The Power of Amortized Recourse for Online Graph Problems

Abstract

In this work, we study online graph problems with monotone-sum objectives. We propose a general two-fold greedy algorithm that references yardstick algorithms to achieve $t$-competitiveness while incurring at most $\frac{w_{\text{max}}\cdot(t+1)}{\min\{1, w_\text{min}\}\cdot(t-1)}$ amortized recourse, where $w_{\text{max}}$ and $w_{\text{min}}$ are the largest value and the smallest positive value that can be assigned to an element in the sum. We further show that the general algorithm can be improved for three classical graph problems by carefully choosing the referenced algorithm and tuning its detailed behavior. For Independent Set, we refine the analysis of our general algorithm and show that $t$-competitiveness can be achieved with $\frac{t}{t-1}$ amortized recourse. For Maximum Cardinality Matching, we limit our algorithm's greed to show that $t$-competitiveness can be achieved with $\frac{(2-t^*)}{(t^*-1)(3-t^*)}+\frac{t^*-1}{3-t^*}$ amortized recourse, where $t^*$ is the largest number such that $t^*= 1 +\frac{1}{j} \leq t$ for some integer $j$. For Vertex Cover, we show that our algorithm guarantees a competitive ratio strictly smaller than $2$ for any finite instance in polynomial time while incurring at most $3.33$ amortized recourse. We beat the almost unbreakable $2$-approximation in polynomial time by using the optimal solution as the reference without computing it. We remark that this online result can be used as an offline approximation result (without violating the unique games conjecture) to partially improve upon the constructive algorithm of Monien and Speckenmeyer.