Étienne Bamas, Andreas Maggiori, Ola Svensson
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.
Etienne Bamas, Sai Ganesh Nagarajan, Ola Svensson
One of the most popular clustering algorithms is the celebrated $D^α$ seeding algorithm (also know as $k$-means++ when $α=2$) by Arthur and Vassilvitskii (2007), who showed that it guarantees in expectation an $O(2^{2α}\cdot \log k)$-approximate solution to the ($k$,$α$)-means cost (where euclidean distances are raised to the power $α$) for any $α\ge 1$. More recently, Balcan, Dick, and White (2018) observed experimentally that using $D^α$ seeding with $α>2$ can lead to a better solution with respect to the standard $k$-means objective (i.e. the $(k,2)$-means cost). In this paper, we provide a rigorous understanding of this phenomenon. For any $α>2$, we show that $D^α$ seeding guarantees in expectation an approximation factor of $$ O_α\left((g_α)^{2/α}\cdot \left(\frac{σ_{\mathrm{max}}}{σ_{\mathrm{min}}}\right)^{2-4/α}\cdot (\min\{\ell,\log k\})^{2/α}\right)$$ with respect to the standard $k$-means cost of any underlying clustering; where $g_α$ is a parameter capturing the concentration of the points in each cluster, $σ_{\mathrm{max}}$ and $σ_{\mathrm{min}}$ are the maximum and minimum standard deviation of the clusters around their means, and $\ell$ is the number of distinct mixing weights in the underlying clustering (after rounding them to the nearest power of $2$). We complement these results by some lower bounds showing that the dependency on $g_α$ and $σ_{\mathrm{max}}/σ_{\mathrm{min}}$ is tight. Finally, we provide an experimental confirmation of the effects of the aforementioned parameters when using $D^α$ seeding. Further, we corroborate the observation that $α>2$ can indeed improve the $k$-means cost compared to $D^2$ seeding, and that this advantage remains even if we run Lloyd's algorithm after the seeding.
Etienne Bamas, Marina Drygala, Ola Svensson
The Matching Augmentation Problem (MAP) has recently received significant attention as an important step towards better approximation algorithms for finding cheap $2$-edge connected subgraphs. This has culminated in a $\frac{5}{3}$-approximation algorithm. However, the algorithm and its analysis are fairly involved and do not compare against the problem's well-known LP relaxation called the cut LP. In this paper, we propose a simple algorithm that, guided by an optimal solution to the cut LP, first selects a DFS tree and then finds a solution to MAP by computing an optimum augmentation of this tree. Using properties of extreme point solutions, we show that our algorithm always returns (in polynomial time) a better than $2$-approximation when compared to the cut LP. We thereby also obtain an improved upper bound on the integrality gap of this natural relaxation.
Étienne Bamas, Lars Rohwedder
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 $\mathrm{poly}(\log\log n)$-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.
Étienne Bamas, Marina Drygala, Andreas Maggiori
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 $\{\{s_1,t_1\},\ldots ,\{s_k,t_k\}\}$ 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 $\{s_i,t_i\}$, connect $s_i$ and $t_i$ 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 long-standing open problem in the area of online algorithms. The last progress on this question is due to an elegant analysis by Awerbuch, Azar, and Bartal [SODA~1996], who showed that greedy is $O(\log^2(k))$-competitive. Our main result is to show that greedy is in fact $O(\log(k)\log\log(k))$-competitive on a wide class of instances. In particular, this wide class of instances contains all the instances that were exhibited in the literature until now.
Etienne Bamas, Shi Li, Lars Rohwedder
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, for example, polynomial in the problem size. Technically, we reinterpret the dynamic programming subproblems and their solutions as a network design problem. Inspired by techniques from, for example, 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 $(n^ε \mathrm{polylog}\ n)$-approximation in time $n^{O(1/ε)}$, for any $ε> 0$. By setting $ε=\log \log n/\log n$, we obtain a polylogarithmic approximation in quasi-polynomial time, or by setting $ε$ as a constant, an $n^ε$-approximation in polynomial time. While there are necessary assumptions on the form of the DP, it is general enough to capture many textbook dynamic programs from Shortest Path to Longest Common Subsequence. Our algorithm then implies that we can impose additional constraints on the solutions to these problems. This allows us to model various problems from the literature in approximation algorithms, many of which were not thought to be connected to dynamic programming. In fact, our result can even be applied indirectly to some problems that involve covering instead of packing constraints, for example, the Directed Steiner Tree problem, or those that do not directly follow a recurrence relation, for example, variants of the Matching problem.
Étienne Bamas, Alexander Lindermayr, Nicole Megow, Lars Rohwedder, Jens Schlöter
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. While the two problems are intuitively related and history has shown that techniques can often be transferred between them, no formal reductions are known. We first show that an affirmative answer to the open question for makespan minimization implies the same for the Santa Claus problem by reducing the latter problem to the former. We also prove that for problem instances with only two input values both questions are equivalent. We then move to a special case called ``restricted assignment'', which is well studied in both problems. Although our reductions do not maintain the characteristics of this special case, we give a reduction in a slight generalization, where the jobs or resources are assigned to multiple machines or players subject to a matroid constraint and in addition we have only two values. This draws a similar picture as before: equivalence for two values and the general case of Santa Claus can only be easier than makespan minimization. To complete the picture, we give an algorithm for our new matroid variant of the Santa Claus problem using a non-trivial extension of the local search method from restricted assignment. Thereby we unify, generalize, and improve several previous results. We believe that this matroid generalization may be of independent interest and provide several sample applications.
Étienne Bamas, Andreas Maggiori, Lars Rohwedder, Ola Svensson
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.
Etienne Bamas
This paper is devoted to the study of the MaxMinDegree Arborescence (MMDA) problem in layered directed graphs of depth $\ell\le O(\log n/\log \log n)$, 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]. The only way we have to solve the MMDA problem within a polylogarithmic factor is via an elegant recursive rounding of the $(\ell-1)^{th}$ level of the Sherali-Adams hierarchy, which needs time $n^{O(\ell)}$ to solve. However, it remains plausible that one could obtain a polylogarithmic approximation in polynomial time by using the same rounding with only $1$ round of the Sherali-Adams hierarchy. As a main result, we rule out this possibility by constructing an MMDA instance of depth $3$ for which an integrality gap of $n^{Ω(1)}$ survives $1$ round of the Sherali-Adams hierarchy. This result is tight since it is known that after only $2$ rounds the gap is at most polylogarithmic on depth-3 graphs. Second, we show that our instance can be ``lifted'' via a simple trick to MMDA instances of any depth $\ell\in Ω(1)\cap o(\log n/\log \log n)$ (the whole range of interest), for which we conjecture that an integrality gap of $n^{Ω(1/\ell)}$ survives $Ω(\ell)$ rounds of Sherali-Adams. We show a number of intermediate results towards this conjecture, which also suggest that our construction is a significant challenge to the techniques used so far for Santa Claus.
Etienne Bamas, Paritosh Garg, Lars Rohwedder
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 $S_1,\dotsc,S_m$ is a partition of the resources. The result by Goemans et al. implies a polynomial time $O(n^{1/2 +\varepsilon})$-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 $f_i(S) = f(S \cap Γ_i)$ 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. Namely, if $f$ is a monotone submodular function, we can in polynomial time compute an $O(\log\log(n))$-approximate solution.
Étienne Bamas, Louis Esperet
This paper is devoted to the distributed complexity of finding an approximation of the maximum cut in graphs. A classical algorithm consists in letting each vertex choose its side of the cut uniformly at random. This does not require any communication and achieves an approximation ratio of at least $\tfrac12$ in average. When the graph is $d$-regular and triangle-free, a slightly better approximation ratio can be achieved with a randomized algorithm running in a single round. Here, we investigate the round complexity of deterministic distributed algorithms for MAXCUT in regular graphs. We first prove that if $G$ is $d$-regular, with $d$ even and fixed, no deterministic algorithm running in a constant number of rounds can achieve a constant approximation ratio. We then give a simple one-round deterministic algorithm achieving an approximation ratio of $\tfrac1{d}$ for $d$-regular graphs with $d$ odd. We show that this is best possible in several ways, and in particular no deterministic algorithm with approximation ratio $\tfrac1{d}+ε$ (with $ε>0$) can run in a constant number of rounds. We also prove results of a similar flavour for the MAXDICUT problem in regular oriented graphs, where we want to maximize the number of arcs oriented from the left part to the right part of the cut.
Kirill Brilliantov, Etienne Bamas, Emmanuel Abbé
We introduce a code-based challenge for automated, open-ended mathematical discovery based on the $k$-server conjecture, a central open problem in competitive analysis. The task is to discover a potential function satisfying a large graph-structured system of simple linear inequalities. The resulting evaluation procedure is sound but incomplete: any violated inequality definitively refutes a candidate, whereas satisfying all inequalities does not by itself constitute a proof of the corresponding conjecture's special case. Nevertheless, a candidate that passes all constraints would be strong evidence toward a valid proof and, to the best of our knowledge, no currently known potential achieves this under our formulation in the open $k=4$ circle case. As such, a successful candidate would already be an interesting contribution to the $k$-server conjecture, and could become a substantial theoretical result when paired with a full proof. Experiments on the resolved $k=3$ regime show that current agentic methods can solve nontrivial instances, and in the open $k=4$ regime they reduce the number of violations relative to existing potentials without fully resolving the task. Taken together, these results suggest that the task is challenging but plausibly within reach of current methods. Beyond its relevance to the $k$-server community, where the developed tooling enables researchers to test new hypotheses and potentially improve on the current record, the task also serves as a useful \emph{benchmark} for developing code-based discovery agents. In particular, our $k=3$ results show that it mitigates important limitations of existing open-ended code-based benchmarks, including early saturation and the weak separation between naive random baselines and more sophisticated methods.
Etienne Bamas, Sarah Morell, Lars Rohwedder
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. In the case where each player has a potentially different additive valuation function, Chakrabarty, Chuzhoy, and Khanna [FOCS'09] gave an $O(n^ε)$-approximation algorithm with polynomial running time for any constant $ε> 0$ and a polylogarithmic approximation algorithm in quasi-polynomial time. We show that the same can be achieved for monotone submodular valuation functions, improving over the previously best algorithm due to Goemans, Harvey, Iwata, and Mirrokni [SODA'09], which has an approximation ratio of more than $\sqrt{n}$. Our result builds up on a sophisticated LP relaxation, which has a recursive block structure that allows us to solve it despite having exponentially many variables and constraints.
Etienne Bamas, Paritosh Garg, Lars Rohwedder
We consider hypergraphs on vertices $P\cup R$ where each hyperedge contains exactly one vertex in $P$. Our goal is to select a matching that covers all of $P$, but we allow each selected hyperedge to drop all but an $(1/α)$-fraction of its intersection with $R$ (thus relaxing the matching constraint). Here $α$ is to be minimized. We dub this problem the Combinatorial Santa Claus problem, since we show in this paper that this problem and the Santa Claus problem are almost equivalent in terms of their approximability. The non-trivial observation that any uniform regular hypergraph admits a relaxed matching for $α= O(1)$ was a major step in obtaining a constant approximation rate for a special case of the Santa Claus problem, which received great attention in literature. It is natural to ask if the uniformity condition can be omitted. Our main result is that every (non-uniform) regular hypergraph admits a relaxed matching for $α= O(\log\log(|R|))$, when all hyperedges are sufficiently large (a condition that is necessary). In particular, this implies an $O(\log\log(|R|))$-approximation algorithm for the Combinatorial Santa Claus problem with large hyperedges.
Étienne Bamas, Louis Esperet
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.