We consider submodular maximization under increasing cardinality constraint and ask for a good incremental solution, i.e., an ordering of the ground set such that each prefix of the ordering yields a good solution for its respective cardinality. A classical result in this setting is that the greedy algorithm achieves a competitive ratio, i.e., an approximation guarantee across all cardinalities, of e/(e-1) ≈ 1.582. No better general guarantee was previously known. We present an adaptive scaling algorithm achieving a competitive ratio of 1.373. We complement our result by a lower bound of 1.25 on the best possible deterministic competitive ratio for incremental submodular maximization.

Vizing’s theorem states that any \(n\)-vertex \(m\)-edge graph of maximum degree \(\Delta\) can be edge colored using at most \(\Delta + 1\) different colors. Vizing’s original proof is easily translated into a deterministic \(O(mn)\) time algorithm. This deterministic time bound was subsequently improved to \(\tilde{O}(m\sqrt{n})\) time, independently by [Arjomandi, 1982] and by [Gabow et al., 1985].

Vizing’s theorem states that any n -vertex m -edge graph of maximum degree Δ can be edge colored using at most Δ + 1 different colors [Vizing, 1964]. Vizing’s original proof is algorithmic and shows that such an edge coloring can be found in O(mn) time. This was subsequently improved to \(\tilde{O}(m\sqrt {n})\) time, independently by [Arjomandi, 1982] and by [Gabow et al., 1985]. 1 Very recently, independently and concurrently, using randomization, this runtime bound was further improved to \(\tilde{O}(n^2)\) by [Assadi, 2024] and \(\tilde{O}(mn^{1/3})\) by [Bhattacharya, Carmon, Costa, Solomon and Zhang, 2024] (and subsequently to \(\tilde{O}(mn^{1/4})\) by [Bhattacharya, Costa, Solomon and Zhang, 2024]). In this article, we present a randomized algorithm that computes a Δ + 1-edge coloring in near-linear time—in fact, only O(m log Δ) time—with high probability, giving a near-optimal algorithm for this fundamental problem .

In this paper, we consider the *metric $k$-center* problem in the fully dynamic setting, where we are given a metric space $(V,d)$ evolving via a sequence of point insertions and deletions and our task is to maintain a subset $S \subseteq V$ of at most $k$ points that minimizes the objective $\max_{x \in V} \min_{y \in S}d(x, y)$. We want to design our algorithm so that we minimize its *approximation ratio*, *recourse* (the number of changes it makes to the solution $S$) and *update time* (the time it takes to handle an update). We give a simple algorithm for dynamic $k$-center that maintains a $O(1)$-approximate solution with $O(1)$ amortized recourse and $\tilde O(k)$ amortized update time, *obtaining near-optimal approximation, recourse and update time simultaneously*. We obtain our result by combining a variant of the dynamic $k$-center algorithm of Bateni et al. [SODA'23] with the dynamic sparsifier of Bhattacharya et al. [NeurIPS'23].

Vizing’s Theorem from 1964 states that any n-vertex m-edge graph with maximum degree Δ can be edge colored using at most Δ + 1 colors. For over 40 years, the state-of-the-art running time for computing such a coloring, obtained independently by Arjomandi [1982] and by Gabow, Nishizeki, Kariv, Leven and Terada [1985], was . Very recently, this time bound was improved in two independent works, by Bhattacharya, Carmon, Costa, Solomon and Zhang to , and by Assadi to Õ (n2).

Vizing’s theorem states that any n-vertex m-edge graph of maximum degree Δ can be edge colored using at most Δ + 1 different colors [Vizing, 1964]. Vizing’s original proof is algorithmic and shows that such an edge coloring can be found in O(mn) time. This was subsequently improved to Õ(m√n) time, independently by [Arjomandi, 1982] and by [Gabow et al., 1985]. Very recently, independently and concurrently, using randomization, this runtime bound was further improved to Õ(n2) by [Assadi, 2024] and Õ(mn1/3) by [Bhattacharya, Carmon, Costa, Solomon and Zhang, 2024] (and subsequently to Õ(mn1/4) by [Bhattacharya, Costa, Solomon and Zhang, 2024]). In this paper, we present a randomized algorithm that computes a (Δ+1)-edge coloring in near-linear time—in fact, only O(mlogΔ) time—with high probability, giving a near-optimal algorithm for this fundamental problem.

Vizing's theorem asserts the existence of a $(Δ+1)$-edge coloring for any graph $G$, where $Δ= Δ(G)$ denotes the maximum degree of $G$. Several polynomial time $(Δ+1)$-edge coloring algorithms are known, and the state-of-the-art running time (up to polylogarithmic factors) is $\tilde{O}(\min\{m \cdot \sqrt{n}, m \cdot Δ\})$, by Gabow et al.\ from 1985, where $n$ and $m$ denote the number of vertices and edges in the graph, respectively. (The $\tilde{O}$ notation suppresses polylogarithmic factors.) Recently, Sinnamon shaved off a polylogarithmic factor from the time bound of Gabow et al. The {arboricity} $α= α(G)$ of a graph $G$ is the minimum number of edge-disjoint forests into which its edge set can be partitioned, and it is a measure of the graph's "uniform density". While $α\le Δ$ in any graph, many natural and real-world graphs exhibit a significant separation between $α$ and $Δ$. In this work we design a $(Δ+1)$-edge coloring algorithm with a running time of $\tilde{O}(\min\{m \cdot \sqrt{n}, m \cdot Δ\})\cdot \fracαΔ$, thus improving the longstanding time barrier by a factor of $\fracαΔ$. In particular, we achieve a near-linear runtime for bounded arboricity graphs (i.e., $α= \tilde{O}(1)$) as well as when $α= \tilde{O}(\fracΔ{\sqrt{n}})$. Our algorithm builds on Sinnamon's algorithm, and can be viewed as a density-sensitive refinement of it.

Vizing's theorem states that any n-vertex m-edge graph of maximum degree$\Delta$can be edge colored using at most$\Delta+1$different colors [Diskret. Analiz, '64]. Vizing's original proof is algorithmic and shows that such an edge coloring can be found in$\tilde{O}(mn)$time. This was subsequently improved to$\tilde{O}(m\sqrt{n})$, independently by Arjomandi [1982] and by Gabow et al. [1985]. In this paper we present an algorithm that computes such an edge coloring in$\tilde{O}(mn^{1/3})$, time, giving the first polynomial improvement for this fundamental problem in over 40 years.

In the dynamic metric$k-\mathbf{median}$problem, we wish to maintain a set of$k$centers$S\subseteq V$in an input metric space$(V, d)$that gets updated via point insertions/deletions, so as to minimize the objective$\sum\nolimits_{x\in V}\min\nolimits_{y\in S}d(x, y)$. The quality of a dynamic algorithm is measured in terms of its approximation ratio, “recourse” (the number of changes in$S$per update) and “update time” (the time it takes to handle an update). The ultimate goal in this line of research is to obtain a dynamic$O(1)$approximation algorithm with$\tilde{O}(1)$recourse and$\tilde{O}(k)$update time. Dynamic$k-\mathbf{median}$is a canonical example of a class of problems known as dynamic$k-\mathbf{clustering}$, that has received significant attention in recent years [Fichtenberger et al, SODA'21], [Bateni et al, SODA'23], [Lacki et al, SODA'24]. To the best of our knowledge, however, all these previous papers either attempt to minimize the algorithm's recourse while ignoring its update time, or minimize the algorithm's update time while ignoring its recourse. For dynamic$k-\mathbf{median}$in particular, the state-of-the-art results get$\tilde{O}(k^{2})$update time and$O(k)$recourse [Cohen-Addad et al, ICML'19], [Henzinger and Kale, ESA'20], [Bhattacharya et al, NeurIPS'23]. But, this recourse bound of$O(k)$can be trivially obtained by recomputing an optimal solution from scratch after every update, provided we ignore the update time. In addition, the update time of$\tilde{O}(k^{2})$is polynomially far away from the desired bound of$\tilde{O}(k)$. We come arbitrarily close to resolving the main open question on this topic, with the following results. (I) We develop a new framework of randomized local search that is suitable for adaptation in a dynamic setting. For every$\epsilon > 0$, this gives us a dynamic$k-\mathbf{median}$algorithm with$O(k^{\epsilon})$approximation ratio,$\tilde{O}(k^{\epsilon})$recourse and$\tilde{O}(k^{1+\epsilon})$update time. This framework also generalizes to dynamic$k-\mathbf{clustering}$with$\ell^{p}$-norm objectives. As a corollary, we obtain similar bounds for the dynamic$k-\mathbf{means}$problem, and a new trade-off between approximation ratio, recourse and update time for the dynamic$k-\mathbf{center}$problem. (II) If it suffices to maintain only an estimate of the value of the optimal$k-\mathbf{median}$objective, then we obtain a$O(1)$approximation algorithm with$\tilde{O}(k)$update time. We achieve this result via adapting the Lagrangian Relaxation framework of [Jain and Vazirani, JACM'01], and a facility location algorithm of [Mettu and Plaxton, FOCS'00] in the dynamic setting.

We consider the problem of maintaining a (1 + ɛ)∆-edge coloring in a dynamic graph G with n nodes and maximum degree at most Δ. The state-of-the-art update time is Oɛ(polylog(n)), by Duan, He and Zhang [SODA’19] and by Christiansen [STOC’23], and more precisely O(log7 n/ɛ2), where Δ = Ω(log2 n/ɛ2).