Tropical Carathéodory with Matroids

Bárány’s colorful generalization of Carathéodory’s Theorem combines geometrical and combinatorial constraints. Kalai–Meshulam (2005) and Holmsen (2016) generalized Bárány’s theorem by replacing color classes with matroid constraints. In this note, we obtain corresponding results in tropical convexity, generalizing the Tropical Colorful Carathéodory Theorem of Gaubert–Meunier (2010). Our proof is inspired by geometric arguments and is reminiscent of matroid intersection. Moreover, we show that the topological approach fails in this setting. We also discuss tropical colorful linear programming and show that it is NP-complete. We end with thoughts and questions on generalizations to polymatroids, anti-matroids as well as examples and matroid simplicial depth

Programmation linéaire colorée, équilibre de Nash et pivots

International audienceConsidérons k ensembles de points S_1,...,S_k dans Q^d. Le problème de la programmation linéaire colorée, défini par Barany et Onn (Mathematics of Operations Research, 22 (1997), 550--567), consiste à décider s'il existe un sous-ensemble T dans l'union des S_i tel que T instersecte chaque S_i au plus une fois et contient 0 dans son enveloppe convexe. Dans leur article,Barany et Onn prouvent que ce problème est NP-complet quand k=d. La complexité du cas k=d+1 est laissée en question ouverte dans ce même article. Contrairement au cas k=d ce dernier cas ne devient pas trivial quand les points sont en position générique. Nous résolvons la question en montrant que ce cas est encore NP-complet. Nous montrons également que si P=NP, alors il existe un algorithme polynomial simple calculant un équilibre de Nash dans un jeu bimatriciel à partir de tout algorithme polynomial résolvant la programmation linéaire coloré pour le cas k=d+1, utilisé comme un sous-programme. Enfin nous proposons une adaptation de l'algorithme de Barany et Onn calculant une solution T dans un cas particulier. Cette adaptation peut être interprétée comme une "Phase I" de la méthode du simplexe

The Carath\'eodory-Fej\'er Interpolation Problems and the von-Neumann Inequality

The validity of the von-Neumann inequality for commuting $n$ - tuples of $3\times 3$ matrices remains open for $n\geq 3$. We give a partial answer to this question, which is used to obtain a necessary condition for the Carath\'{e}odory-Fej\'{e}r interpolation problem on the polydisc $\mathbb D^n.$ In the special case of $n=2$ (which follows from Ando's theorem as well), this necessary condition is made explicit. An alternative approach to the Carath\'{e}odory-Fej\'{e}r interpolation problem, in the special case of $n=2,$ adapting a theorem of Kor\'{a}nyi and Puk\'{a}nzsky is given. As a consequence, a class of polynomials are isolated for which a complete solution to the Carath\'{e}odory-Fej\'{e}r interpolation problem is easily obtained. A natural generalization of the Hankel operators on the Hardy space of $H^2(\mathbb T^2)$ then becomes apparent. Many of our results remain valid for any $n\in \mathbb N,$ however, the computations are somewhat cumbersome for $n>2$ and are omitted. The inequality $\lim_{n\to \infty}C_2(n)\leq 2 K^\mathbb C_G$, where $K_G^\mathbb C$ is the complex Grothendieck constant and $$C_2(n)=\sup\big\{\|p(\boldsymbol T)\|:\|p\|_{\mathbb D^n,\infty}\leq 1, \|\boldsymbol T\|_{\infty} \leq 1 \big\}$$ is due to Varopoulos. Here the supremum is taken over all complex polynomials $p$ in $n$ variables of degree at most $2$ and commuting $n$ - tuples $\boldsymbol T:=(T_1,\ldots,T_n)$ of contractions. We show that $$\lim_{n\to \infty}C_2(n)\leq \frac{3\sqrt{3}}{4} K^\mathbb C_G$$ obtaining a slight improvement in the inequality of Varopoulos. We show that the normed linear space $\ell^1(n),$ $n>1,$ has no isometric embedding into $k\times k$ complex matrices for any $k\in \mathbb N$ and discuss several infinite dimensional operator space structures on it.Comment: This is my thesis submitted to Indian Institute of Science, Bangalore on 20th July, 201

No-Dimensional Tverberg Theorems and Algorithms

Tverberg's theorem is a classic result in discrete geometry. It states that for any integer $k \ge 2$ and any finite $d$-dimensional point set $P \subseteq \mathbb{R}^d$ of at least $(d + 1)(k - 1) + 1$ points, we can partition $P$ into $k$ subsets whose convex hulls have a non-empty intersection. The computational problem of finding such a partition lies in the complexity class $\mathrm{PPAD} \cap \mathrm{PLS}$, but no hardness results are known. Tverberg's theorem also has a colorful variant: the points in $P$ have colors, and under certain conditions, $P$ can be partitioned into colorful sets, i.e., sets in which each color appears exactly once such that the convex hulls of the sets intersect. Recently, Adiprasito, Barany, and Mustafa [SODA 2019] proved a no-dimensional version of Tverberg's theorem, in which the convex hulls of the sets in the partition may intersect in an approximate fashion, relaxing the requirement on the cardinality of $P$. The argument is constructive, but it does not result in a polynomial-time algorithm. We present an alternative proof for a no-dimensional Tverberg theorem that leads to an efficient algorithm to find the partition. More specifically, we show an deterministic algorithm that finds for any set $P \subseteq \mathbb{R}^d$ of $n$ points and any $k \in \{2, \dots, n\}$ in $O(nd \log k )$ time a partition of $P$ into $k$ subsets such that there is a ball of radius $O\left(\frac{k}{\sqrt{n}}\textrm{diam}(P)\right)$ intersecting the convex hull of each subset. A similar result holds also for the colorful version. To obtain our result, we generalize Sarkaria's tensor product constructions [Israel Journal Math., 1992] that reduces the Tverberg problem to the Colorful Caratheodory problem. By carefully choosing the vectors used in the tensor products, we implement the reduction in an efficient manner.Comment: A shorter version will appear at SoCG 202

No-Dimensional Tverberg Theorems and Algorithms

Tverberg’s theorem states that for any k≥2 and any set P⊂Rd of at least (d+1)(k−1)+1 points in d dimensions, we can partition P into k subsets whose convex hulls have a non-empty intersection. The associated search problem of finding the partition lies in the complexity class CLS=PPAD∩PLS, but no hardness results are known. In the colorful Tverberg theorem, the points in P have colors, and under certain conditions, P can be partitioned into colorful sets, in which each color appears exactly once and whose convex hulls intersect. To date, the complexity of the associated search problem is unresolved. Recently, Adiprasito, Bárány, and Mustafa (SODA 2019) gave a no-dimensional Tverberg theorem, in which the convex hulls may intersect in an approximate fashion. This relaxes the requirement on the cardinality of P. The argument is constructive, but does not result in a polynomial-time algorithm. We present a deterministic algorithm that finds for any n-point set P⊂Rd and any k∈{2,…,n} in O(nd⌈logk⌉) time a k-partition of P such that there is a ball of radius O((k/n−−√)diam(P)) that intersects the convex hull of each set. Given that this problem is not known to be solvable exactly in polynomial time, our result provides a remarkably efficient and simple new notion of approximation. Our main contribution is to generalize Sarkaria’s method (Israel Journal Math., 1992) to reduce the Tverberg problem to the colorful Carathéodory problem (in the simplified tensor product interpretation of Bárány and Onn) and to apply it algorithmically. It turns out that this not only leads to an alternative algorithmic proof of a no-dimensional Tverberg theorem, but it also generalizes to other settings such as the colorful variant of the problem