We show that under mild assumptions for a problem whose solutions admit a dynamic programming-like recurrence relation, we can still find a solution under additional packing constraints, which need to be satisfied approximately. The number of additional constraints can be very large, e.g., polynomial in the problem size. Technically, we reinterpret the dynamic programming subproblems and their solutions as a network design problem. Inspired by techniques from, e.g., the Directed Steiner Tree problem, we construct a strong LP relaxation, on which we then apply randomized rounding. Our approximation guarantees on the packing constraints have roughly the form of a (nє polylog n)-approximation in time nO(1/є), for any є > 0. By setting є=loglogn/logn, we obtain a polylogarithmic approximation in quasi-polynomial time, or by setting є as a constant, an nє-approximation in polynomial time.

This paper is devoted to the study of the MaxMinDegree Arborescence (MMDA) problem in layered directed graphs of depth ℓ ≤ O (log n/ log log n ), which is an important special case of the Santa Claus problem. Obtaining a polylogarithmic approximation for MMDA in polynomial time is of high interest as it is a necessary condition to improve upon the well-known 2-approximation for makespan scheduling on unrelated machines by Lenstra, Shmoys, and Tardos [FOCS’87].

We consider the problem of allocating indivisible resources to players so as to maximize the minimum total value any player receives. This problem is sometimes dubbed the Santa Claus problem and its different variants have been subject to extensive research towards approximation algorithms over the past two decades.

One of the most popular clustering algorithms is the celebrated $D^\alpha$ seeding algorithm (also know as $k$-means++ when $\alpha=2$) by Arthur and Vassilvitskii (2007), who showed that it guarantees in expectation an $O(2^{2\alpha}\cdot \log k)$-approximate solution to the ($k$,$\alpha$)-clustering cost (where distances are raised to the power $\alpha$) for any $\alpha\ge 1$. More recently, Balcan, Dick, and White (2018) observed experimentally that using $D^\alpha$ seeding with $\alpha>2$ can lead to a better solution with respect to the standard $k$-means objective (i.e. the $(k,2)$-clustering cost). In this paper, we provide a rigorous understanding of this phenomenon. For any $\alpha>2$, we show that $D^\alpha$ seeding guarantees in expectation an approximation factor of \begin{equation*} O_\alpha \left(\left(\frac{\sigma_{\textrm{max}}}{\sigma_{\textrm{min}}}\right)^{2-4/\alpha}\cdot (g_\alpha \cdot \min \lbrace\ell,\log k\rbrace)^{2/\alpha}\right) \end{equation*} with respect to the standard $k$-means cost of any underlying clustering; where $g_\alpha$ is a parameter capturing the concentration of the points in each cluster, $\sigma_{\textrm{max}}$ and $\sigma_{\textrm{min}}$ are the maximum and minimum standard deviation of the clusters around their center, and $\ell$ is the number of distinct mixing weights in the underlying clustering (after rounding them to the nearest power of $2$). For instance, if the underlying clustering is defined by a mixture of $k$ Gaussian distributions with equal cluster variance (up to a constant-factor), then our result implies that: (1) if there are a constant number of mixing weights, any constant $\alpha>2$ yields a constant-factor approximation; (2) if the mixing weights are arbitrary, any constant $\alpha>2$ yields an $O\left(\log^{2/\alpha}k\right)$-approximation, and $\alpha=\Theta(\log\log k)$ yields an $O(\log\log k)^3$-approximation. We complement these results by some lower bounds showing that the dependency on $g_\alpha$ and $\sigma_{\textrm{max}}/\sigma_{\textrm{min}}$ is tight. Finally, we provide an experimental validation of the effects of the aforementioned parameters when using $D^\alpha$ seeding.

In this paper we study the relation of two fundamental problems in scheduling and fair allocation: makespan minimization on unrelated parallel machines and max-min fair allocation, also known as the Santa Claus problem. For both of these problems the best approximation factor is a notorious open question; more precisely, whether there is a better-than-2 approximation for the former problem and whether there is a constant approximation for the latter.

We revisit the problem max-min degree arborescence, which was introduced by Bateni et al. [STOC’09] as a central special case of the general Santa Claus problem, which constitutes a notorious open question in approximation algorithms. In the former problem we are given a directed graph with sources and sinks and our goal is to find vertex disjoint arborescences rooted in the sources such that at each non-sink vertex of an arborescence the out-degree is at least k, where k is to be maximized. This problem is of particular interest, since it appears to capture much of the difficulty of the Santa Claus problem: (1) like in the Santa Claus problem the configuration LP has a large integrality gap in this case and (2) previous progress by Bateni et al. was quickly generalized to the Santa Claus problem (Chakrabarty et al. [FOCS’09]). These results remain the state-of-the-art both for the Santa Claus problem and for max-min degree arborescence and they yield a polylogarithmic approximation in quasi-polynomial time. We present an exponential improvement to this, a poly(loglogn)-approximation in quasi-polynomial time for the max-min degree arborescence problem. To the best of our knowledge, this is the first example of breaking the logarithmic barrier for a special case of the Santa Claus problem, where the configuration LP cannot be utilized. The main technical novelty of our result are locally good solutions: informally, we show that it suffices to find a poly(logn)-approximation that locally has stronger guarantees. We use a lift-and-project type of LP and randomized rounding, which were also used by Bateni et al., but unlike previous work we integrate careful pruning steps in the rounding. In the proof we extensively apply Lovász Local Lemma and a local search technique, both of which were previously used only in the context of the configuration LP.

This paper considers the classic Online Steiner Forest problem where one is given a (weighted) graph G and an arbitrary set of k terminal pairs {{s1, t1}, …, {sk, tk}} that are required to be connected. The goal is to maintain a minimum-weight sub-graph that satisfies all the connectivity requirements as the pairs are revealed one by one. It has been known for a long time that no algorithm (even randomized) can be better than Ω(log(k))-competitive for this problem. Interestingly, a simple greedy algorithm is already very efficient for this problem. This algorithm can be informally described as follows: Upon arrival of a new pair {si, ti}, connect si and ti with the shortest path in the current metric, contract the metric along the chosen path and wait for the next pair. Although simple and intuitive, greedy proved itself challenging to analyze and its competitive ratio is a longstanding open problem in the area of online algorithms. The last progress on this problem is due to an elegant analysis by Awerbuch, Azar, and Bartal [SODA 1996], who showed that greedy is O(log2(k))-competitive. In this paper, we identify a natural measure of the “efficiency” of greedy that we call the contraction. The contraction of a pair {si, ti} is the ratio between the distance dG (si, ti) in the graph G and the actual cost that greedy pays for connecting the pair {si, ti}. Intuitively, a worst-case instance should be an instance on which greedy is very “inefficient”, i.e. an instance for which all pairs have a relatively small contraction. Indeed, one can remark that all hard instances that appeared in the literature are such that all pairs have a contraction of exactly 1 (which is the smallest contraction possible). Our main result, among others, is to show that greedy is O(log(k) log log(k))-competitive on such instances. At the heart of this new result lies an original use of dual fitting, in which we use the dual solution not only to lower bound the optimum as it is usually the case in competitive analysis, but also to recursively partition the global instance into several disjoint instances of much smaller complexity.

The submodular Santa Claus problem was introduced in a seminal work by Goemans, Harvey, Iwata, and Mirrokni (SODA'09) as an application of their structural result. In the mentioned problem n unsplittable resources have to be assigned to m players, each with a monotone submodular utility function f_i. The goal is to maximize min_i f_i(S_i) where S₁,...,S_m is a partition of the resources. The result by Goemans et al. implies a polynomial time O(n^{1/2 +ε})-approximation algorithm. Since then progress on this problem was limited to the linear case, that is, all f_i are linear functions. In particular, a line of research has shown that there is a polynomial time constant approximation algorithm for linear valuation functions in the restricted assignment case. This is the special case where each player is given a set of desired resources Γ_i and the individual valuation functions are defined as f_i(S) = f(S ∩ Γ_i) for a global linear function f. This can also be interpreted as maximizing min_i f(S_i) with additional assignment restrictions, i.e., resources can only be assigned to certain players. In this paper we make comparable progress for the submodular variant: If f is a monotone submodular function, we can in polynomial time compute an O(log log(n))-approximate solution.

As power management has become a primary concern in modern data centers, computing resources are being scaled dynamically to minimize energy consumption. We initiate the study of a variant of the classic online speed scaling problem, in which machine learning predictions about the future can be integrated naturally. Inspired by recent work on learning-augmented online algorithms, we propose an algorithm which incorporates predictions in a black-box manner and outperforms any online algorithm if the accuracy is high, yet maintains provable guarantees if the prediction is very inaccurate. We provide both theoretical and experimental evidence to support our claims.

The extension of classical online algorithms when provided with predictions is a new and active research area. In this paper, we extend the primal-dual method for online algorithms in order to incorporate predictions that advise the online algorithm about the next action to take. We use this framework to obtain novel algorithms for a variety of online covering problems. We compare our algorithms to the cost of the true and predicted offline optimal solutions and show that these algorithms outperform any online algorithm when the prediction is accurate while maintaining good guarantees when the prediction is misleading.

This paper studies sufficient conditions to obtain efficient distributed algorithms coloring graphs optimally (i.e.\ with the minimum number of colors) in the LOCAL model of computation. Most of the work on distributed vertex coloring so far has focused on coloring graphs of maximum degree $Δ$ with at most $Δ+1$ colors (or $Δ$ colors when some simple obstructions are forbidden). When $Δ$ is sufficiently large and $c\ge Δ-k_Δ+1$, for some integer $k_Δ\approx \sqrtΔ-2$, we give a distributed algorithm that given a $c$-colorable graph $G$ of maximum degree $Δ$, finds a $c$-coloring of $G$ in $\min\{O((\logΔ)^{1/12}\log n), 2^{O(\log Δ+\sqrt{\log \log n})}\}$ rounds, with high probability. The lower bound $Δ-k_Δ+1$ is best possible in the sense that for infinitely many values of $Δ$, we prove that when $χ(G)\le Δ-k_Δ$, finding an optimal coloring of $G$ requires $Ω(n)$ rounds. Our proof is a light adaptation of a remarkable result of Molloy and Reed, who proved that for $Δ$ large enough, for any $c\ge Δ- k_Δ$ deciding whether $χ(G)\le c$ is in {\textsf{P}}, while Embden-Weinert \emph{et al.}\ proved that for $c\le Δ-k_Δ-1$, the same problem is {\textsf{NP}}-complete. Note that the sequential and distributed thresholds differ by one. We also show that for any sufficiently large $Δ$, and $Ω(\log Δ)\le k \le Δ/100$, every graph of maximum degree $Δ$ and clique number at most $Δ-k$ can be efficiently colored with at most $Δ-\varepsilon k$ colors, for some absolute constant $\varepsilon >0$, with a randomized algorithm running in $O(\log n/\log \log n)$ rounds with high probability.