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Woodruff, David P. (Ed.)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, \emph{$k$fold matroid union}), computing $k$ disjoint bases (a.k.a.\ \emph{matroid base packing}), and covering the elements by $k$ bases (a.k.a.\ \emph{matroid base covering}). These problems generalize naturally to integral and realvalued 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 \emph{pushrelabel} algorithm for the integercapacitated versions of these problems, building on previous work by [FM12]. 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 pushrelabel algorithm, gives a faster exact algorithm for some parameters of $k$. In particular we obtain a subquadraticquery running time in the uncapacitated setting for the three basic problems listed above. We also obtain faster approximation algorithms for these problems with realvalued 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.more » « lessFree, publiclyaccessible full text available January 7, 2025

Woodruff, David P. (Ed.)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 kcuts 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 [BK15] 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 [CKN20], that also preserves all kcuts. 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.more » « lessFree, publiclyaccessible full text available January 7, 2025

Woodruff, David P. (Ed.)We give improved algorithms for maintaining edgeorientations of a fullydynamic graph, such that the maximum outdegree is bounded. On one hand, we show how to orient the edges such that maximum outdegree is proportional to the arboricity $\alpha$ of the graph, in, either, an amortised update time of $O(\log^2 n \log \alpha)$, or a worstcase update time of $O(\log^3 n \log \alpha)$. On the other hand, motivated by applications including dynamic maximal matching, we obtain a different tradeoff. Namely, the improved update time of either $O(\log n \log \alpha)$, amortised, or $O(\log ^2 n \log \alpha)$, worstcase, for the problem of maintaining an edgeorientation with at most $O(\alpha + \log n)$ outedges per vertex. Finally, all of our algorithms naturally limit the recourse to be polylogarithmic in $n$ and $\alpha$. Our algorithms adapt to the current arboricity of the graph, and yield improvements over previous work: Firstly, we obtain deterministic algorithms for maintaining a $(1+\varepsilon)$ approximation of the maximum subgraph density, $\rho$, of the dynamic graph. Our algorithms have update times of $O(\varepsilon^{6}\log^3 n \log \rho)$ worstcase, and $O(\varepsilon^{4}\log^2 n \log \rho)$ amortised, respectively. We may output a subgraph $H$ of the input graph where its density is a $(1+\varepsilon)$ approximation of the maximum subgraph density in time linear in the size of the subgraph. These algorithms have improved update time compared to the $O(\varepsilon^{6}\log ^4 n)$ algorithm by Sawlani and Wang from STOC 2020. Secondly, we obtain an $O(\varepsilon^{6}\log^3 n \log \alpha)$ worstcase update time algorithm for maintaining a $(1~+~\varepsilon)\textnormal{OPT} + 2$ approximation of the optimal outorientation of a graph with adaptive arboricity $\alpha$, improving the $O(\varepsilon^{6}\alpha^2 \log^3 n)$ algorithm by Christiansen and Rotenberg from ICALP 2022. This yields the first worstcase polylogarithmic dynamic algorithm for decomposing into $O(\alpha)$ forests. Thirdly, we obtain arboricityadaptive fullydynamic deterministic algorithms for a variety of problems including maximal matching, $\Delta+1$ colouring, and matrix vector multiplication. All update times are worstcase $O(\alpha+\log^2n \log \alpha)$, where $\alpha$ is the current arboricity of the graph. For the maximal matching problem, the stateoftheart deterministic algorithms by Kopelowitz, Krauthgamer, Porat, and Solomon from ICALP 2014 runs in time $O(\alpha^2 + \log^2 n)$, and by Neiman and Solomon from STOC 2013 runs in time $O(\sqrt{m})$. We give improved running times whenever the arboricity $\alpha \in \omega( \log n\sqrt{\log\log n})$.more » « lessFree, publiclyaccessible full text available January 7, 2025

Free, publiclyaccessible full text available September 11, 2024

Megow, Nicole ; Smith, Adam (Ed.)Maximum weight independent set (MWIS) admits a 1/kapproximation in inductively kindependent graphs [Karhan Akcoglu et al., 2002; Ye and Borodin, 2012] and a 1/(2k)approximation in kperfectly orientable graphs [Kammer and Tholey, 2014]. These are a parameterized class of graphs that generalize kdegenerate graphs, chordal graphs, and intersection graphs of various geometric shapes such as intervals, pseudodisks, 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 nonnegative 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 lowadaptivity) approximations. Motivated by the goal of designing efficient and deterministic algorithms, we describe two other algorithms for inductively kindependent graphs that are inspired by work on streaming algorithms: a preemptive greedy algorithm and a primaldual 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 pseudodisks.more » « lessFree, publiclyaccessible full text available September 4, 2024

Free, publiclyaccessible full text available September 4, 2024

Gørtz, Inge Li ; FarachColton, Martin ; Puglisi, Simon J. ; Herman, Grzegorz (Ed.)Boob et al. [Boob et al., 2020] described an iterative peeling algorithm called Greedy++ for the Densest Subgraph Problem (DSG) and conjectured that it converges to an optimum solution. Chekuri, Qaunrud and Torres [Chandra Chekuri et al., 2022] extended the algorithm to supermodular density problems (of which DSG is a special case) and proved that the resulting algorithm SuperGreedy++ (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 [Satoru Fujishige, 1980], and a noisy version of the FrankWolfe method from convex optimization [Frank and Wolfe, 1956; Jaggi, 2013]. This yields a simpler convergence proof, and also shows a stronger property that SuperGreedy++ converges to the optimal dense decomposition vector, answering a question raised in Harb et al. [Harb et al., 2022]. A second contribution of the paper is to understand Thorup’s work on ideal tree packing and greedy tree packing [Thorup, 2007; Thorup, 2008] via the FrankWolfe 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 optimization.more » « less

Abstract We consider the problem of covering multiple submodular constraints. Given a finite ground set
N , a weight function ,$$w: N \rightarrow \mathbb {R}_+$$ $w:N\to {R}_{+}$r monotone submodular functions over$$f_1,f_2,\ldots ,f_r$$ ${f}_{1},{f}_{2},\dots ,{f}_{r}$N and requirements the goal is to find a minimum weight subset$$k_1,k_2,\ldots ,k_r$$ ${k}_{1},{k}_{2},\dots ,{k}_{r}$ such that$$S \subseteq N$$ $S\subseteq N$ for$$f_i(S) \ge k_i$$ ${f}_{i}\left(S\right)\ge {k}_{i}$ . We refer to this problem as$$1 \le i \le r$$ $1\le i\le r$MultiSubmodCover and it was recently considered by HarPeled and Jones (Few cuts meet many point sets. CoRR.arxiv:abs1808.03260 HarPeled and Jones 2018) who were motivated by an application in geometry. Even with$$r=1$$ $r=1$MultiSubmodCover generalizes the wellknown Submodular Set Cover problem (SubmodSC ), and it can also be easily reduced toSubmodSC . A simple greedy algorithm gives an approximation where$$O(\log (kr))$$ $O(log(kr\left)\right)$ and this ratio cannot be improved in the general case. In this paper, motivated by several concrete applications, we consider two ways to improve upon the approximation given by the greedy algorithm. First, we give a bicriteria approximation algorithm for$$k = \sum _i k_i$$ $k={\sum}_{i}{k}_{i}$MultiSubmodCover that covers each constraint to within a factor of while incurring an approximation of$$(11/e\varepsilon )$$ $(11/e\epsilon )$ in the cost. Second, we consider the special case when each$$O(\frac{1}{\epsilon }\log r)$$ $O(\frac{1}{\u03f5}logr)$ is a obtained from a truncated coverage function and obtain an algorithm that generalizes previous work on partial set cover ($$f_i$$ ${f}_{i}$PartialSC ), covering integer programs (CIPs ) and multiple vertex cover constraints Bera et al. (Theoret Comput Sci 555:2–8 Bera et al. 2014). Both these algorithms are based on mathematical programming relaxations that avoid the limitations of the greedy algorithm. We demonstrate the implications of our algorithms and related ideas to several applications ranging from geometric covering problems to clustering with outliers. Our work highlights the utility of the highlevel model and the lens of submodularity in addressing this class of covering problems.