Shetty and Sohan Shetty. Even though I have not managed to keep in regular contact with all of you because of the geographical distance, I really miss you all. I thank my family for their support over the years and I am forever indebt of my mother's unconditional love. Finally, Heli thank you for your existence in my life.

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In this paper we study a family of pattern-detection problems in vertex-colored temporal graphs. In particular, given a vertex-colored temporal graph and a multi-set of colors as a query, we search for temporal paths in the graph that contain the colors specified in the query. These types of problems have several interesting applications, for example, recommending tours for tourists, or searching for abnormal behavior in a network of financial transactions.For the family of pattern-detection problems we define, we establish complexity results and design an algebraic-algorithmic framework based on constrained multilinear sieving. We demonstrate that our solution can scale to massive graphs with up to hundred million edges, despite the problems being NP-hard. Our implementation, which is publicly available, exhibits practical edge-linear scalability and highly optimized. For example, in a real-world graph dataset with more than six million edges and a multi-set query with ten colors, we can extract an optimal solution in less than eight minutes on a haswell desktop with four cores.

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We introduce a novel problem for diversity-aware clustering. We assume that the potential cluster centers belong to a set of groups defined by protected attributes, such as ethnicity, gender, etc. We then ask to find a minimum-cost clustering of the data into k clusters so that a specified minimum number of cluster centers are chosen from each group. We thus require that all groups are represented in the clustering solution as cluster centers, according to specified requirements. More precisely, we are given a set of clients C, a set of facilities F , a collection F = {F1, . . . , Ft} of facility groups Fi ⊆ F , budget k, and a set of lower-bound thresholds R = {r1, . . . , rt}, one for each group in F . The diversity-aware k-median problem asks to find a set S of k facilities in F such that |S ∩ Fi| ≥ ri, that is, at least ri centers in S are from group Fi, and the k-median cost c∈C mins∈S d(c, s) is minimized. We show that in the general case where the facility groups may overlap, the diversityaware k-median problem is NP-hard, fixed-parameter intractable, and inapproximable to any multiplicative factor. On the other hand, when the facility groups are disjoint, approximation algorithms can be obtained by reduction to the matroid median and red-blue median problems. Experimentally, we evaluate our approximation methods for the tractable cases, and present a relaxation-based heuristic for the theoretically intractable case, which can provide high-quality and efficient solutions for real-world datasets.

We study a family of temporal reachability problems under waiting-time restrictions. In particular, given a temporal graph and a set of source vertices, we find the set of vertices that are reachable from a source via a time-respecting path, and such that the difference in timestamps between consecutive edges is at most a resting time. This kind of problems have several interesting applications in understanding the spread of a disease in a network, tracing contacts in epidemic outbreaks, and finding signaling pathways in the brain network.We present an algebraic algorithm based on constrained multilinear sieving for solving the restless reachability problems we propose. With an open-source implementation we demonstrate that the algorithm can scale to large temporal graphs with tens of millions of edges, despite the problem being NP-hard. The implementation is efficiently engineered and highly optimized. For instance, we can solve the restless reachability problem by restricting the path length to 9 in a real-world graph dataset with over 36 million directed edges in less than one hour on a 4-core Haswell desktop.

We study a variant of the classical k-median problem known as diversity-aware k-median (introduced by Thejaswi et al. 2021), where we are given a collection of facility subsets, and a solution must contain at least a specified number of facilities from each subset. We investigate the fixed-parameter tractability of this problem and show several negative hardness and inapproximability results, even when we afford exponential running time with respect to some parameters of the problem.Motivated by these results we present a fixed parameter approximation algorithm with approximation ratio (1 + 2 e + ), and argue that this ratio is essentially tight assuming the gap-exponential time hypothesis. We also present a simple, practical local-search algorithm that gives a bicriteria (2k, 3 + ) approximation with better running time bounds.