The $\ell_p$-norm objectives for correlation clustering present a fundamental trade-off between minimizing total disagreements (the $\ell_1$-norm) and ensuring fairness to individual nodes (the $\ell_\infty$-norm). Surprisingly, in the offline setting it is possible to simultaneously approximate all $\ell_p$-norms with a single clustering. Can this powerful guarantee be achieved in an online setting? This paper provides the first affirmative answer. We present a single algorithm for the online-with-a-sample (AOS) model that, given a small constant fraction of the input as a sample, produces one clustering that is simultaneously $O(\log^4 n)$-competitive for all $\ell_p$-norms with high probability, $O(\log n)$-competitive for the $\ell_\infty$-norm with high probability, and $O(1)$-competitive for the $\ell_1$-norm in expectation. This work successfully translates the offline "all-norms" guarantee to the online world. Our setting is motivated by a new hardness result that demonstrates a fundamental separation between these objectives in the standard random-order (RO) online model. Namely, while the $\ell_1$-norm is trivially $O(1)$-approximable in the RO model, we prove that any algorithm in the RO model for the fairness-promoting $\ell_\infty$-norm must have a competitive ratio of at least $Ω(n^{1/3})$. This highlights the necessity of a different beyond-worst-case model. We complement our algorithm with lower bounds, showing our competitive ratios for the $\ell_1$- and $\ell_\infty$- norms are nearly tight in the AOS model.

We study the online busy time scheduling model on heterogeneous machines. In our setting, jobs with uniform length arrive online with a deadline that becomes known to the algorithm at the job's arrival time. An algorithm has access to machines, each with different associated capacities and costs. The goal is to schedule jobs on machines by their deadline, so that the total cost incurred by the scheduling algorithm is minimized. While busy time scheduling has been well-studied, relatively little is known when machines are heterogeneous (i.e., have different costs and capacities), despite this natural theoretical generalization being the most practical model for clients using cloud computing services. We make significant progress in understanding this model by designing an 8-competitive algorithm for the problem on unit-length jobs and providing a lower bound of 2 on the competitive ratio. The lower bound is tight in the setting when jobs form non-nested intervals. Our 8-competitive algorithm generalizes to one with competitive ratio $8(2p-1)/p < 16$ when all jobs have uniform length $p$.

This paper considers correlation clustering on unweighted complete graphs. We give a combinatorial algorithm that returns a single clustering solution that is simultaneously $O(1)$-approximate for all $\ell_p$-norms of the disagreement vector; in other words, a combinatorial $O(1)$-approximation of the all-norms objective for correlation clustering. This is the first proof that minimal sacrifice is needed in order to optimize different norms of the disagreement vector. In addition, our algorithm is the first combinatorial approximation algorithm for the $\ell_2$-norm objective, and more generally the first combinatorial algorithm for the $\ell_p$-norm objective when $1 < p < \infty$. It is also faster than all previous algorithms that minimize the $\ell_p$-norm of the disagreement vector, with run-time $O(n^ω)$, where $O(n^ω)$ is the time for matrix multiplication on $n \times n$ matrices. When the maximum positive degree in the graph is at most $Δ$, this can be improved to a run-time of $O(nΔ^2 \log n)$.

Push-Relabel is one of the most celebrated network flow algorithms. Maintaining a pre-flow that saturates a cut, it enjoys better theoretical and empirical running time than other flow algorithms, such as Ford-Fulkerson. In practice, Push-Relabel is even faster than what theoretical guarantees can promise, in part because of the use of good heuristics for seeding and updating the iterative algorithm. However, it remains unclear how to run Push-Relabel on an arbitrary initialization that is not necessarily a pre-flow or cut-saturating. We provide the first theoretical guarantees for warm-starting Push-Relabel with a predicted flow, where our learning-augmented version benefits from fast running time when the predicted flow is close to an optimal flow, while maintaining robust worst-case guarantees. Interestingly, our algorithm uses the gap relabeling heuristic, which has long been employed in practice, even though prior to our work there was no rigorous theoretical justification for why it can lead to run-time improvements. We then show our algorithmic framework works well in practice, as our warm-start version of Push-Relabel improves over the cold-start version by a larger and larger percentage as the size of the image increases.

Abstract. Since the seminal result of Karger, Motwani, and Sudan, algorithms for approximate 3-coloring have primarily centered around rounding the solution to a Semidefinite Program. However, it is likely that important combinatorial or algebraic insights are needed in order to break the [Formula: see text] threshold. One way to develop new understanding in graph coloring is to study special subclasses of graphs. For instance, Blum studied the 3-coloring of random graphs, and Arora and Ge studied the 3-coloring of graphs with low threshold-rank. In this work, we study graphs that arise from a tensor product, which appear to be novel instances of the 3-coloring problem. We consider graphs of the form [Formula: see text] with [Formula: see text] and [Formula: see text], where [Formula: see text] is any edge set such that no vertex has more than an [Formula: see text]-fraction of its edges in [Formula: see text]. We show that one can construct [Formula: see text] with [Formula: see text] that is close to [Formula: see text]. For arbitrary [Formula: see text], [Formula: see text] satisfies [Formula: see text]. Additionally, when [Formula: see text] is a mild expander, we provide a 3-coloring for [Formula: see text] in polynomial time. These results partially generalize an exact tensor factorization algorithm of Imrich. On the other hand, without any assumptions on [Formula: see text], we show that it is NP -hard to 3-color [Formula: see text].

We introduce fast algorithms for correlation clustering with respect to the Min Max objective that provide constant factor approximations on complete graphs. Our algorithms are the first purely combinatorial approximation algorithms for this problem. We construct a novel semi-metric on the set of vertices, which we call the correlation metric, that indicates to our clustering algorithms whether pairs of nodes should be in the same cluster. The paper demonstrates empirically that, compared to prior work, our algorithms sacrifice little in the objective quality to obtain significantly better run-time. Moreover, our algorithms scale to larger networks that are effectively intractable for known algorithms.

Recent work has shown that leveraging learned predictions can improve the running time of algorithms for bipartite matching and similar combinatorial problems. In this work, we build on this idea to improve the performance of the widely used Ford-Fulkerson algorithm for computing maximum flows by seeding Ford-Fulkerson with predicted flows. Our proposed method offers strong theoretical performance in terms of the quality of the prediction. We then consider image segmentation, a common use-case of flows in computer vision, and complement our theoretical analysis with strong empirical results.

Mixtures of high dimensional Gaussian distributions have been studied extensively in statistics and learning theory. While the total variation distance appears naturally in the sample complexity of distribution learning, it is analytically difficult to obtain tight lower bounds for mixtures. Exploiting a connection between total variation distance and the characteristic function of the mixture, we provide fairly tight functional approximations. This enables us to derive new lower bounds on the total variation distance between two-component Gaussian mixtures with a shared covariance matrix.

In the problem of scheduling with non-uniform communication delays, the input is a set of jobs with precedence constraints. Associated with every precedence constraint between a pair of jobs is a communication delay, the time duration the scheduler has to wait between the two jobs if they are scheduled on different machines. The objective is to assign the jobs to machines to minimize the makespan of the schedule. Despite being a fundamental problem in theory and a consequential problem in practice, the approximability of scheduling problems with communication delays is not very well understood. One of the top ten open problems in scheduling theory, in the influential list by Schuurman and Woeginger and its latest update by Bansal, asks if the problem admits a constant-factor approximation algorithm. In this paper, we answer this question in the negative by proving a logarithmic hardness for the problem under the standard complexity theory assumption that NP-complete problems do not admit quasi-polynomial-time algorithms. Our hardness result is obtained using a surprisingly simple reduction from a problem that we call Unique Machine Precedence constraints Scheduling (UMPS). We believe that this problem is of central importance in understanding the hardness of many scheduling problems and we conjecture that it is very hard to approximate. Among other things, our conjecture implies a logarithmic hardness of related machine scheduling with precedences, a long-standing open problem in scheduling theory and approximation algorithms.

In this paper, we study the following batch scheduling model: find a schedule that minimizes total flow time for n uniform length jobs, with release times and deadlines, where the machine is only actively processing jobs in at most k synchronized batches of size at most B. Prior work on such batch scheduling models has considered only feasibility with no regard to the flow time of the schedule. However, algorithms that minimize the cost from the scheduler's perspective---such as ones that minimize the active time of the processor---can result in schedules where the total flow time is arbitrarily high [15]. Such schedules are not valuable from the perspective of the client. In response, our work provides dynamic programs which minimize flow time subject to active time constraints. Our main contribution focuses on jobs with agreeable deadlines; for such job instances, we introduce dynamic programs that achieve runtimes of O(B ․ k ․ n) for unit jobs and O(B ․ O(B ․ n5) for uniform length jobs. These results improve upon our modification of a different, classical dynamic programming approach by Baptiste. While the modified DP works when deadlines are non-agreeable, this solution is more expensive, with runtime O(B ․ k2 ․ n7) [7].

We introduce approximate trace reconstruction, a relaxed version of the trace reconstruction problem. Here, instead of learning a binary string perfectly from noisy samples, as in the original trace reconstruction problem, the goal is to output a string that is close in edit distance to the original string using few traces. We present several algorithms that can approximately reconstruct strings that belong to certain classes, where the estimate is within$n$/ polylog (n) edit distance and where we only use polylog (n) traces (or sometimes just a single trace). These classes contain strings that require a linear number of traces for exact reconstruction and that are quite different from a typical random string. From a technical point of view, our algorithms approximately reconstruct consecutive substrings of the unknown string by aligning dense regions of traces and using a run of a suitable length to approximate each region. A full version of this paper is accessible at: https://arxiv.org/abs/2012.06713.pdf

We consider the problem of scheduling jobs with precedence constraints on related machines to minimize the weighted sum of completion times, in the presence of communication delays. In this setting, denoted by Q | prec, c | ΣwjCj, if two dependent jobs are scheduled on different machines, then at least c units of communication delay time must pass between their executions. Our main result is an O(log4 n)-approximation algorithm for the problem. As a byproduct of our result, we also obtain an O(log3 n)-approximation algorithm for the problem of minimizing makespan Q | prec, c | Cmax, which improves upon the O(log5 n/ log log n)-approximation algorithm due to a recent work of Maiti et al. [MRS+20].

We consider the classic problem of scheduling jobs with precedence constraints on identical machines to minimize makespan, in the presence of communication delays. In this setting, denoted by P | prec, c | Cmax, if two dependent jobs are scheduled on different machines, then at least c units of time must pass between their executions. Despite its relevance to many applications, this model remains one of the most poorly understood in scheduling theory. Even for a special case where an unlimited number of machines is available, the best known approximation ratio is 2/3·(c+1), whereas Graham's greedy list scheduling algorithm already gives a ( c+1) -approximation in that setting. An outstanding open problem in the top-10 list by Schuurman and Woeginger and its recent update by Bansal asks whether there exists a constant-factor approximation algorithm. In this work we give a polynomial-time O(logc·logm)-approximation algorithm for this problem, where m is the number of machines and c is the communication delay. Our approach is based on a Sherali-Adams lift of a linear programming relaxation and a randomized clustering of the semimetric space induced by this lift. The full version of this paper is available on arXiv.

A well-known problem in scheduling and approximation algorithms is the Santa Claus problem. Suppose that Santa Claus has a set of gifts, and he wants to distribute them among a set of children so that the least happy child is made as happy as possible. Here, the value that a child i has for a present j is of the form pij ϵ {0, pj}. A polynomial time algorithm by Annamalai et al. gives a 12.33-approximation and is based on a modification of Haxell's hypergraph matching argument. In this paper, we introduce a matroid version of the Santa Claus problem. Our algorithm is also based on Haxell's augmenting tree, but with the introduction of the matroid structure we solve a more general problem with cleaner methods. Our result can then be used as a blackbox to obtain a (4 + ϵ)-approximation for Santa Claus. This factor also compares against a natural, compact LP for Santa Claus.

We study the problem of learning a node-labeled tree given independent traces from an appropriately defined deletion channel. This problem, tree trace reconstruction, generalizes string trace reconstruction, which corresponds to the tree being a path. For many classes of trees, including complete trees and spiders, we provide algorithms that reconstruct the labels using only a polynomial number of traces. This exhibits a stark contrast to known results on string trace reconstruction, which require exponentially many traces, and where a central open problem is to determine whether a polynomial number of traces suffice. Our techniques combine novel combinatorial and complex analytic methods.