Classes of Cycle Bases

In the last years, new variants of the minimum cycle basis (MCB) problem and new classes of cycle basis have been introduced, as motivated by several applications from disparate areas of scientific and technological inquiries. At present, the complexity status of the MCB problem has been settled only for undirected, directed, and strictly fundamental cycle basis

Facets of the p-cycle polytope

The purpose of this study is to provide a polyhedral analysis of the p-cycle polytope, which is the convex hull of the incidence vectors of all the p-cycles (simple directed cycles consisting of p arcs) of the complete directed graph Kn. We first determine the dimension of the p-cycle, polytope, characterize the bases of its equality set, and prove two lifting results. We then describe several classes of valid inequalities for the case 2<p<n, together with necessary and sufficient conditions for these inequalities to induce facets of the p-cycle polytope. We also briefly discuss the complexity of the associated separation problems. Finally, we investigate the relationship between the p-cycle polytope and related polytopes, including the p-circuit polytope. Since the undirected versions of symmetric inequalities which induce facets of the p-cycle polytope are facet-inducing for the p-circuit polytope, we obtain new classes of facet-inducing inequalities for the p-circuit polytope

Determinants of latin squares of a given pattern

Cycle structures of autotopisms of Latin squares determine all possible patterns of this kind of design. Moreover, given any isotopism, the number of Latin squares containing it in their autotopism group only depends on the cycle structure of this isotopism. This number has been studied in for Latin squares of order up to 7, by following the classification given in. Specifically, regarding each symbol of a Latin square as a variable, any Latin square can be seen as the vector space associated with the solution of an algebraic system of polynomial equations, which can be solved using Gröbner bases, by following the ideas implemented by Bayer to solve the problem of n-colouring a graph. However, computations for orders higher than 7 have been shown to be very difficult without using some other combinatorial tools. In this sense, we will see in this paper the possibility of studying the determinants of those Latin squares related to a given cycle structure. Specifically, since the determinant of a Latin square can be seen as a polynomial of degree n in n variables, it will determine a new polynomial equation that can be included into the previous system. Moreover, since determinants of Latin squares of order up to 7 determine their isotopic classes, we will study the set of isotopic classes of Latin squares of these orders related to each cycle structure

Minimum cycle and homology bases of surface embedded graphs

We study the problems of finding a minimum cycle basis (a minimum weight set of cycles that form a basis for the cycle space) and a minimum homology basis (a minimum weight set of cycles that generates the $1$-dimensional ($\mathbb{Z}_2$)-homology classes) of an undirected graph embedded on a surface. The problems are closely related, because the minimum cycle basis of a graph contains its minimum homology basis, and the minimum homology basis of the $1$-skeleton of any graph is exactly its minimum cycle basis. For the minimum cycle basis problem, we give a deterministic $O(n^\omega+2^{2g}n^2+m)$-time algorithm for graphs embedded on an orientable surface of genus $g$. The best known existing algorithms for surface embedded graphs are those for general graphs: an $O(m^\omega)$ time Monte Carlo algorithm and a deterministic $O(nm^2/\log n + n^2 m)$ time algorithm. For the minimum homology basis problem, we give a deterministic $O((g+b)^3 n \log n + m)$-time algorithm for graphs embedded on an orientable or non-orientable surface of genus $g$ with $b$ boundary components, assuming shortest paths are unique, improving on existing algorithms for many values of $g$ and $n$. The assumption of unique shortest paths can be avoided with high probability using randomization or deterministically by increasing the running time of the homology basis algorithm by a factor of $O(\log n)$.Comment: A preliminary version of this work was presented at the 32nd Annual International Symposium on Computational Geometr

Extensions of tautological rings and motivic structures in the cohomology of M‾g,n\overline{\mathcal{M}}_{g,n}

We study collections of subrings of $H^*(\overline{\mathcal{M}}_{g,n})$ that are closed under the tautological operations that map cohomology classes on moduli spaces of smaller dimension to those on moduli spaces of larger dimension and contain the tautological subrings. Such extensions of tautological rings are well-suited for inductive arguments and flexible enough for a wide range of applications. In particular, we confirm predictions of Chenevier and Lannes for the $\ell$-adic Galois representations and Hodge structures that appear in $H^k(\overline{\mathcal{M}}_{g,n})$ for $k = 13$, $14$, and $15$. We also show that $H^4(\overline{\mathcal{M}}_{g,n})$ is generated by tautological classes for all $g$ and $n$, confirming a prediction of Arbarello and Cornalba from the 1990s. In order to establish the final bases cases needed for the inductive proofs of our main results, we use Mukai's construction of canonically embedded pentagonal curves of genus 7 as linear sections of an orthogonal Grassmannian and a decomposition of the diagonal to show that the pure weight cohomology of $\mathcal{M}_{7,n}$ is generated by algebraic cycle classes, for $n \leq 3$.Comment: 27 pages, comments welcome

A transform of complementary aspects with applications to entropic uncertainty relations

Even though mutually unbiased bases and entropic uncertainty relations play an important role in quantum cryptographic protocols they remain ill understood. Here, we construct special sets of up to 2n+1 mutually unbiased bases (MUBs) in dimension d=2^n which have particularly beautiful symmetry properties derived from the Clifford algebra. More precisely, we show that there exists a unitary transformation that cyclically permutes such bases. This unitary can be understood as a generalization of the Fourier transform, which exchanges two MUBs, to multiple complementary aspects. We proceed to prove a lower bound for min-entropic entropic uncertainty relations for any set of MUBs, and show that symmetry plays a central role in obtaining tight bounds. For example, we obtain for the first time a tight bound for four MUBs in dimension d=4, which is attained by an eigenstate of our complementarity transform. Finally, we discuss the relation to other symmetries obtained by transformations in discrete phase space, and note that the extrema of discrete Wigner functions are directly related to min-entropic uncertainty relations for MUBs.Comment: 16 pages, 2 figures, v2: published version, clarified ref [30

The Bergman complex of a matroid and phylogenetic trees

We study the Bergman complex B(M) of a matroid M: a polyhedral complex which arises in algebraic geometry, but which we describe purely combinatorially. We prove that a natural subdivision of the Bergman complex of M is a geometric realization of the order complex of its lattice of flats. In addition, we show that the Bergman fan B'(K_n) of the graphical matroid of the complete graph K_n is homeomorphic to the space of phylogenetic trees T_n.Comment: 15 pages, 6 figures. Reorganized paper and updated references. To appear in J. Combin. Theory Ser.