Jiratchaphat Nanta | Search arXiv

arXiv2026

Approximation Algorithms for the $b$-Matching and List-Restricted Variants of MaxQAP

We study approximation algorithms for two natural generalizations of the Maximum Quadratic Assignment Problem (MaxQAP). In the Maximum List-Restricted Quadratic Assignment Problem, each node in one partite set may only be matched to nodes from a prescribed list. For instances on $n$ nodes where every list has size at least $n - k$, we design a randomized $O(\sqrt{n}+k)$-approximation algorithm based on the linear-programming relaxation and randomized rounding framework of Makarychev, Manokaran, and Sviridenko. In the Maximum Quadratic $b$-Matching Assignment Problem, we seek a $b$-matching that maximizes the MaxQAP objective. We refine the standard MaxQAP relaxation and combine randomized rounding over $b$ independent iterations with a polynomial-time algorithm for maximum-weight $b$-matching problem to obtain an $O(\sqrt{bn})$-approximation. When $b$ is constant and all lists have size $n - O(\sqrt{n})$, our guarantees asymptotically match the best known approximation factor for MaxQAP, yielding the first approximation algorithms for these two variants.

cs.DS↗

arXiv2025

Multipartite and Structural Results on Transparent Rectangle Visibility Graphs

We consider a graph representation in the plane, called the transparent rectangle visibility graph (TRVG), where each vertex is represented by a rectangle in the plane with sides parallel to the plane axes, in a way that any two vertices are adjacent if and only if a vertical or horizontal line can be drawn from the interior of one rectangle to the other. Expanding upon previously done work by Juntarapomdach and Kittipassorn, we show that $K_{3,3,3}$ is not a TRVG, and classify complete $k$-partite TRVGs. We also prove that the complement of $C^2_n$ is not a TRVG whenever $n \geq 15$, and that every $k$-partite TRVG with $n$ vertices has at most $2(k-1)n-k(k-1)$ edges. Furthermore, we introduce a novel representation, the intersecting transparent rectangle visibility graph (ITRVG), and show that there exists a graph that is an ITRVG but not a TRVG.

math.CO↗