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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

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

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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