Deterministic online matching under short augmenting paths and restricted vertex reassignments

Abstract

We study deterministic online bipartite matching with local recourse. Online vertices arrive one by one and reveal edges to a fixed offline set. After each arrival, the algorithm maintains a matching and may update it only by an augmenting path of length $1$ or $3$ starting at the new vertex. In the budgeted model $\operatorname{OMP}_3(s,t)$, each offline vertex can be reassigned at most $s$ times and each online vertex at most $t$ times. Our main result is a deterministic algorithm, Lowest-Cost-Path ($\operatorname{LCP}$), that achieves the optimal competitive ratio for every choice of budgets. If $s=0$ or $t=0$, the best possible ratio is $1/2$. For every $s\ge 1$ and finite $t$, the optimal ratio is $$ γ_t=\frac{2\cdot 2^t-1}{3\cdot 2^t-1}. $$ This value equals $3/5$ for $t=1$ and converges to $2/3$ as $t$ increases. For $t=\infty$, the optimal ratio is exactly $2/3$. In particular, allowing a single reassignment per offline vertex already matches the best guarantee achievable with any larger offline budget. We also prove matching upper bounds for all deterministic algorithms. Our analysis is primal--dual and uses a structural description of the history graph of $\operatorname{LCP}$, which explains how exhausted reassignment budgets can block length-$3$ augmentations. We also study offline-weighted variants. In particular, we determine the exact optimal deterministic ratios for two unit-offline-budget cases: $2-\sqrt2$ for $\operatorname{WOMP}_3(1,1)$, and $(\sqrt5-1)/2$ for $\operatorname{WOMP}_3(1,\infty)$. Both are strictly below their unweighted counterparts. If weights are placed on online vertices instead, no deterministic algorithm has a positive competitive ratio.