The Power of Amortized Recourse for Online Graph Problems

In this work, we study graph problems with monotone-sum objectives. We propose a general two-fold greedy algorithm that references $\alpha$-approximation algorithms (where $\alpha \ge 1$) to achieve $(t \cdot \alpha)$-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 refine this trade-off between competitive ratio and amortized recourse for three classical graph problems. 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 Matching, we use an existing algorithm with limited 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 introduce a polynomial-time algorithm that further limits greed to show that $(2 - \frac{2}{\texttt{OPT}})$-competitiveness, where $\texttt{OPT}$ is the size of the optimal vertex cover, can be achieved with at most $\frac{10}{3}$ amortized recourse by a potential function argument. We remark that this online result can be used as an offline approximation result (without violating the unique games conjecture) to improve upon that of Monien and Speckenmeyer for graphs containing odd cycles of length no less than $2k + 3$, using an algorithm that is also constructive.