any weak composition $s$. They proved that it has a lattice structure and further conjectured that it can be realized as the $1$-skeleton of a polyhedral…
Scientists form hypotheses and experimentally test them. If a hypothesis fails (is refuted), scientists try to explain the failure to eliminate other hypotheses. The more precise the failure analysis the more hypotheses can be eliminated.…