Section: New Results

Combinatorics and combinatorial geometry

Participant : Xavier Goaoc.

In a joint work with Jiří Matoušek, Pavel Paták, Zuzana Safernová, Martin Tancer (Charles University, Prague, Czech republic), we worked on computing simplified inclusion-exclusion formulas. Let $ℱ=\{F_1,F_2,...,F_n\}$ be a family of $n$ sets on a ground set $S$, such as a family of balls in ${ℝ}^d$. For every finite measure $\mu$ on $S$, such that the sets of $ℱ$ are measurable, the classical inclusion-exclusion formula asserts that $\mu (F_1\cup F_2\cup \cdots \cup F_n)={\sum }_{I:\varnothing \ne I\subseteq \left[n\right]}{(-1)}^{\left|I\right|+1}\mu \left({\bigcap }_{i\in I}{F}_i\right)$; that is, the measure of the union is expressed using measures of various intersections. The number of terms in this formula is exponential in $n$, and a significant amount of research, originating in applied areas, has been devoted to constructing simpler formulas for particular families $ℱ$. We provide an upper bound valid for an arbitrary $ℱ$: we show that every system $ℱ$ of $n$ sets with $m$ nonempty fields in the Venn diagram admits an inclusion-exclusion formula with $m^{O\left({log}^{2}n\right)}$ terms and with $\pm 1$ coefficients, and that such a formula can be computed in $m^{O\left({log}^{2}n\right)}$ expected time. We also construct systems of $n$ sets on $n$ points for which every valid inclusion-exclusion formula has the sum of absolute values of the coefficients at least $\Omega \left(n^{3/2}\right)$. This work was presented at the EUROCOMB conference [21] in September 2013.

In a joint work with Éric Colin de Verdière (CNRS-ENS) and Grégory Ginot (IMJ-UPMC), we worked on applications of algebraic topology to combinatorial geometry, and more precisely on extending classical results on nerve complexes. The nerve complex of a family is an abstract simplicial complex that encode its intersection patterns. Nerves are widely used in computational geometry and topology, in particular in reconstruction problems where one aims at inferring the geometry of an object from a point sample while guaranteeing that the topology is correct. Indeed, the nerve theorem ensures that the nerve of a family of geometric objects has the same “topology” (formally: homotopy type) as the union of the objects whenever they form a “good cover”, that is, when any subset of the objects has an empty or contractible intersection. We relaxed this “good cover” condition to allow for families of non-connected sets. We defined an analogue of the nerve, called the multinerve, that is suitable for general acyclic families, and we proved that this combinatorial structure enjoys an analogue of the nerve theorem. Using multinerve, we could derive a new topological Helly-type theorem for acyclic families that generalizes previous results of Amenta, Kalai and Meshulam, and Matoušek. We finally used this new Helly-type theorem to (re)prove, in a unified way, bounds on transversal Helly numbers in geometric transversal theory. This article was submitted to the journal Advances in mathematics in 2012; it was accepted in 2013 and will appear in 2014 [16] .

In a joint work with Otfried Cheong (KAIST, South Korea) and Cyril Nicaud (Univ. Marne-La-Vallée), we studied two problems of the following flavor: how large can a family of combinatorial objects defined on a finite set be if its number of distinct “projections” on any small subset is bounded? We first consider set systems, where the “projections” is the standard notion of trace, and for which we generalized Sauer's Lemma on the size of set systems with bounded VC-dimension. We then studied families of permutations, where the “projections” corresponds to the notion of containment used in the study of permutations with excluded patterns, and for which we delineated the main growth rates ensured by projection conditions. One of our motivations for considering these questions is the “geometric permutation problem” in geometric transversal theory, a question that has been open for two decades. This work was submitted to the European Journal of Combinatorics in 2012 and published in 2013 [12] .