The complete classification of five-dimensional Dirichlet-Voronoi polyhedra of translational lattices

In this paper we report on the full classification of Dirichlet-Voronoi polyhedra and Delaunay subdivisions of five-dimensional translational lattices. We obtain a complete list of $110244$ affine types (L-types) of Delaunay subdivisions and it turns out that they are all combinatorially inequivalent, giving the same number of combinatorial types of Dirichlet-Voronoi polyhedra. Using a refinement of corresponding secondary cones, we obtain $181394$ contraction types. We report on details of our computer assisted enumeration, which we verified by three independent implementations and a topological mass formula check.Comment: 16 page

Index appearance record with preorders

Transforming ω-automata into parity automata is traditionally done using appearance records. We present an efficient variant of this idea, tailored to Rabin automata, and several optimizations applicable to all appearance records. We compare the methods experimentally and show that our method produces significantly smaller automata than previous approaches

The complete classification of five-dimensional Dirichlet–Voronoi polyhedra of translational lattices

This paper reports on the full classification of Dirichlet–Voronoi polyhedra and Delaunay subdivisions of five-dimensional translational lattices. A complete list is obtained of 110 244 affine types (L-types) of Delaunay subdivisions and it turns out that they are all combinatorially inequivalent, giving the same number of combinatorial types of Dirichlet–Voronoi polyhedra. Using a refinement of corresponding secondary cones, 181 394 contraction types are obtained. The paper gives details of the computer-assisted enumeration, which was verified by three independent implementations and a topological mass formula check

Existence and Complexity of Approximate Equilibria in Weighted Congestion Games

We study the existence of approximate pure Nash equilibria ($\alpha$-PNE) in weighted atomic congestion games with polynomial cost functions of maximum degree $d$. Previously it was known that $d$-approximate equilibria always exist, while nonexistence was established only for small constants, namely for $1.153$-PNE. We improve significantly upon this gap, proving that such games in general do not have $\tilde{\Theta}(\sqrt{d})$-approximate PNE, which provides the first super-constant lower bound. Furthermore, we provide a black-box gap-introducing method of combining such nonexistence results with a specific circuit gadget, in order to derive NP-completeness of the decision version of the problem. In particular, deploying this technique we are able to show that deciding whether a weighted congestion game has an $\tilde{O}(\sqrt{d})$-PNE is NP-complete. Previous hardness results were known only for the special case of exact equilibria and arbitrary cost functions. The circuit gadget is of independent interest and it allows us to also prove hardness for a variety of problems related to the complexity of PNE in congestion games. For example, we demonstrate that the question of existence of $\alpha$-PNE in which a certain set of players plays a specific strategy profile is NP-hard for any $\alpha < 3^{d/2}$, even for unweighted congestion games. Finally, we study the existence of approximate equilibria in weighted congestion games with general (nondecreasing) costs, as a function of the number of players $n$. We show that $n$-PNE always exist, matched by an almost tight nonexistence bound of $\tilde\Theta(n)$ which we can again transform into an NP-completeness proof for the decision problem

Pathogenic Germline Variants in 10,389 Adult Cancers

We conducted the largest investigation of predisposition variants in cancer to date, discovering 853 pathogenic or likely pathogenic variants in 8% of 10,389 cases from 33 cancer types. Twenty-one genes showed single or cross-cancer associations, including novel associations of SDHA in melanoma and PALB2 in stomach adenocarcinoma. The 659 predisposition variants and 18 additional large deletions in tumor suppressors, including ATM, BRCA1, and NF1, showed low gene expression and frequent (43%) loss of heterozygosity or biallelic two-hit events. We also discovered 33 such variants in oncogenes, including missenses in MET, RET, and PTPN11 associated with high gene expression. We nominated 47 additional predisposition variants from prioritized VUSs supported by multiple evidences involving case-control frequency, loss of heterozygosity, expression effect, and co-localization with mutations and modified residues. Our integrative approach links rare predisposition variants to functional consequences, informing future guidelines of variant classification and germline genetic testing in cancer. A pan-cancer analysis identifies hundreds of predisposing germline variants

LNCS

Transforming deterministic ω -automata into deterministic parity automata is traditionally done using variants of appearance records. We present a more efficient variant of this approach, tailored to Rabin automata, and several optimizations applicable to all appearance records. We compare the methods experimentally and find out that our method produces smaller automata than previous approaches. Moreover, the experiments demonstrate the potential of our method for LTL synthesis, using LTL-to-Rabin translators. It leads to significantly smaller parity automata when compared to state-of-the-art approaches on complex formulae