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Abstract The Survivable Network Design problem (SNDP) is a well-studied problem, (partly) motivated by the design of networks that are robust to faults under the assumption that any subset of edges up to a specific number can fail. We considernon-uniformfault models where the subset of edges that fail can be specified in different ways. We consider three models: the flexible graph connectivity model (Adjiashvili, D.: Fault-tolerant shortest paths - beyond the uniform failure model (unpublished).) (Adjiashvili, D. et al. in Math. Program.192(1–2), 409–441 (2022)) (Boyd, S. et al. in Math. Program. pp. 1–24 (2023)) which was our initial motivation, the bulk-robust model (Adjiashvili, D. in Algorithms and Techniques (APPROX/RANDOM 2015), Leibniz International Proceedings in Informatics (LIPIcs), vol. 40, pp. 61–77. Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl, Germany (2015)) (Adjiashvili, D. et al,: in Math. Program.149(1–2), 361–390 (2015)) and the relative survivable network design model (Dinitz, M. et al.: in Algorithms and Techniques (APPROX/RANDOM 2022), Leibniz International Proceedings in Informatics (LIPIcs),245, pp. 41:1–41:19. Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl, Germany (2022)) (Dinitz, M. et al.: in Algorithms, pp. 190–204. Springer Nature Switzerland, Cham (2023)). While SNDP admits a 2-approximation (Jain, K.: in Combinatorica21(1), 39–60 (2001)), the approximability of problems in these more complex models is much less understood even in special cases. We make two contributions. Our first set of results are in the flexible graph connectivity model. Motivated by a conjecture that a constant factor approximation is feasible when the robustness parameters are fixed constants, we consider two important special cases, namely the single pair case and the global connectivity case. For both these, we obtain constant factor approximations in several parameter ranges of interest. These are based on an augmentation framework and via decomposing the families of cuts that need to be covered into a small number of uncrossable families. Our second set of results are poly-logarithmic approximations for the bulk-robust model (Adjiashvili, D. in Algorithms and Techniques (APPROX/RANDOM 2015), Leibniz International Proceedings in Informatics (LIPIcs), vol. 40, pp. 61–77. Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl, Germany (2015)) when the “width” of the given instance (the maximum number of edges that can fail in any particular scenario) is fixed. Via this, we derive corresponding approximations for the flexible graph connectivity model and the relative survivable network design model. The results are obtained via two algorithmic approaches and they have different tradeoffs in terms of the approximation ratio and generality. 

We consider the following question: given a submodular/supermodular set function f : 2V → R, how should one minimize/maximize its average value f (S)/|S|over non-empty subsets S ⊆ V ? This problem generalizes several well-known objectives, including Densest Subgraph (DSG), Densest Supermodular Set (DSS), and Submodular Function Minimization (SFM). Motivated by recent applications [42, 34], we formalize two new broad problems: the Unrestricted Sparsest Submodular Set (USSS) and Unrestricted Densest Supermodular Set (UDSS), both of which allow negative and non-monotone functions. Using classical results, we show that DSS, SFM, USSS, UDSS, and Minimum Norm Point (MNP) are all equivalent under strongly polynomial-time reductions. This equivalence enables algorithmic cross-over: methods designed for one problem can be repurposed to solve others efficiently. In particular, we use the perspective of the minimum norm point in the base polyhedron of a sub/supermodular function, which, via Fujishige’s results, yields the dense decomposition as a byproduct. Through this perspective, we show that a recent converging heuristic for DSS, SUPERGREEDY++ [17, 32], and Wolfe’s minimum norm point algorithm are both universal solvers for all of these problems. On the theoretical front, we explain the observation made in recent work [42, 34] that SUPERGREEDY++ appears to work well even in settings beyond DSS. Surprisingly, we also show that this simple algorithm can be used for Submodular Function Minimization, including acting as a practical minimum s-t cut algorithm. On the empirical front, we explore the utility of several algorithms for recent problems. We conduct over 400 experiments across seven problem types and largescale synthetic and real-world datasets (up to ≈ 100 million edges). Our results reveal that methods historically considered inefficient, such as convex-programming methods, flow-based solvers, and Fujishige-Wolfe’s algorithm, outperform stateof-the-art task-specific baselines by orders of magnitude on concrete problems like HNSN [42]. These findings challenge prevailing assumptions and demonstrate that with the proper framing, general optimization algorithms can be both scalable and state-of-the-art for supermodular and submodular ratio problems. 

We consider deletion problems in graphs and supermodular functions where the goal is to reduce density. In Graph Density Deletion (GraphDD), we are given a graph G = (V,E) with non-negative vertex costs and a non-negative parameter ρ ≥ 0 and the goal is to remove a minimum cost subset S of vertices such that the densest subgraph in G-S has density at most ρ. This problem has an underlying matroidal structure and generalizes several classical problems such as vertex cover, feedback vertex set, and pseudoforest deletion set for appropriately chosen ρ ≤ 1 and all of these classical problems admit a 2-approximation. In sharp contrast, we prove that for every fixed integer ρ > 1, GraphDD is hard to approximate to within a logarithmic factor via a reduction from SetCover, thus showing a phase transition phenomenon. Next, we investigate a generalization of GraphDD to monotone supermodular functions, termed Supermodular Density Deletion (SupmodDD). In SupmodDD, we are given a monotone supermodular function f:2^V → ℤ_{≥0} via an evaluation oracle with element costs and a non-negative integer ρ ≥ 0 and the goal is remove a minimum cost subset S ⊆ V such that the densest subset according to f in V-S has density at most ρ. We show that SupmodDD is approximation equivalent to the well-known Submodular Cover problem; this implies a tight logarithmic approximation and hardness for SupmodDD; it also implies a logarithmic approximation for GraphDD, thus matching our inapproximability bound. Motivated by these hardness results, we design bicriteria approximation algorithms for both GraphDD and SupmodDD. 

We consider the problem of covering multiple submodular constraints. Given a finite ground set N, a cost function c: N → ℝ_+, r monotone submodular functions f_1,f_2,…,f_r over N and requirements b_1,b_2,…,b_r the goal is to find a minimum cost subset S ⊆ N such that f_i(S) ≥ b_i for 1 ≤ i ≤ r. When r = 1 this is the well-known Submodular Set Cover problem. Previous work [Chekuri et al., 2022] considered the setting when r is large and developed bi-criteria approximation algorithms, and approximation algorithms for the important special case when each f_i is a weighted coverage function. These are fairly general models and capture several concrete and interesting problems as special cases. The approximation ratios for these problem are at least Ω(log r) which is unavoidable when r is part of the input. In this paper, motivated by some recent applications, we consider the problem when r is a fixed constant and obtain two main results. When the f_i are weighted coverage functions from a deletion-closed set system we obtain a (1+ε)(e/(e-1))(1+β)-approximation where β is the approximation ratio for the underlying set cover instances via the natural LP. Second, for covering multiple submodular constraints we obtain a randomized bi-criteria approximation algorithm that for any given integer α ≥ 1 outputs a set S such that f_i(S) ≥ (1-1/e^α-ε)b_i for each i ∈ [r] and 𝔼[c(S)] ≤ (1+ε)α ⋅ OPT. These results show that one can obtain nearly as good an approximation for any fixed r as what one would achieve for r = 1. We also demonstrate applications of our results to implicit covering problems such as fair facility location. 

Finding the maximum number of disjoint spanning trees in a given graph is a well-studied problem with several applications and connections. The Tutte-Nash-Williams theorem provides a min-max relation for this problem which also extends to disjoint bases in a matroid and leads to efficient algorithms [Schrijver, 2003]. Several other packing problems such as element disjoint Steiner trees, disjoint set covers, and disjoint dominating sets are NP-Hard but admit an O(log n)-approximation [Feige et al., 2002; Cheriyan and Salavatipour, 2007]. Călinescu, Chekuri, and Vondrák [G. Călinescu et al., 2009] viewed all these packing problems as packing bases of a polymatroid and provided a unified perspective. Motivated by applications in wireless networks, recent works have studied the problem of packing set covers in the online model [Pananjady et al., 2015; Emek et al., 2019; Bienkowski et al., 2025]. The online model poses new challenges for packing problems. In particular, it is not clear how to pack a maximum number of disjoint spanning trees in a graph when edges arrive online. Motivated by these applications and theoretical considerations, we formulate an online model for packing bases of a polymatroid, and describe a randomized algorithm with a polylogarithmic competitive ratio. Our algorithm is based on interesting connections to the notion of quotients of a polymatroid that has recently seen applications in polymatroid sparsification [Quanrud, 2024]. We generalize the previously known result for the online disjoint set cover problem [Emek et al., 2019] and also address several other packing problems in a unified fashion. For the special case of packing disjoint spanning trees in a graph (or a hypergraph) whose edges arrive online, we provide an alternative to our general algorithm that is simpler and faster while achieving the same poly-logarithmic competitive ratio. 

We consider the Survivable Network Design problem (SNDP) in the single-pass insertion-only streaming model. The input to SNDP is an edge-weighted graph G = (V, E) and an integer connectivity requirement r(uv) for each u, v ∈ V. The objective is to find a minimum-weight subgraph H ⊆ G such that, for every pair of vertices u, v ∈ V, u and v are r(uv)-edge/vertex-connected. Recent work by [Ce Jin et al., 2024] obtained approximation algorithms for edge-connectivity augmentation, and via that, also derived algorithms for edge-connectivity SNDP (EC-SNDP). In this work we consider vertex-connectivity setting (VC-SNDP) and obtain several results for it as well as improved results for EC-SNDP. - We provide a general framework for solving connectivity problems including SNDP and others in streaming; this is based on a connection to fault-tolerant spanners. For VC-SNDP we provide an O(tk)-approximation in Õ(k^{1-1/t}n^{1 + 1/t}) space, where k is the maximum connectivity requirement, assuming an exact algorithm at the end of the stream. Using a refined LP-based analysis, we provide an O(β t)-approximation where β is the integrality gap of the natural cut-based LP relaxation. These are the first approximation algorithms in the streaming model for VC-SNDP. When applied to the EC-SNDP, our framework provides an O(t)-approximation in Õ(k^{1/2-1/(2t)}n^{1 + 1/t} + kn) space, improving the O(t log k)-approximation of [Ce Jin et al., 2024] using Õ(kn^{1+1/t}) space; this also extends to element-connectivity SNDP. - We consider vertex connectivity-augmentation in the link-arrival model. The input is a k-vertex-connected spanning subgraph G, and additional weighted links L arrive in the stream; the goal is to store the min-weight set of links such that G ∪ L is (k+1)-vertex-connected. We obtain constant-factor approximations in near-linear space for k = 1, 2. Our result for k = 2 is based on using the SPQR tree, a novel application for this well-known representation of 2-connected graphs.