Graph algorithms are central to large-scale applications such as navigation systems, social networks, and data analysis platforms. This thesis studies two important challenges in such systems: robustness to failures and fairness in clustering outcomes. In the first part, we investigate fault-tolerant reachability preservers in directed graphs. We present the first non-trivial constructions of dual fault-tolerant pairwise reachability preservers that remain resilient to two edge or vertex failures, achieving a sparse construction of size $O(n^{4/3}|\mathcal{P}|^{1/3})$. In the second part, we study fair clustering algorithms that ensure balanced representation of protected groups. We develop approximation algorithms for fair consensus clustering and introduce the framework of closest fair clustering, establishing hardness results and efficient algorithms for multi-group settings. Building on this framework, we obtain improved guarantees for fair correlation clustering and design the first streaming algorithm for fair consensus clustering using only logarithmic memory. Together, these results contribute toward the design of graph algorithms that are both robust and socially responsible.

翻译：图算法对于导航系统、社交网络和数据分析平台等大规模应用至关重要。本论文研究了此类系统中的两个重要挑战：对故障的鲁棒性以及聚类结果的公平性。在第一部分中，我们探讨了有向图中的容错可达性保持器。我们提出了首个针对双故障成对可达性保持器的非平凡构造，该构造能抵抗两条边或两个顶点的故障，实现了规模为 $O(n^{4/3}|\mathcal{P}|^{1/3})$ 的稀疏结构。在第二部分中，我们研究了确保受保护群体均衡代表性的公平聚类算法。我们为公平共识聚类开发了近似算法，并引入了最邻近公平聚类框架，建立了多组设置下的难解性结果和高效算法。基于此框架，我们获得了公平相关聚类的改进保证，并设计了首个仅使用对数级内存的公平共识聚类流式算法。这些成果共同促进了既具鲁棒性又具社会责任的图算法设计。