Andrey Kupavskii

Grigorii Dakhno, Dmitry Gribanov, Nikita Kasianov et al.

In our work, we consider the problem of computing a vector $x \in Z^n$ of minimum $\|\cdot\|_p$-norm such that $a^\top x \not= a_0$, for any vector $(a,a_0)$ from a given subset of $Z^n$ of size $m$. In other words, we search for a vector of minimum norm that avoids a given finite set of hyperplanes, which is natural to call as the $\textit{Hyperplanes Avoiding Problem}$. This problem naturally appears as a subproblem in Barvinok-type algorithms for counting integer points in polyhedra. We show that: 1) With respect to $\|\cdot\|_1$, the problem admits a feasible solution $x$ with $\|x\|_1 \leq (m+n)/2$, and show that such solution can be constructed by a deterministic polynomial-time algorithm with $O(n \cdot m)$ operations. Moreover, this inequality is the best possible. This is a significant improvement over the previous randomized algorithm, which computes $x$ with a guaranty $\|x\|_{1} \leq n \cdot m$. The original approach of A.~Barvinok can guarantee only $\|x\|_1 = O\bigl((n \cdot m)^n\bigr)$. To prove this result, we use a newly established algorithmic variant of the Combinatorial Nullstellensatz; 2) The problem is NP-hard with respect to any norm $\|\cdot\|_p$, for $p \in \bigl(R_{\geq 1} \cup \{\infty\}\bigr)$. 3) As an application, we show that the problem to count integer points in a polytope $P = \{x \in R^n \colon A x \leq b\}$, for given $A \in Z^{m \times n}$ and $b \in Q^m$, can be solved by an algorithm with $O\bigl(ν^2 \cdot n^3 \cdot Δ^3 \bigr)$ operations, where $ν$ is the maximum size of a normal fan triangulation of $P$, and $Δ$ is the maximum value of rank-order subdeterminants of $A$. As a further application, it provides a refined complexity bound for the counting problem in polyhedra of bounded codimension. For example, in the polyhedra of the Unbounded Subset-Sum problem.

Sergei Kiselev, Andrey Kupavskii, Oleg Verbitsky et al.

We study vulnerability of a uniformly distributed random graph to an attack by an adversary who aims for a global change of the distribution while being able to make only a local change in the graph. We call a graph property $A$ anti-stochastic if the probability that a random graph $G$ satisfies $A$ is small but, with high probability, there is a small perturbation transforming $G$ into a graph satisfying $A$. While for labeled graphs such properties are easy to obtain from binary covering codes, the existence of anti-stochastic properties for unlabeled graphs is not so evident. If an admissible perturbation is either the addition or the deletion of one edge, we exhibit an anti-stochastic property that is satisfied by a random unlabeled graph of order $n$ with probability $(2+o(1))/n^2$, which is as small as possible. We also express another anti-stochastic property in terms of the degree sequence of a graph. This property has probability $(2+o(1))/(n\ln n)$, which is optimal up to factor of 2.

Andrey Kupavskii

In this short note, we show that the VC-dimension of the class of $k$-vertex polytopes in $\mathbb R^d$ is at most $8d^2k\log_2k$, answering an old question of Long and Warmuth.

Monika Csikos, Andrey Kupavskii, Nabil H. Mustafa

The VC-dimension of a set system is a way to capture its complexity and has been a key parameter studied extensively in machine learning and geometry communities. In this paper, we resolve two longstanding open problems on bounding the VC-dimension of two fundamental set systems: $k$-fold unions/intersections of half-spaces, and the simplices set system. Among other implications, it settles an open question in machine learning that was first studied in the 1989 foundational paper of Blumer, Ehrenfeucht, Haussler and Warmuth as well as by Eisenstat and Angluin and Johnson.

Andrey Kupavskii, Nikita Zhivotovskiy

In many interesting situations the size of epsilon-nets depends only on $ε$ together with different complexity measures. The aim of this paper is to give a systematic treatment of such complexity measures arising in Discrete and Computational Geometry and Statistical Learning, and to bridge the gap between the results appearing in these two fields. As a byproduct, we obtain several new upper bounds on the sizes of epsilon-nets that generalize/improve the best known general guarantees. In particular, our results work with regimes when small epsilon-nets of size $o(\frac{1}ε)$ exist, which are not usually covered by standard upper bounds. Inspired by results in Statistical Learning we also give a short proof of the Haussler's upper bound on packing numbers.

Andrey Kupavskii, Emo Welzl

Suppose we are sending out $k$ robots from $0$ to search the real line at constant speed (with turns) to find a target at an unknown location; $f$ of the robots are faulty, meaning that they fail to report the target although visiting its location (called crash type). The goal is to find the target in time at most $λ|d|$, if the target is located at $d$, $|d| \ge 1$, for $λ$ as small as possible. We show that this cannot be achieved for $$λ< 2\frac{ρ^ρ}{(ρ-1)^{ρ-1}}+1,~~ ρ:= \frac{2(f+1)}{k}~, $$ which is tight due to earlier work (see J. Czyzowitz, E. Kranakis, D. Krizanc, L. Narayanan, J. Opatrny, PODC'16, where this problem was introduced). This also gives some better than previously known lower bounds for so-called Byzantine-type faulty robots that may actually wrongly report a target. In the second part of the paper, we deal with the $m$-rays generalization of the problem, where the hidden target is to be detected on $m$ rays all emanating at the same point. Using a generalization of our methods, along with a useful relaxation of the original problem, we establish a tight lower for this setting as well (as above, with $ρ:= m(f+1)/k$). When specialized to the case $f=0$, this resolves the question on parallel search on $m$ rays, posed by three groups of scientists some 15 to 30 years ago: by Baeza-Yates, Culberson, and Rawlins; by Kao, Ma, Sipser, and Yin; and by Bernstein, Finkelstein, and Zilberstein. The $m$-rays generalization is known to have connections to other, seemingly unrelated, problems, including hybrid algorithms for on-line problems, and so-called contract algorithms.