Bounded-Support Additive Latin Transversals via Color-Counted Matching

2026-07-13

AI Summary

This work addresses the additive Latin transversal problem: given a multiset \( A \) and an equinumerous subset \( B \), how to order the elements of \( B \) such that the sums \( a_i + b_i \) are pairwise distinct. The authors propose a matching algorithm incorporating color counting, reducing the problem to finding a matching in a graph with constraints on the number of edges of specific colors. Through a \((q+1)\)-ary reduction, this is further transformed into an exact red-edge matching problem, which is then solved efficiently using the Mulmuley–Vazirani–Vazirani algorithm. The resulting randomized algorithm runs in time \((k + \log m)^{O(s)}\), where \( s \) denotes the support size of \( A \). This establishes, for the first time, that the problem admits a constructive randomized polynomial-time solution when \( s \) is fixed, significantly outperforming black-box approaches based on integer programming.

Abstract

We consider the following additive Latin transversal problem. Given a multiset $A=(a_1,\dots,a_k)$ of elements of $\mathbb Z_m$ and a set $B\subseteq\mathbb Z_m$ of cardinality $k$, the task is to order $B$ as $b_1,\dots,b_k$ so that the sums $a_i+b_i$ are pairwise distinct. When $k=m$, Hall proved that a solution exists if and only if $\sum_{i=1}^m a_i\equiv 0 \pmod m$; moreover, his theorem yields a polynomial-time construction. Alon proved that a solution always exists when $m$ is prime and $k<m$, but no polynomial-time construction is known in general. Our main algorithmic contribution is a direct randomized algorithm for Color-Counted Matching: given an edge-colored graph and prescribed target counts for the colors, find a matching using exactly the prescribed number of edges of each color. If $q$ is the sum of the target counts and $h$ is the number of colors, our base-$(q+1)$ reduction to Exact Red Matching, combined with the algorithm of Mulmuley-Vazirani-Vazirani, gives a randomized algorithm with running time $\left(|V|^2+|E|(q+1)^{h-1}\right)^{O(1)} $ for an input graph $(V,E)$. Thus the dependence on the target matching size is $q^{O(h)}$, up to polynomial factors in the graph size. In contrast, applying the general matching-ILP theorem of Lassota and Ligthart as a black box yields a $q^{O(h^2)}$ dependence for the corresponding fixed-size color-counted instances. Applying this primitive to additive Latin transversals with $s=|\operatorname{supp}(A)|$, we obtain an algorithm in randomized time $(k+\log m)^{O(s)}$. In particular, additive Latin transversals are randomized polynomial-time constructible for every fixed support size.

Problem

• additive Latin transversal
• bounded support
• color-counted matching
• distinct sums
• combinatorial design

Innovation

• Color-Counted Matching
• Additive Latin Transversals
• Randomized Algorithm
• Exact Red Matching
• Bounded Support