Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2311.00992

Computing random $r$-orthogonal Latin squares

Sergey Bereg
View original source ↗Read source document ↗Export metadata ↓

Abstract

Two Latin squares of order $n$ are $r$-orthogonal if, when superimposed, there are exactly $r$ distinct ordered pairs. The spectrum of all values of $r$ for Latin squares of order $n$ is known. A Latin square $A$ of order $n$ is $r$-self-orthogonal if $A$ and its transpose are $r$-orthogonal. The spectrum of all values of $r$ is known for all orders $n\ne 14$. We develop randomized algorithms for computing pairs of $r$-orthogonal Latin squares of order $n$ and algorithms for computing $r$-self-orthogonal Latin squares of order $n$.

Explore related subjects

cs.DMmath.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Sergey Bereg. 2024-02-14. Computing random $r$-orthogonal Latin squares. https://arxiv.org/abs/2311.00992

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Oblivious Self-Distance Symmetric Rendezvous on the Integer Line

Symmetric rendezvous on the line is a search problem in which two agents, initially placed at distance $2d$, must follow the same randomized strategy to meet as quickly as possible. In the standard model, agents may condition their actions on the entire execution history, and both the known- and unknown-distance variants admit expected rendezvous time $Θ(d)$. We study the role of memory by introducing oblivious self-distance strategies, in which an agent's decision depends only on her position relative to her own starting location. For an initial separation of $2d$, let $R_d$ denote the optimal oblivious expected rendezvous time in the known-distance setting. We develop two finite-state frameworks based on absorbing Markov chains. Truncated chains give computable upper bounds through finite-support strategies, while weak-peek chains give lower bounds through a revealed-information relaxation. Together, they provide a mechanism for certifying optimality. Using that mechanism, we determine $R_1$ exactly and prove that it is attained by a finite-support strategy. For $d=2,\ldots,6$, numerical optimization gives the same truncation structure and objective values, yielding rigorous upper bounds below $7.83d^2$. We do not prove that the computed weak-peek minimizers are global, but the stability of the computations leads us to conjecture that they are, in which case the corresponding truncated strategies are optimal. We also prove that $R_d=Θ(d^2)$. In the unknown-distance setting, we construct a universal strategy, independent of $d$, with expected rendezvous time $O(d^{2+η})$ for every fixed $η>0$. Thus, under the memory restriction, the known-distance rendezvous time becomes quadratic, while near-quadratic performance remains possible even without knowing $d$. The asymptotic analysis uses birth-death Markov chains and their electrical-network interpretation.

cs.DM↗
arXiv2026

Tournaments not inducible by five voters

A tournament T is k-inducible if there are k linear orders on its vertex set such that, for every arc $i \to j$ of T, a majority of the orders rank i above j. For odd k, let N(k) be the least order at which some tournament is not k-inducible. Only N(3) = 8 is known exactly; for N(5) the best bounds were $12 \le N(5) \le 38$, from our previous paper [2], which also gave the first explicit example of moderate order, the Paley tournament $P_{43}$. Results. A bespoke search algorithm improves both ends: $13 \le N(5) \le 23$. The upper bound comes from proving that $P_{23}$ is not 5-inducible, the case Bachmeier et al. [1] reported they could not decide, their SAT solver not having terminated within a cumulative six weeks; ours takes 22 hours on one laptop. The lower bound comes from an analysis at order 12. We also show that $P_{31}$ is not 5-inducible, while $P_{19}$ is 5-inducible but not with unit margin, that is, not by a profile in which every arc is carried by exactly three voters against two. Both $P_{19}$ and $P_{23}$ are arc-critical for their respective properties, whereas $P_{31}$ and $P_{43}$ are not vertex-critical: deleting a vertex leaves a tournament that is still not 5-inducible. Method. The search places one vertex at a time, always choosing the vertex with the fewest options left, and propagates the consequences. Together with the automorphisms of the tournament, this decides on a single laptop instances that neither integer programming nor a general-purpose SAT solver can settle. The refutations for $P_{19}$ and $P_{23}$ are certified as well: the search is split into independent subproblems, a SAT solver emits a machine-checkable proof for each, and a separate program rechecks every proof. All results, subject to two human-checked lemmas, are reproducible from https://github.com/Leonardini/TournamentsBeyond5Voters.

cs.DM↗
arXiv2026

Two-Machine Flow Shop with a Fixed Non-Availability Interval on the Second Machine

This paper investigates a two-machine permutation flow shop in which the second machine is unavailable during one fixed interval $[s,t]$. We consider the non-resumable setting: an operation interrupted by the interval must restart from the beginning after the machine becomes available. The objective is to minimize the makespan. We establish three results. First, we give a polynomial-time $10/7$-approximation algorithm. Second, we develop a pseudopolynomial-time exact dynamic program. Third, we prove that the problem does not admit a fully polynomial-time approximation scheme (FPTAS) unless $\mathrm{P}=\mathrm{NP}$, even when the non-availability interval has unit length. Together, these results characterize a distinctive complexity profile: exact optimization is possible in pseudopolynomial time, whereas the usual route from such an algorithm to an FPTAS is impossible unless $\mathrm{P}=\mathrm{NP}$. They also reveal an approximability separation from the corresponding non-resumable problem with the interval on the first machine.

cs.DM↗