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application/pdf en text Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2025-10-19 without embargo terms The student, Shubhang Kulkarni, accepted the attached license on 2025-04-20 at 11:53. The student, Shubhang Kulkarni, submitted this Dissertation for approval on 2025-04-21 at 16:17. This Dissertation was approved for publication on 2025-04-24 at 09:38. DSpace SAF Submission Ingestion Package generated from Vireo submission #21842 on 2025-10-19 at 18:18:27 Algorithmic aspects of connectivity and density in graphs and hypergraphs Kulkarni, Shubhang M 2025-04-24 Chandrasekaran, Karthekeyan Chandrasekaran, Karthekeyan Chekuri, Chandra Har-Peled, Sariel Bérczi, Kristóf Combinatorial Optimization Graph Optimization Hypergraph Optimization Submodular Functions Polyhedral Combinatorics Approximation Algorithms Linear Programming Randomized Algorithms Connectivity Augmentation Densest Subgraph Hypergraph Splitting-Off Feedback Vertex Set eng This thesis investigates algorithmic problems in combinatorial optimization centered around modifying a given network—via deletion, augmentation, or reconfiguration—to achieve tar- get connectivity or density bounds. We study associated optimization problems on graphs, hypergraphs, and submodular functions. Our main contributions are: • Hypergraph Splitting-off. We introduce a splitting-off operation in hypergraphs and prove an analogue of Mader’s theorem: in every hypergraph, a vertex can be re- moved via splitting-off while preserving all pairwise edge-connectivities. We give a strongly polynomial-time algorithm in weighted hypergraphs, with applications including a constructive characterization of k-hyperedge-connected hypergraphs and an alternate proof of an approximate min-max relation for Steiner rooted-connected orientations. Our framework extends to symmetric skew-supermodular functions. • Hypergraph Connectivity Augmentation. We study augmentation to achieve target pairwise connectivities subject to vertex degree constraints. We give a strongly polynomial time algorithm, improving prior pseudo-polynomial results. Our method extends to generating near-uniform hypergraphs, simultaneously augmenting two hypergraphs, and to covering skew-supermodular functions. Applications include strongly polyno- mial time algorithms for node-to-area and mixed-hypergraph connectivity augmentation. • Graph Density Deletion. We study vertex deletion on graphs where the goal is to delete a minimum-cost subset of vertices so that the densest subgraph has density at most a given target ρ. When ρ ≤ 1, this problem is 2-approximable. In contrast, we show logarithmic hardness of approximation for all fixed integers ρ > 1. We also study a generalization to monotone supermodular functions, show approximation equivalence to Submodular Set Cover, and design bicriteria approximation algorithms. • Feedback Vertex Set (FVS) and Pseudoforest Deletion Set (PFDS). We undertake a polyhedral study of FVS and PFDS, special cases of graph density deletion for appropriate ρ ≤ 1. Both problems are 2-approximable, but lacked polynomial-time solvable LP relaxations with matching approximation guarantees. We establish the first such LP formulations for both problems. For PFDS, we resolve a question of Bodlaender, Ono and Otachi by exhibiting an extreme point property of an associated polytope. 2025-05 Thesis https://hdl.handle.net/2142/129413 Copyright 2025 Shubhang Kulkarni Siebel School Comp & Data Sci Computer Science University of Illinois Urbana-Champaign Ph.D. Dissertation