arXiv subjects

Tomáš Flídr

Publications and source records attributed to Tomáš Flídr.

4 recordsLinked to original sources

arXiv2026

Optimal Embeddings of Posets in Hypercubes

Given a finite poset $\mathcal P$, the hypercube-height, denoted by $h^*(\mathcal P)$, is defined to be the minimum $h$ such that there exists a natural number $n$ for which the subsets of $[n]$ of size at most $h$ contain an induced copy of $\mathcal P$. The hypercube-width, denoted by $w^*(\mathcal P)$, is the smallest $w$ such that the subsets of $[w]$ of size at most $h^*(\mathcal P)$ contain an induced copy of $\mathcal P$. In other words, $h^*(\mathcal P)$ asks how `low' can a poset be embedded, and $w^*(\mathcal P)$ asks for the first hypercube in which such an `optimal' embedding occurs. These notions were introduced by Bastide, Groenland, Ivan and Johnston in connection to upper bounds for the poset saturation numbers. While it is not hard to see that $h^*(\mathcal P)\leq |\mathcal P|-1$ (and this bound can be tight), the hypercube-width has proved to be much more elusive. It was shown by the authors mentioned above that $w^*(\mathcal P)\leq|\mathcal P|^2/4$, but they conjectured that in fact $w^*(\mathcal P)\leq |\mathcal P|$ for any finite poset $\mathcal P$. In this paper we prove this conjecture. The proof uses Hall's theorem for bipartite graphs as a precision tool for modifying an existing copy of our poset.

math.CO↗

arXiv2026

Node-private community estimation in stochastic block models: Tractable algorithms and lower bounds

We study the classical problem of community recovery in stochastic block models with a fixed number of communities, with a twist: We seek algorithms that are stable with respect to node-wise changes in the graph structure, formally defined as a differential privacy constraint. The algorithms we develop are based on spectral clustering, where we introduce privacy to the community recovery pipeline in the form of directly privatizing the adjacency matrix; private PCA; private convex optimization; private low-rank matrix estimation; and private approximate subspace estimation. Straightforward applications of existing private algorithms lead to a rapid increase in the privacy parameter $ε$ in order to ensure consistent estimation under node differential privacy, in contrast with the simpler setting of edge privacy. To alleviate these issues, we develop novel algorithms based on (1) sampling from an exponential mechanism with a Lipschitz extension and (2) a general framework for constructing smooth projections from the space of undirected graphs to the space of bounded-degree graphs, which can then be combined with various edge-private algorithms. Importantly, the methods we develop are all computable in polynomial-time as a function of the number of nodes in the graph. We also develop novel lower bounds on the growth rate of $ε$ required in order to achieve consistent community estimation under node privacy. On a technical note, our paper highlights the complications that arise when analyzing private algorithms under the non-standard scaling $ε\rightarrow \infty$ and proposes some solutions. We also provide a novel application of the HGR maximal correlation from information theory in the context of accuracy amplification in PAC learning, which may be of independent interest.

math.ST↗

arXiv2026

All Ordinals are Cop-Robber Ordinals

The game of cops and robbers, played on a fixed graph $G$, is a two-player game, where the cop and the robber (the players) take turns in moving to adjacent vertices. The game finishes if the cop lands on the robber's vertex. In that case we say that the cop wins. If the cop can always win, regardless of the starting positions, we say that $G$ is a cop-win graph. For a finite cop-win graph $G$ we can ask for the minimum number $n$ such that, regardless of the starting positions, the game will end in at most $n$ steps. This number is called the maximum capture time of $G$. By looking at finite paths, we see that any non-negative integer is the maximum capture time for a cop-win graph. What about infinite cop-win graphs? In this case, the notion of capture time is nicely generalised if one works with ordinals, and so the question becomes which ordinals can be the maximum capture time of a cop-win graph? These ordinals are called CR (Cop-Robber)-ordinals. In this paper we fully settle this by showing that all ordinals are CR-ordinals, answering a question of Bonato, Gordinowicz and Hahn.

math.CO↗

arXiv2025

A Cop-Win Graph with Maximum Capture Time $ω$

The game of cops and robbers is played on a fixed (finite or infinite) graph $G$. The cop chooses his starting position, then the robber chooses his. After that, they take turns and move to adjacent vertices, or stay at their current vertex, with the cop moving first. The game finishes if the cop lands on the robber's vertex. In that case we say that the cop wins, while if the robber is never caught then we say that the robber wins. The graph $G$ is called cop-win if the cop has a winning strategy. In this paper we construct an infinite cop-win graph in which, for any two given starting positions of the cop and the robber, we can name in advance a finite time in which the cop can capture the robber, but these finite times are not bounded above. This shows that this graph has maximum capture time (CR-ordinal) $ω$, disproving a conjecture of Bonato, Gordinowicz and Hahn that no such graph should exist.

math.CO↗