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Extremal Planar Matchings of Inhomogenous Random Bipartite Graphs

Ghurumuruhan Ganesan

TL;DR

This work analyzes extremal properties of planar matchings in inhomogeneous random bipartite graphs, focusing on the maximum size under a length constraint and the minimum weight under a fixed-edge count. It develops a segmentation-based and martingale/variance framework to derive concentration, deviation, and $L^2$ convergence results for both the size and weight problems, including precise bounds and saturation phenomena. A key insight is that, under appropriate scaling, the minimum-weight problem exhibits a smooth, convex limiting behavior $\theta(\rho)$ with saturation as $\rho$ grows, which informs redundancy optimization in networked systems. Collectively, the results advance understanding of extremal planar matchings in nonuniform random settings and have potential applications in data subset selection and relay-network design where planarity constraints model non-intersecting pairings.

Abstract

In this paper we study maximum size and minimum weight planar matchings of inhomogenous random bipartite graphs. Our motivation for this study comes from efficient usage of cross edges in relay networks for overall improvement in network performance. We first consider Bernoulli planar matchings with a constraint on the edge length and obtain deviation estimates for the maximum size of a planar matching. We then equip each edge of the complete bipartite graph with a positive random weight and obtain bounds on the minimum weight of a planar matching containing a given number of edges. We also use segmentation and martingale methods to obtain~\(L^2-\)convergence of the minimum weight, appropriately scaled and centred.

Extremal Planar Matchings of Inhomogenous Random Bipartite Graphs

TL;DR

This work analyzes extremal properties of planar matchings in inhomogeneous random bipartite graphs, focusing on the maximum size under a length constraint and the minimum weight under a fixed-edge count. It develops a segmentation-based and martingale/variance framework to derive concentration, deviation, and convergence results for both the size and weight problems, including precise bounds and saturation phenomena. A key insight is that, under appropriate scaling, the minimum-weight problem exhibits a smooth, convex limiting behavior with saturation as grows, which informs redundancy optimization in networked systems. Collectively, the results advance understanding of extremal planar matchings in nonuniform random settings and have potential applications in data subset selection and relay-network design where planarity constraints model non-intersecting pairings.

Abstract

In this paper we study maximum size and minimum weight planar matchings of inhomogenous random bipartite graphs. Our motivation for this study comes from efficient usage of cross edges in relay networks for overall improvement in network performance. We first consider Bernoulli planar matchings with a constraint on the edge length and obtain deviation estimates for the maximum size of a planar matching. We then equip each edge of the complete bipartite graph with a positive random weight and obtain bounds on the minimum weight of a planar matching containing a given number of edges. We also use segmentation and martingale methods to obtain~convergence of the minimum weight, appropriately scaled and centred.

Paper Structure

This paper contains 3 sections, 3 theorems, 78 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Length Constrained Planar Matchings
  3. Minimum Weight Planar Matchings

Key Result

theorem 1

We have and

Figures (2)

  • Figure 1: Illustration of a planar matching.
  • Figure 2: Determining the range of edges that intersect the edge $e$ with end-vertices $u_i$ and $v_j.$

Theorems & Definitions (5)

  • definition 1
  • theorem 1
  • definition 2
  • theorem 2
  • theorem 3