Trimmed Moebius Inversion and Graphs of Bounded Degree

We study ways to expedite Yates's algorithm for computing the zeta and Moebius transforms of a function defined on the subset lattice. We develop a trimmed variant of Moebius inversion that proceeds point by point, finishing the calculation at a subset before considering its supersets. For an $n$-element universe $U$ and a family \scr F of its subsets, trimmed Moebius inversion allows us to compute the number of packings, coverings, and partitions of $U$ with $k$ sets from \scr F in time within a polynomial factor (in $n$) of the number of supersets of the members of \scr F. Relying on an intersection theorem of Chung et al. (1986) to bound the sizes of set families, we apply these ideas to well-studied combinatorial optimisation problems on graphs of maximum degree $\Delta$. In particular, we show how to compute the Domatic Number in time within a polynomial factor of (2^{\Delta+1-2)^{n/(\Delta+1) and the Chromatic Number in time within a polynomial factor of (2^{\Delta+1-\Delta-1)^{n/(\Delta+1). For any constant $\Delta$, these bounds are $O\bigl((2-\epsilon)^n\bigr)$ for $\epsilon>0$ independent of the number of vertices $n$

Constructions of q-Ary Constant-Weight Codes

This paper introduces a new combinatorial construction for q-ary constant-weight codes which yields several families of optimal codes and asymptotically optimal codes. The construction reveals intimate connection between q-ary constant-weight codes and sets of pairwise disjoint combinatorial designs of various types.Comment: 12 page

Packing and covering of crossing families of cuts

AbstractLet C be a crossing family of subsets of the finite set V (i.e., if T, U ∈ C and T ⋔ U ≠ ⊘, T ⌣ U ≠ V, then T ⋔ U ∈ C and T ⌣ U ∈ C). If D = (V, A) is a directed graph on V, then a cut induced by C is the set of arcs entering some set in C. A covering for C is a set of arcs entering each set in C, i.e., intersecting all cuts induced by C. It is shown that the following three conditions are equivalent for any given crossing family C: 1.(P1) For every directed graph D = (V, A), the minimum cardinality of a cut induced by C is equal to the maximum number of pairwise disjoint coverings for C.2.(P2) For every directed graph D = (V, A), and for every length function l: A → Z+, the minimum length of a covering for C is equal to the maximum number t of cuts C1,…, Ct induced by C (repetition allowed) such that no arc a is in more than l(a) of these cuts.3.(P3) ⊘ ∈ C, or V ∈ C, or there are no V1, V2, V3, V4, V5 in C such that V1 ⊆ V2 ⋔ V3, V2 ⌣ V3 = V, V3 ⌣ V4 ⊆ V5, V3 ⋔ V4 = ⊘.Directed graphs are allowed to have parallel arcs, so that (P1) is equivalent to its capacity version. (P1) and (P2) assert that certain hypergraphs, as well as their blockers, have the “Z+-max-flow min-cut property”. The equivalence of (P1), (P2), and (P3) implies Menger's theorem, the König-Egerváry theorem, the König-Gupta edge-colouring theorem for bipartite graphs, Fulkerson's optimum branching theorem, Edmonds' disjoint branching theorem, and theorems of Frank, Feofiloff and Younger, and the present author

Discrete Geometry

The workshop on Discrete Geometry was attended by 53 participants, many of them young researchers. In 13 survey talks an overview of recent developments in Discrete Geometry was given. These talks were supplemented by 16 shorter talks in the afternoon, an open problem session and two special sessions. Mathematics Subject Classification (2000): 52Cxx. Abstract regular polytopes: recent developments. (Peter McMullen) Counting crossing-free configurations in the plane. (Micha Sharir) Geometry in additive combinatorics. (József Solymosi) Rigid components: geometric problems, combinatorial solutions. (Ileana Streinu) • Forbidden patterns. (János Pach) • Projected polytopes, Gale diagrams, and polyhedral surfaces. (Günter M. Ziegler) • What is known about unit cubes? (Chuanming Zong) There were 16 shorter talks in the afternoon, an open problem session chaired by Jesús De Loera, and two special sessions: on geometric transversal theory (organized by Eli Goodman) and on a new release of the geometric software Cinderella (Jürgen Richter-Gebert). On the one hand, the contributions witnessed the progress the field provided in recent years, on the other hand, they also showed how many basic (and seemingly simple) questions are still far from being resolved. The program left enough time to use the stimulating atmosphere of the Oberwolfach facilities for fruitful interaction between the participants

Covering Radius 1985-1994

We survey important developments in the theory of covering radius during the period 1985-1994. We present lower bounds, constructions and upper bounds, the linear and nonlinear cases, density and asymptotic results, normality, specific classes of codes, covering radius and dual distance, tables, and open problems

Discrete Geometry and Convexity in Honour of Imre Bárány

This special volume is contributed by the speakers of the Discrete Geometry and Convexity conference, held in Budapest, June 19–23, 2017. The aim of the conference is to celebrate the 70th birthday and the scientific achievements of professor Imre Bárány, a pioneering researcher of discrete and convex geometry, topological methods, and combinatorics. The extended abstracts presented here are written by prominent mathematicians whose work has special connections to that of professor Bárány. Topics that are covered include: discrete and combinatorial geometry, convex geometry and general convexity, topological and combinatorial methods. The research papers are presented here in two sections. After this preface and a short overview of Imre Bárány’s works, the main part consists of 20 short but very high level surveys and/or original results (at least an extended abstract of them) by the invited speakers. Then in the second part there are 13 short summaries of further contributed talks. We would like to dedicate this volume to Imre, our great teacher, inspiring colleague, and warm-hearted friend