[Paper Review] A Tight Lower Bound for the Weights of Maximum Weight Matching in Bipartite Graphs

Шибсанкар Дас|arXiv (Cornell University)|May 2, 2016

Advanced Graph Theory Research14 references3 citations

TL;DR

This paper establishes a tight lower bound of $\lceil\frac{m - \sigma}{\sigma}\rceil + 1$ for the weight of the maximum weight bipartite matching (MWBM) in any weighted bipartite graph with total weight $m$ and $\sigma$ vertices in each partition. The bound is derived through combinatorial analysis of graph partitions and proven tight via induction and extremal graph construction, with applications in network design and string matching.

ABSTRACT

Let $\Ga$ be the collection of all weighted bipartite graphs each having $σ$ and $m$, as the size of a vertex partition and the total weight, respectively. We give a tight lower bound $\lceil \frac{m-σ}σ ceil+1$ for the set $\{ extit{Wt}( extit{mwm}(G))~|~G \in \Ga\}$ which denotes the collection of weights of maximum weight bipartite matchings of all graphs in $\Ga$.

Motivation & Objective

• To determine the minimum possible weight of a maximum weight bipartite matching across all weighted bipartite graphs with fixed total weight $m$ and $\sigma$ vertices per partition.
• To establish a tight lower bound for the weight of the MWBM in such graphs, applicable across all valid configurations of $m$ and $\sigma$.
• To demonstrate the tightness of the bound through constructive examples and inductive proof techniques.
• To enable practical applications in areas like optical networking, stringology, and network optimization by providing a theoretical foundation for minimum matching weight.

Proposed method

• Partition the set of all such graphs $\mathcal{G}_{m,\sigma}$ into two subclasses: $\mathcal{G}_{m \geq \sigma}$ and $\mathcal{G}_{m < \sigma}$ based on the relationship between total weight $m$ and partition size $\sigma$.
• For $m \geq \sigma$, express $m = q\sigma + r$ with $q \in \mathbb{N}_0$, $0 < r \leq \sigma$, and show that the minimum MWBM weight is $q + 1 = \lceil\frac{m - \sigma}{\sigma}\rceil + 1$.
• Construct extremal graphs achieving this bound by maximizing edge weights under constraints, ensuring the MWBM weight is minimized yet still feasible.
• Use induction on the total weight $m$ to prove the bound is tight, showing that any attempt to reduce the MWBM weight leads to contradiction.
• Analyze edge adjacency and matching structure to rule out configurations that would allow a lower MWBM weight than the proposed bound.
• Prove the bound holds for $m < \sigma$ by showing the minimum MWBM weight is always 1, consistent with the general formula.

Experimental results

Research questions

• RQ1 What is the minimum possible weight of a maximum weight bipartite matching across all weighted bipartite graphs with total weight $m$ and $\sigma$ vertices per partition?
• RQ2 Is the bound $\lceil\frac{m - \sigma}{\sigma}\rceil + 1$ tight, and can it be achieved by actual graphs in $\mathcal{G}_{m,\sigma}$?
• RQ3 How does the structure of the graph—particularly edge distribution and adjacency—affect the minimum achievable MWBM weight?
• RQ4 Can the bound be proven uniformly across all $m$ and $\sigma$, including the case $m < \sigma$?
• RQ5 What structural properties of the graph prevent the MWBM weight from being lower than the proposed bound?

Key findings

• The tight lower bound for the weight of the maximum weight bipartite matching in any graph $G \in \mathcal{G}_{m,\sigma}$ is $\lceil\frac{m - \sigma}{\sigma}\rceil + 1$.
• This bound is achievable, as demonstrated by explicit construction of graphs in $\mathcal{G}_{m \geq \sigma}$ where the MWBM weight equals $q + 1$ when $m = q\sigma + r$.
• For $m < \sigma$, the minimum MWBM weight is exactly 1, which matches the general formula $\lceil\frac{m - \sigma}{\sigma}\rceil + 1 = 1$.
• The bound is proven tight via strong induction on $m$, showing that no graph in $\mathcal{G}_{m,\sigma}$ can have an MWBM weight less than the stated value.
• The proof relies on structural analysis of matchings and edge adjacency, ruling out configurations that would allow a lower MWBM weight.
• The result holds uniformly across all positive integers $m$ and $\sigma$, with the bound being optimal and unimprovable.