The textbook algorithm for real-weighted single-source shortest paths takes \(O(mn)\) time on a graph with \(m\) edges and \(n\) vertices. The breakthrough algorithm by Fineman takes \(\tilde{O}(mn^{8/9})\) randomized time. The running time was subsequently improved to \(\tilde{O}(mn^{4/5})\) by Huang, Jin, and Quanrud.

We present randomized algorithms that compute $(1+ε)$-approximate minimum global edge and vertex cuts in weighted directed graphs in $O(\log^4(n) / ε)$ and $O(\log^5(n)/ε)$ single-commodity flows, respectively. With the almost-linear time flow algorithm of [CKL+22], this gives almost linear time approximation schemes for edge and vertex connectivity. By setting $ε$ appropriately, this also gives faster exact algorithms for small vertex connectivity. At the heart of these algorithms is a divide-and-conquer technique called "shrink-wrapping" for a certain well-conditioned rooted Steiner connectivity problem. Loosely speaking, for a root $r$ and a set of terminals, shrink-wrapping uses flow to certify the connectivity from a root $r$ to some of the terminals, and for the remaining uncertified terminals, generates an $r$-cut where the sink component both (a) contains the sink component of the minimum $(r,t)$-cut for each uncertified terminal $t$ and (b) has size proportional to the number of uncertified terminals. This yields a divide-and-conquer scheme over the terminals where we can divide the set of terminals and compute their respective minimum $r$-cuts in smaller, contracted subgraphs.

The textbook algorithm for real-weighted single-source shortest paths takes O(m n) time on a graph with m edges and n vertices. A recent breakthrough algorithm by Fineman [STOC 2024] takes Õ(m n8/9) randomized time. The running time was subsequently improved by Huang, Jin, and Quanrud [SODA 2025, 2026] to Õ(mn4/5) and then Õ(m n3/4 + m4/5 n).

The textbook algorithm for single-source shortest paths with real-valued edge weights runs in O (mn ) time on a graph with m edges and n vertices. A recent breakthrough algorithm by Fineman [11] takes Õ(mn8/9) randomized time. We present an Õ(mn4/5) randomized time algorithm building on ideas from [11].

We give improved algorithms for maintaining edge-orientations of a fully-dynamic graph, such that the maximum out-degree is bounded. On one hand, we show how to orient the edges such that maximum out- degree is proportional to the arboricity α of the graph, in, either, an amortised update time of 𝒪(log2 n log α), or a worst-case update time of 𝒪 (log3 n log α). On the other hand, motivated by applications including dynamic maximal matching, we obtain a different trade-off. Namely, the improved update time of either 𝒪 (log n log α), amortised, or 𝒪(log2 n log α), worst-case, for the problem of maintaining an edge-orientation with at most 𝒪 (α + log n) out-edges per vertex. Finally, all of our algorithms naturally limit the recourse to be polylogarithmic in n and α. Our algorithms adapt to the current arboricity of the graph, and yield improvements over previous work:

Matroids are a fundamental object of study in combinatorial optimization. Three closely related and important problems involving matroids are maximizing the size of the union of k independent sets (that is, k-fold matroid union), computing k disjoint bases (a.k.a. matroid base packing), and covering the elements by k bases (a.k.a. matroid base covering). These problems generalize naturally to integral and real-valued capacities on the elements. This work develops faster exact and/or approximation problems for these and some other closely related problems such as optimal reinforcement and matroid membership. We obtain improved running times both for general matroids in the independence oracle model and for the graphic matroid. The main thrust of our improvements comes from developing a faster and unifying push-relabel algorithm for the integer-capacitated versions of these problems, building on previous work by Frank and Miklós [24]. We then build on this algorithm in two directions. First we develop a faster augmenting path subroutine for k-fold matroid union that, when appended to an approximation version of the push-relabel algorithm, gives a faster exact algorithm for some parameters of k. In particular we obtain a subquadratic-query running time in the uncapacitated setting for the three basic problems listed above. We also obtain faster approximation algorithms for these problems with real-valued capacities by reducing to small integral capacities via randomized rounding. To this end, we develop a new randomized rounding technique for base covering problems in matroids that may also be of independent interest.

Graph sparsification has been an important topic with many structural and algorithmic consequences. Recently hypergraph sparsification has come to the fore and has seen exciting progress. In this paper we take a fresh perspective and show that they can be both be derived as corollaries of a general theorem on sparsifying matroids and monotone submodular functions. Quotients of matroids and monotone submodular functions generalize k-cuts in graphs and hypergraphs. We show that a weighted ground set of a monotone submodular function f can be sparsified while approximately preserving the weight of every quotient of f with high probability in randomized polynomial time. This theorem conceptually unifies cut sparsifiers for undirected graphs [7] with other interesting applications. One basic application is to reduce the number of elements in a matroid while preserving the weight of every quotient of the matroid. For hypergraphs, the theorem gives an alternative approach to the hypergraph cut sparsifiers obtained recently in [12], that also preserves all k-cuts. Another application is to reduce the number of points in a set system while preserving the weight of the union of every collection of sets. We also present algorithms that sparsify hypergraphs and set systems in nearly linear time, and sparsify matroids in nearly linear time and queries in the rank oracle model. * Dept. of Computer Science, Purdue University, West Lafayette, IN 47907. Supported in part by NSF grant CCF-2129816.

Maximum weight independent set (MWIS) admits a 1/k-approximation in inductively k-independent graphs [Karhan Akcoglu et al., 2002; Ye and Borodin, 2012] and a 1/(2k)-approximation in k-perfectly orientable graphs [Kammer and Tholey, 2014]. These are a parameterized class of graphs that generalize k-degenerate graphs, chordal graphs, and intersection graphs of various geometric shapes such as intervals, pseudo-disks, and several others [Ye and Borodin, 2012; Kammer and Tholey, 2014]. We consider a generalization of MWIS to a submodular objective. Given a graph G = (V,E) and a non-negative submodular function f: 2^V → ℝ_+, the goal is to approximately solve max_{S ∈ ℐ_G} f(S) where ℐ_G is the set of independent sets of G. We obtain an Ω(1/k)-approximation for this problem in the two mentioned graph classes. The first approach is via the multilinear relaxation framework and a simple contention resolution scheme, and this results in a randomized algorithm with approximation ratio at least 1/e(k+1). This approach also yields parallel (or low-adaptivity) approximations. Motivated by the goal of designing efficient and deterministic algorithms, we describe two other algorithms for inductively k-independent graphs that are inspired by work on streaming algorithms: a preemptive greedy algorithm and a primal-dual algorithm. In addition to being simpler and faster, these algorithms, in the monotone submodular case, yield the first deterministic constant factor approximations for various special cases that have been previously considered such as intersection graphs of intervals, disks and pseudo-disks.

Boob et al. [1] described an iterative peeling algorithm called Greedy++ for the Densest Subgraph Problem (DSG) and conjectured that it converges to an optimum solution. Chekuri, Quanrud, and Torres [2] extended the algorithm to general supermodular density problems (of which DSG is a special case) and proved that the resulting algorithm Super-Greedy++ (and hence also Greedy++) converges. In this paper, we revisit the convergence proof and provide a different perspective. This is done via a connection to Fujishige's quadratic program for finding a lexicographically optimal base in a (contra)polymatroid [3], and a noisy version of the Frank-Wolfe method from convex optimisation [4,5]. This gives us a simpler convergence proof, and also shows a stronger property that Super-Greedy++ converges to the optimal dense decomposition vector, answering a question raised in Harb et al. [6]. A second contribution of the paper is to understand Thorup's work on ideal tree packing and greedy tree packing [7,8] via the Frank-Wolfe algorithm applied to find a lexicographically optimum base in the graphic matroid. This yields a simpler and transparent proof. The two results appear disparate but are unified via Fujishige's result and convex optimisation.

We study the densest subgraph problem (DSG) and the densest subgraph local decomposition problem (DSG-LD) in undirected graphs. We also consider supermodular generalizations of these problems. For large scale graphs simple iterative algorithms perform much better in practice than theoretically fast algorithms based on network-flow or LP solvers. Boob et al [1] recently gave a fast iterative algorithm called Greedy++ for DSG. It was shown in [2] that it converges to a $(1-\epsilon)$ relative approximation to the optimum density in $O(\frac{1}{\epsilon^2} \frac{\Delta(G)}{\lambda^*})$ iterations where $\Delta(G)$ is the maximum degree and $\lambda^*$ is the optimum density. Danisch et al. [3] gave an iterative algorithm based on the Frank-Wolfe algorithm for DSG-LD that takes $O(\frac{m\Delta(G) }{\epsilon^2})$ iterations to converge to an $\epsilon$-additive approximate local decomposition vector $\hat{b}$, where $m$ is number of edges in the graph.In this paper we give a new iterative algorithm for both problems that takes at most $O(\frac{\sqrt{m\Delta(G)}}{\epsilon})$ iterations to converge to an $\epsilon$-additive approximate local decomposition vector; each iteration can be implemented in $O(m)$ time. We describe a fractional peeling technique which has strong empirical performance as well as theoretical guarantees. The algorithm is scalable and simple, and can be applied to graphs with hundreds of millions of edges. We test our algorithm on real and synthetic data sets and show that it provides a significant benefit over previous algorithms. The algorithm and analysis extends to hypergraphs.

The densest subgraph problem in a graph (DSG), in the simplest form, is the following. Given an undirected graph G = (V, E) find a subset S ⊆ V of vertices that maximizes the ratio |E(S)|/|S| where E(S) is the set of edges with both endpoints in S. DSG and several of its variants are well-studied in theory and practice and have many applications in data mining and network analysis. In this paper we study fast algorithms and structural aspects of DSG via the lens of supermodularity. For this we consider the densest supermodular subset problem (DSS): given a non-negative supermodular function f : 2V → ℝ+, maximize f(S)/|S|. For DSG we describe a simple flow-based algorithm that outputs a (1–∊)-approximation in deterministic Õ(m/∊) time where m is the number of edges. Our algorithm is the first to have a near-linear dependence on m and 1/∊ and improves previous methods based on an LP relaxation. It generalizes to hypergraphs, and also yields a faster algorithm for directed DSG. Greedy peeling algorithms have been very popular for DSG and several variants due to their efficiency, empirical performance, and worst-case approximation guarantees. We describe a simple peeling algorithm for DSS and analyze its approximation guarantee in a fashion that unifies several existing results. Boob et al. [12] developed an iterative peeling algorithm for DSG which appears to work very well in practice, and made a conjecture about its convergence to optimality. We affirmatively answer their conjecture, and in fact prove that a natural generalization of their algorithm converges to a (1–∊)-approximation for any supermodular function f; the key to our proof is to consider an LP formulation that is derived via the Lovász extension of a supermodular function. For DSG the bound on the number of iterations we prove is where Δ is the maximum degree and λ∗ is the optimum value. Our work suggests that iterative peeling can be an effective heuristic for several objectives considered in the literature. Finally, we show that the 2-approximation for densest-at-least-k subgraph [37] extends to the supermodular setting. We also give a unified analysis of the peeling algorithm for this problem, and via this analysis derive an approximation guarantee for a generalization of DSS to maximize f(S)/g(|S|) for a concave function g.

We present online algorithms for directed spanners and Steiner forests. These problems fall under the unifying framework of online covering linear programming formulations, developed by Buchbinder and Naor (MOR, 34, 2009), based on primal-dual techniques. Our results include the following: For the pairwise spanner problem, in which the pairs of vertices to be spanned arrive online, we present an efficient randomized $\tilde{O}(n^{4/5})$-competitive algorithm for graphs with general lengths, where $n$ is the number of vertices. With uniform lengths, we give an efficient randomized $\tilde{O}(n^{2/3+ε})$-competitive algorithm, and an efficient deterministic $\tilde{O}(k^{1/2+ε})$-competitive algorithm, where $k$ is the number of terminal pairs. These are the first online algorithms for directed spanners. In the offline setting, the current best approximation ratio with uniform lengths is $\tilde{O}(n^{3/5 + ε})$, due to Chlamtac, Dinitz, Kortsarz, and Laekhanukit (TALG 2020). For the directed Steiner forest problem with uniform costs, in which the pairs of vertices to be connected arrive online, we present an efficient randomized $\tilde{O}(n^{2/3 + ε})$-competitive algorithm. The state-of-the-art online algorithm for general costs is due to Chakrabarty, Ene, Krishnaswamy, and Panigrahi (SICOMP 2018) and is $\tilde{O}(k^{1/2 + ε})$-competitive. In the offline version, the current best approximation ratio with uniform costs is $\tilde{O}(n^{4/7 + ε})$, due to Abboud and Bodwin (SODA 2018). A small modification of the online covering framework by Buchbinder and Naor implies a polynomial-time primal-dual approach with separation oracles, which a priori might perform exponentially many calls. We convert the online spanner problem and the online Steiner forest problem into online covering problems and round in a problem-specific fashion.

We give an algorithm to find a minimum cut in an edge-weighted directed graph with$n$vertices and$m$edges in$\tilde{O}(n\cdot\max\{m^{2/3},\ n\})$time. This improves on the 30 year old bound of$\tilde{O}(nm)$obtained by Hao and Orlin for this problem. Using similar techniques, we also obtain$\tilde{O}(n^{2}/\epsilon^{2})$-time$(1+{\epsilon})$-approximation algorithms for both the minimum edge and minimum vertex cuts in directed graphs, for any fixed$\epsilon$. Before our work, no (1 +$\epsilon)$-approximation algorithm better than the exact runtime of$\tilde{O}(nm)$is known for either problem. Our algorithms follow a two-step template. In the first step, we employ a partial sparsification of the input graph to preserve a critical subset of cut values approximately. In the second step, we design algorithms to find the (edge/vertex) mincut among the preserved cuts from the first step. For edge mincut, we give a new reduction to$\tilde{O}(\min\{{n}/m^{1/3}, \sqrt{n}\}){-}$calls of any maxflow subroutine, via packing arborescences in the sparsifier. For vertex mincut, we develop new local flow algorithms to identify small unbalanced cuts in the sparsified graph.

We consider the fundamental problems of determining the rooted and global edge and vertex connectivities (and computing the corresponding cuts) in directed graphs. For rooted (and hence also global) edge connectivity with small integer capacities we give a new randomized Monte Carlo algorithm that runs in time Õ(n²). For rooted edge connectivity this is the first algorithm to improve on the Ω(n³) time bound in the dense-graph high-connectivity regime. Our result relies on a simple combination of sampling coupled with sparsification that appears new, and could lead to further tradeoffs for directed graph connectivity problems. We extend the edge connectivity ideas to rooted and global vertex connectivity in directed graphs. We obtain a (1+ε)-approximation for rooted vertex connectivity in Õ(nW/ε) time where W is the total vertex weight (assuming integral vertex weights); in particular this yields an Õ(n²/ε) time randomized algorithm for unweighted graphs. This translates to a Õ(KnW) time exact algorithm where K is the rooted connectivity. We build on this to obtain similar bounds for global vertex connectivity. Our results complement the known results for these problems in the low connectivity regime due to work of Gabow [Harold N. Gabow, 1995] for edge connectivity from 1991, and the very recent work of Nanongkai et al. [Nanongkai et al., 2019] and Forster et al. [Sebastian Forster et al., 2020] for vertex connectivity.

Li and Panigrahi [Jason Li and Debmalya Panigrahi, 2020], in recent work, obtained the first deterministic algorithm for the global minimum cut of a weighted undirected graph that runs in time o(mn). They introduced an elegant and powerful technique to find isolating cuts for a terminal set in a graph via a small number of s-t minimum cut computations. In this paper we generalize their isolating cut approach to the abstract setting of symmetric bisubmodular functions (which also capture symmetric submodular functions). Our generalization to bisubmodularity is motivated by applications to element connectivity and vertex connectivity. Utilizing the general framework and other ideas we obtain significantly faster randomized algorithms for computing global (and subset) connectivity in a number of settings including hypergraphs, element connectivity and vertex connectivity in graphs, and for symmetric submodular functions.

A set of n points in a geometric space implicitly induces a complete graph where the weight of an edge between two points is a function of the distance between the endpoints. There are many natural problems that arise from such a geometric graph and many standard geometric problems can be recast as simple properties of this graph. A basic algorithmic obstacle that arises is that the explicit size of the graph is quadratic in the size of the input. There is a long line of research overcoming this obstacle in low-dimensional spaces, as well as some positive results in high-dimensional and more abstract models for specific applications. Here we consider graph problems in general and address the issue of constructing the geometric graph. Rather than constructing these graphs exactly, we ask if it is possible to explicitly construct a sparse approximation of these geometric graphs in nearly linear time. We consider geometric graphs where the edge weights are given as either as a metric (via an oracle), or given by a smooth kernel function in a Euclidean space. For both of these settings, we show that for any ∊ > 0, one can compute an explicit (1 + ∊)-approximate spectral approximation of the geometric graph with Õ(n/∊2) edges in Õ(n/∊2) randomized time. Some of these algorithms are extremely simple. Composed with nearly linear time graph algorithms, this allows for a broad class of applications on geometric graphs with running times proportional to the number of points.

We derive fast approximation schemes for LP relaxations of several well-studied geometric optimization problems that include packing, covering, and mixed packing and covering constraints. Previous work in computational geometry concentrated mainly on the rounding stage to prove approximation bounds, assuming that the underlying LPs can be solved efficiently. This work demonstrates that many of those results can be made to run in nearly linear time. In contrast to prior work on this topic our algorithms handle weights and capacities, side constraints, and also apply to mixed packing and covering problems, in a unified fashion. Our framework relies crucially on the properties of a randomized MWU algorithm of [41]; we demonstrate that it is well-suited for range spaces that admit efficient approximate dynamic data structures for emptiness oracles. Our framework cleanly separates the MWU algorithm for solving the LP from the key geometric data structure primitives, and this enables us to handle side constraints in a simple way. Combined with rounding algorithms that can also be implemented efficiently, we obtain the first near-linear constant factor approximation algorithms for several problems.

The Circle Packing Theorem states that every planar graph can be represented as the tangency graph of a family of internally-disjoint circles. A well-known generalization is the Primal-Dual Circle Packing Theorem for 3-connected planar graphs. The existence of these representations has widespread applications in theoretical computer science and mathematics; however, the algorithmic aspect has received relatively little attention. In this work, we present an algorithm based on convex optimization for computing a primal-dual circle packing representation of maximal planar graphs, i.e. triangulations. This in turn gives an algorithm for computing a circle packing representation of any planar graph. Both take Õ(n log(R/ε)) expected run-time to produce a solution that is ϵ close to a true representation, where R is the ratio between the maximum and minimum circle radius in the true representation.

Karger used spanning tree packings [D. R. Karger, J. ACM, 47 (2000), pp. 46--76] to derive a near linear-time randomized algorithm for the global minimum cut problem as well as a bound on the number of approximate minimum cuts. This is a different approach from his well-known random contraction algorithm [D. R. Karger, Random Sampling in Graph Optimization Problems, Ph.D. thesis, Stanford University, Stanford, CA, 1995, D. R. Karger and C. Stein, J. ACM, 43 (1996), pp. 601--640]. Thorup developed a fast deterministic algorithm for the minimum $k$-cut problem via greedy recursive tree packings [M. Thorup, Minimum $k$-way cuts via deterministic greedy tree packing, in Proceedings of the Fortieth Annual ACM Symposium on Theory of Computing, ACM, 2008, pp. 159--166]. In this paper we revisit properties of an LP relaxation for cͅut proposed by Naor and Rabani [ Tree packing and approximating $k$-cuts, in Proceedings of the Twelfth Annual ACM-SIAM Symposium on Discrete Algorithms, Vol. 103, SIAM, Philadelphia, 2001, pp. 26--27], and analyzed in [C. Chekuri, S. Guha, and J. Naor, SIAM J. Discrete Math., 20 (2006), pp. 261--271]. We show that the dual of the LP yields a tree packing that, when combined with an upper bound on the integrality gap for the LP, easily and transparently extends Karger's analysis for mincut to the $k$-cut problem. In addition to the simplicity of the algorithm and its analysis, this allows us to improve the running time of Thorup's algorithm by a factor of $n$. We also improve the bound on the number of $\alpha$-approximate $k$-cuts. Second, we give a simple proof that the integrality gap of the LP is $2(1-1/n)$. Third, we show that an optimum solution to the LP relaxation, for all values of $k$, is fully determined by the principal sequence of partitions of the input graph. This allows us to relate the LP relaxation to the Lagrangean relaxation approach of Barahona [ Oper. Res. Lett., 26 (2000), pp. 99--105] and Ravi and Sinha [ European J. Oper. Res., 186 (2008), pp. 77--90]; it also shows that the idealized recursive tree packing considered by Thorup gives an optimum dual solution to the LP.

In an undirected graph, a k-cut is a set of edges whose removal breaks the graph into at least k connected components. The minimum weight k-cut can be computed in n^O(k) time, but when k is treated as part of the input, computing the minimum weight k-cut is NP-Hard [Goldschmidt and Hochbaum, 1994]. For poly(m,n,k)-time algorithms, the best possible approximation factor is essentially 2 under the small set expansion hypothesis [Manurangsi, 2017]. Saran and Vazirani [1995] showed that a (2 - 2/k)-approximately minimum weight k-cut can be computed via O(k) minimum cuts, which implies a O~(km) randomized running time via the nearly linear time randomized min-cut algorithm of Karger [2000]. Nagamochi and Kamidoi [2007] showed that a (2 - 2/k)-approximately minimum weight k-cut can be computed deterministically in O(mn + n^2 log n) time. These results prompt two basic questions. The first concerns the role of randomization. Is there a deterministic algorithm for 2-approximate k-cuts matching the randomized running time of O~(km)? The second question qualitatively compares minimum cut to 2-approximate minimum k-cut. Can 2-approximate k-cuts be computed as fast as the minimum cut - in O~(m) randomized time? We give a deterministic approximation algorithm that computes (2 + eps)-minimum k-cuts in O(m log^3 n / eps^2) time, via a (1 + eps)-approximation for an LP relaxation of k-cut.

Balkanski and Singer [4] recently initiated the study of adaptivity (or parallelism) for constrained submodular function maximization, and studied the setting of a cardinality constraint. Subsequent improvements for this problem by Balkanski, Rubinstein, and Singer [6] and Ene and Nguyen [21] resulted in a near-optimal (1 – 1/e – ∊)-approximation in O(log n/∊2) rounds of adaptivity. Partly motivated by the goal of extending these results to more general constraints, we describe parallel algorithms for approximately maximizing the multilinear relaxation of a monotone submodular function subject to packing constraints. Formally our problem is to maximize F(x) over x ∊ [0, 1]n subject to where F is the multilinear relaxation of a monotone submodular function. Our algorithm achieves a near-optimal (1 – 1/e – ∊)-approximation in O(log2 m log n/∊4) rounds where n is the cardinality of the ground set and m is the number of packing constraints. For many constraints of interest, the resulting fractional solution can be rounded via known randomized rounding schemes that are oblivious to the specific submodular function. We thus derive randomized algorithms with poly-logarithmic adaptivity for a number of constraints including partition and laminar matroids, matchings, knapsack constraints, and their intersections. Our algorithm takes a continuous view point and combines several ideas ranging from the continuous greedy algorithm of [38, 13], its adaptation to the MWU framework for packing constraints [20], and parallel algorithms for packing LPs [31, 41]. For the basic setting of cardinality constraints, this viewpoint gives rise to an alternative, simple to understand algorithm that matches recent results [6, 21]. Our algorithm to solve the multilinear relaxation is deterministic if it is given access to a value oracle for the multilinear extension and its gradient; this is possible in some interesting cases such as the coverage function of an explicitly given set system.

We consider approximation algorithms for covering integer programs of the form min 〈c, x〉 over x ∊ ℤ≥0n s.t. Ax ≥ b and x ≤ d; where A ∊ ℝ≥0m×n, b ∊ ℝ≥0m, and c, d ∊ ℝ≥0n all have nonnegative entries. We refer to this problem as CIP, and the special case without the multiplicity constraints x < d as CIP∞. These problems generalize the well-studied Set Cover problem. We make two algorithmic contributions. First, we show that a simple algorithm based on randomized rounding with alteration improves or matches the best known approximation algorithms for CIP and CIP∞ in a wide range of parameter settings, and these bounds are essentially optimal. As a byproduct of the simplicity of the alteration algorithm and analysis, we can derandomize the algorithm without any loss in the approximation guarantee or efficiency. Previous work by Chen, Harris and Srinivasan [13] which obtained near-tight bounds is based on a resampling-based randomized algorithm whose analysis is complex. Non-trivial approximation algorithms for CIP are based on solving the natural LP relaxation strengthened with knapsack cover (KC) inequalities [5, 26, 13]. Our second contribution is a fast (essentially near-linear time) approximation scheme for solving the strengthened LP with a factor of n speed up over the previous best running time [5]. To achieve this fast algorithm we combine recent work on accelerating the multiplicative weight update framework with a partially dynamic approach to the knapsack covering problem. Together, our contributions lead to near-optimal (deterministic) approximation bounds with near-linear running times for CIP and CIP∞.

We consider parallel, or low adaptivity, algorithms for submodular function maximization. This line of work was recently initiated by Balkanski and Singer and has already led to several interesting results on the cardinality constraint and explicit packing constraints. An important open problem is the classical setting of matroid constraint, which has been instrumental for developments in submodular function maximization. In this paper we develop a general strategy to parallelize the well-studied greedy algorithm and use it to obtain a randomized (1 / 2 − є)-approximation in O( log2(n) / 2 ) rounds of adaptivity. We rely on this algorithm, and an elegant amplification approach due to Badanidiyuru and Vondrák to obtain a fractional solution that yields a near-optimal randomized ( 1 − 1/e − є )-approximation in O( log2(n) / є3 ) rounds of adaptivity. For non-negative functions we obtain a ( 3−2√2 − є )-approximation and a fractional solution that yields a ( 1 / e − є)-approximation. Our approach for parallelizing greedy yields approximations for intersections of matroids and matchoids, and the approximation ratios are comparable to those known for sequential greedy.

We describe and analyze a simple randomized multiplicative weight update (MWU) based algorithm for approximately solving positive linear programming problems, in particular, mixed packing and covering LPs. Given m explicit linear packing and covering constraints over n variables specified by N nonzero entries, Young [36] gave a deterministic algorithm returning an (1 + ε)-approximate feasible solution (if a feasible solution exists) in Õ(N/ε2) time. We show that a simple randomized implementation matches this bound, and that randomization can be further exploited to improve the running time to Õ(N/ε + m/ε2 + n/ε3) (both with high probability). For instances that are not very sparse (with at least ῶ(1/ε) nonzeroes per column on average), this improves the running time of Õ(N/ε2). The randomized algorithm also gives improved running times for some implicitly defined problems that arise in combinatorial and geometric optimization.

We give a nearly linear-time randomized approximation scheme for the Held-Karp bound [22] for Metric-TSP. Formally, given an undirected edge-weighted graph G = (V, ε) on m edges and ε > 0, the algorithm outputs in O(m log4n/ε2) time, with high probability, a (1 + ε)-approximation to the Held-Karp bound on the Metric-TSP instance induced by the shortest path metric on G. The algorithm can also be used to output a corresponding solution to the Subtour Elimination LP. We substantially improve upon the O(m2log2(m)/ε2) running time achieved previously by Garg and Khandekar.

We consider several implicit fractional packing problems and obtain faster implementations of approximation schemes based on multiplicative-weight updates. This leads to new algorithms with near-linear running times for some fundamental problems in combinatorial optimization. We highlight two concrete applications. The first is to find the maximum fractional packing of spanning trees in a capacitated graph; we obtain a (1 - ∊)-approximation in Õ(m/∊2) time, where m is the number of edges in the graph. Second, we consider the LP relaxation of the weighted unsplittable flow problem on a path and obtain a (1 - ∊)-approximation in O(n/∊2) time, where n is the number of demands.

We investigate the family of intersection graphs of low density objects in low dimensional Euclidean space. This family is quite general, includes planar graphs, and in particular is a subset of the family of graphs that have polynomial expansion. We present efficient $(1+\varepsilon)$-approximation algorithms for polynomial expansion graphs for unweighted Independent Set, Set Cover, and Dominating Set problems, among others, and these results seem to be new. Naturally, PTASs for these problems are known for subclasses of this graph family. These results have immediate applications in the geometric domain. For example, the new algorithms yield the only PTAS known for covering points by fat triangles (that are shallow). We also prove corresponding hardness of approximation for some of these optimization problems, characterizing their intractability with respect to density. For example, we show that there is no PTAS for covering points by fat triangles if they are not shallow, thus matching our PTAS for this problem with respect to depth.

We present an approximation algorithm for the maximum weight matroid intersection problem in the independence oracle model. Given two matroids defined over a common ground set N of n elements, let k be the rank of the matroid intersection and let Q denote the cost of an independence query for either matroid. An exact algorithm for finding a maximum cardinality independent set (the unweighted case), due to Cunningham, runs in O(nk1.5Q) time. For the weighted case, algorithms due to Frank and Brezovec et al. run in O(nk2Q) time. There are also scaling based algorithms that run in time, where W is the maximum weight (assuming all weights are integers), and ellipsoid-style algorithms that run in O((n2 log(n)Q + n3 polylog(n))log(nW)) time. Recently, Huang, Kakimura, and Kamiyama described an algorithm that gives a (1 – ∊)-approximation for the weighted matroid intersection problem in O(nk1.5 log(k)Q/∊) time. We observe that a (1 – ∊)-approximation for the maximum cardinality case can be obtained in O(nkQ/∊) time by terminating Cunningham's algorithm early. Our main contribution is a (1 – ∊) approximation algorithm for the weighted matroid intersection problem with running time O(nk log2 (1/∊)Q/∊2).

We study the performance of standard online learning algorithms when the feedback is delayed by an adversary. We show that \texttt{online-gradient-descent} and \texttt{follow-the-perturbed-leader} achieve regret $O(\sqrt{D})$ in the delayed setting, where $D$ is the sum of delays of each round's feedback. This bound collapses to an optimal $O(\sqrt{T})$ bound in the usual setting of no delays (where $D = T$). Our main contribution is to show that standard algorithms for online learning already have simple regret bounds in the most general setting of delayed feedback, making adjustments to the analysis and not to the algorithms themselves. Our results help affirm and clarify the success of recent algorithms in optimization and machine learning that operate in a delayed feedback model.