We initiate a resource-aware theory of \textit{language generation in the limit} under the minimal constraint of space efficiency. In our framework, a learner observes an adversarial positive stream from a target language $K$ and must eventually output a hallucination-free hypothesis language $L \subseteq K$ while omitting at most $\Delta$ strings of $K$. We focus on $\mathcal{C}_{s,k}$, the collection of languages recognized by DFAs with at most $s$ states over an alphabet of size $k$, as the natural hypothesis class for memory-bounded learners. In the exponential-space regime, we prove that a learner can exactly identify the target $K$. Under a stricter memory budget, we characterize the strongest possible generation guarantees. In particular, we present a streaming algorithm using $\mathrm{poly}(s,k)$ space that converges to a hypothesis with generation gap $\Delta = O(k^{2s-2})$. Moreover, the learned hypothesis captures every string in $K$ of length at least $2s-1$. We complement this result with a near-matching lower bound through a reduction from a standard communication complexity problem. Specifically, achieving generation gap $\Delta \le k^{(1-\varepsilon)s}$ requires $k^{\Omega(\varepsilon s)}$ memory. Together, these results reveal a sharp transition between polynomial-space generation and exponential-space exact identification.

Due to their numerous applications, in particular in Mechanism Design, Prophet Inequalities have experienced a surge of interest. They describe competitive ratios for basic stopping time problems where random variables get revealed sequentially. A key drawback in the classical setting is the assumption of full distributional knowledge of the involved random variables, which is often unrealistic. A natural way to address this is via sample-based approaches, where only a limited number of samples from the distribution of each random variable is available. Recently, Fu, Lu, Gavin Tang, Wu, Wu, and Zhang (2024) showed that sample-based Online Contention Resolution Schemes (OCRS) are a powerful tool to obtain sample-based Prophet Inequalities. They presented the first sample-based OCRS for matroid constraints, which is a heavily studied constraint family in this context, as it captures many interesting settings. This allowed them to get the first sample-based Matroid Prophet Inequality, using \(O(\log^4 n)\) many samples (per ground set element), where \(n\) is the number of random variables, while obtaining a constant competitiveness of \(1/4 - \varepsilon\).

We study online edge coloring, where edges of an n-vertex graph arrive sequentially and must be colored irrevocably so that adjacent edges receive different colors. The goal is to use as few colors as possible as a function of the maximum degree Δ. This talk surveys recent progress that achieves near-optimal guarantees by leveraging martingale concentration arguments. Specifically, we show that near-optimal colorings (using (1+o(1))Δ colors) exhibit sharp threshold phenomena that match long-standing lower bounds, resolving and strengthening a conjecture of Bar‑Noy, Motwani, and Naor [Bar-Noy et al., 1992]. First, while the conjecture posited the existence of a randomized algorithm achieving a (1+o(1))Δ-edge-coloring for maximum degree Δ = ω(log n), we present a deterministic online algorithm that achieves this guarantee in the same regime. This result matches the known impossibility result for deterministic algorithms when Δ = O(log n), establishing a sharp threshold. Second, improving the conditions under which near-optimal coloring is known to be possible with randomness, we present a randomized online algorithm achieving a (1+o(1))Δ-edge-coloring already for graphs with maximum degree Δ = ω(√{log n}). This establishes a sharp threshold for randomized algorithms, matching the lower bound in [Bar-Noy et al., 1992] for the Δ = O(√{log n}) regime. This is joint work with Joakim Blikstad, Radu Vintan, and David Wajc [Joakim Blikstad et al., 2024; Joakim Blikstad et al., 2025].

The goal in the stochastic vertex cover problem is to obtain an approximately minimum vertex cover for a graph G⋆ that is realized by sampling each edge independently with some probability p∈ (0, 1] in a base graph G = (V, E). The algorithm is given the base graph G and the probability p as inputs, but its only access to the realized graph G⋆ is through queries on individual edges in G that reveal the existence (or not) of the queried edge in G⋆. In this paper, we resolve the central open question for this problem: to find a (1+ε)-approximate vertex cover using only Oε(n/p) edge queries. Prior to our work, there were two incomparable state-of-the-art results for this problem: a (3/2+ε)-approximation using Oε(n/p) queries (Derakhshan, Durvasula, and Haghtalab, 2023) and a (1+ε)-approximation using Oε((n/p)· RS(n)) queries (Derakhshan, Saneian, and Xun, 2025), where RS(n) is known to be at least 2Ω(logn/loglogn) and could be as large as n/2Θ(log* n). Our improved upper bound of Oε(n/p) matches the known lower bound of Ω(n/p) for any constant-factor approximation algorithm for this problem (Behnezhad, Blum, and Derakhshan, 2022). A key tool in our result is a new concentration bound for the size of minimum vertex cover on random graphs, which might be of independent interest.

The Weighted Tree Augmentation Problem (WTAP) is a fundamental network design problem where the goal is to find a minimum-cost set of additional edges (links) to make an input tree 2-edge-connected. While a 2-approximation is standard and the integrality gap of the classic Cut LP relaxation is known to be at least 1.5, achieving approximation factors significantly below 2 has proven challenging. Recent advances of Traub and Zenklusen using local search culminated in a ratio of 1.5+є, establishing the state-of-the-art. In this work, we present a randomized approximation algorithm for WTAP with an approximation ratio below 1.49. Our approach is based on designing and rounding a strong linear programming relaxation for WTAP which incorporates variables that represent subsets of edges and the links used to cover them, inspired by lift-and-project methods like Sherali-Adams.

Vizing’s theorem guarantees that every graph with maximum degree $\Delta$ admits an edge coloring using $\Delta+1$ colors. In online settings-where edges arrive one at a time and must be colored immediately-a simple greedy algorithm uses at most $2 \Delta-1$ colors. Over thirty years ago, Bar-Noy, Motwani, and Naor [IPL’92] proved that this guarantee is optimal among deterministic algorithms when $\Delta=O(\log n)$, and among randomized algorithms when $\Delta=O(\sqrt{\log n})$. While deterministic improvements seemed out of reach, they conjectured that for graphs with $\Delta=\omega(\log n)$, randomized algorithms can achieve $(1+o(1)) \Delta$ edge coloring. This conjecture was recently resolved in the affirmative: a $(1+o(1)) \Delta$ coloring is achievable online using randomization for all graphs with $\Delta=\omega(\log n)$ [BSVW STOC’24]. Our results go further, uncovering two findings not predicted by the original conjecture. First, we give a deterministic online algorithm achieving $(1+o(1)) \Delta$-colorings for all $\Delta=\omega(\log n)$. Second, we give a randomized algorithm achieving $(1+o(1)) \Delta$ colorings already when $\Delta=\omega(\sqrt{\log n})$. Our results establish sharp thresholds for when greedy can be surpassed, and nearoptimal guarantees can be achieved - matching the impossibility results of [BNMN IPL’92], both deterministically and randomly.

In this paper, we consider a new problem of portfolio optimization using stochastic information. In a setting where there is some uncertainty, we ask how to best select k potential solutions, with the goal of optimizing the value of the best solution. More formally, given a combinatorial problem Π, a set of value functions 𝒱 over the solutions of Π, and a distribution 𝒟 over 𝒱, our goal is to select k solutions of Π that maximize or minimize the expected value of the best of those solutions. For a simple example, consider the classic knapsack problem: given a universe of elements each with unit weight and a positive value, the task is to select r elements maximizing the total value. Now suppose that each element’s weight comes from a (known) distribution. How should we select k different solutions so that one of them is likely to yield a high value? In this work, we tackle this basic problem, and generalize it to the setting where the underlying set system forms a matroid. On the technical side, it is clear that the candidate solutions we select must be diverse and anti-correlated; however, it is not clear how to do so efficiently. Our main result is a polynomial-time algorithm that constructs a portfolio within a constant factor of the optimal.

We study online bipartite edge coloring, with nodes on one side of the graph revealed sequentially.

In the classical NP-hard (metric) k-median problem, we are given a set of n clients and centers with metric distances between them, along with an integer parameter k ≥ 1. The objective is to select a subset of k open centers that minimizes the total distance from each client to its closest open center. In their seminal work, Jain, Mahdian, Markakis, Saberi, and Vazirani presented the Greedy algorithm for facility location, which implies a 2-approximation algorithm for k-median that opens k centers in expectation. Since then, substantial research has aimed at narrowing the gap between their algorithm and the best achievable approximation by an algorithm guaranteed to open exactly k centers, as required in the k-median problem. During the last decade, all improvements have been achieved by leveraging their algorithm (or a small improvement thereof), followed by a second step called bi-point rounding, which inherently adds an additional factor to the approximation guarantee. Our main result closes this gap: for any > 0, we present a (2+)-approximation algorithm for the k-median problem, improving the previous best-known approximation factor of 2.613. Our approach builds on a combination of two key algorithms. First, we present a non-trivial modification of the Greedy algorithm that operates with only O(logn/2) adaptive phases. Through a novel walk-between-solutions approach, this enables us to construct a (2+)-approximation algorithm for k-median that consistently opens at most k + O(logn/2) centers: via known results, this already implies a (2+)-approximation algorithm that runs in quasi-polynomial time. Second, we develop a novel (2+)-approximation algorithm tailored for stable instances, where removing any center from an optimal solution increases the cost by at least an ω(3/logn) fraction. Achieving this involves several ideas, including a sampling approach inspired by the k-means++ algorithm and a reduction to submodular optimization subject to a partition matroid. This allows us to convert the previous result into a polynomial time algorithm that opens exactly k centers while maintaining the (2+)-approximation guarantee.

In this work, we study online submodular maximization and how the requirement of maintaining a stable solution impacts the approximation. In particular, we seek bounds on the best-possible approximation ratio that is attainable when the algorithm is allowed to make, at most, a constant number of updates per step. We show a tight information-theoretic bound of 2/3 for general monotone submodular functions and an improved (also tight) bound of 3/4 for coverage functions. Since both these bounds are attained by non poly-time algorithms, we also give a poly-time randomized algorithm that achieves a 0.51-approximation. Combined with an information-theoretic hardness of 1/2 for deterministic algorithms from prior work, our work thus shows a separation between deterministic and randomized algorithms, both information theoretically and for poly-time algorithms.

For any 𝜀 > 0, we prove that 𝑘-Dimensional Matching is hard to approximate within a factor of 𝑘/(12+𝜀) for large 𝑘 unless NP ⊆ BPP. Listed in Karp’s 21 NP-complete problems, 𝑘-Dimensional Matching is a benchmark computational complexity problem which we find as a special case of many constrained optimization problems over independence systems including: 𝑘-Set Packing, 𝑘-Matroid Intersection, and Matroid 𝑘-Parity. For all the aforementioned problems, the best-known lower bound was a Ω(𝑘/log(𝑘))-hardness by Hazan, Safra, and Schwartz. In contrast, state-of-the-art algorithms achieve an approximation of 𝑂(𝑘). Our result narrows down this gap to a constant and thus provides a rationale for the observed algorithmic difficulties. The crux of our result hinges on a novel approximation preserving gadget from 𝑅-degree bounded 𝑘-CSPs over alphabet size 𝑅 to 𝑘𝑅-Dimensional Matching. Along the way, we prove that 𝑅-degree bounded 𝑘-CSPs over alphabet size 𝑅 are hard to approximate within a factor Ω𝑘 (𝑅) using known randomised sparsification methods for CSPs.

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.

The classic theorem of Vizing (Diskret. Analiz.’64) asserts that any graph of maximum degree Δ can be edge colored (offline) using no more than Δ+1 colors (with Δ being a trivial lower bound). In the online setting, Bar-Noy, Motwani and Naor (IPL’92) conjectured that a (1+o(1))Δ-edge-coloring can be computed online in n-vertex graphs of maximum degree Δ=ω(logn). Numerous algorithms made progress on this question, using a higher number of colors or assuming restricted arrival models, such as random-order edge arrivals or vertex arrivals (e.g., AGKM FOCS’03, BMM SODA’10, CPW FOCS’19, BGW SODA’21, KLSST STOC’22). In this work, we resolve this longstanding conjecture in the affirmative in the most general setting of adversarial edge arrivals. We further generalize this result to obtain online counterparts of the list edge coloring result of Kahn (J. Comb. Theory. A’96) and of the recent “local” edge coloring result of Christiansen (STOC’23).

In recent years there has been a significant research effort on incorporating predictions into online algorithms. However, work in this area often makes the underlying assumption that predictions come for free (e.g., without any computational or monetary costs). In this paper, we consider a cost associated with making predictions. We show that interesting algorithmic subtleties arise for even the most basic online problems, such as ski rental and its generalization, the Bahncard problem. In particular, we show that with costly predictions, care needs to be taken in (i) asking for the prediction at the right time, (ii) deciding if it is worth asking for the prediction, and (iii) how many predictions we ask for, in settings where it is natural to consider making multiple predictions. Specifically, (i) in the basic ski-rental setting, we compute the optimal delay before asking the predictor, (ii) in the same setting, given apriori information about the true number of ski-days through its mean and variance, we provide a simple algorithm that is near-optimal, under some natural parameter settings, in deciding if it is worth asking for the predictor and (iii) in the setting of the Bahncard problem, we provide a $(1+\varepsilon)$-approximation algorithm and quantify lower bounds on the number of queries required to do so. In addition, we show that solving the problem optimally would require almost complete information of the instance.

Given a set of points in d-dimensional space, an explainable clustering is one where the clusters are specified by a tree of axis-aligned threshold cuts. Dasgupta et al. (ICML 2020) posed the question of the price of explainability: the worst-case ratio between the cost of the best explainable clusterings to that of the best clusterings.We show that the price of explainability for k medians is at most $1+H_{k-1}$; in fact, we show that the popular Random Thresholds algorithm has exactly this price of explainability, matching the known lower bound constructions. We complement our tight analysis of this particular algorithm by constructing instances where the price of explainability (using any algorithm) is at least $(1-o(1)) \ln k$, showing that our result is best possible, up to lower-order terms. We also improve the price of explainability for the k-means problem to $O(k \ln \ln k)$ from the previous $O(k \ln k)$, considerably closing the gap to the lower bounds of $\Omega(k)$. Finally, we study the algorithmic question of finding the best explainable clustering: We show that explainable k medians and k-means cannot be approximated better than $O(\ln k)$, under standard complexity-theoretic conjectures. This essentially settles the approximability of explainable k-medians and leaves open the intriguing possibility to get significantly better approximation algorithms for k-means than its price of explainability.

The Bellman-Ford algorithm is a basic primitive for computing single source shortest paths in graphs with negative weight edges. Its running time is governed by the order the algorithm examines vertices for iterative updates on the value of their shortest path. In this work we study this problem through the lens of ’Algorithms with predictions,’ and show how to leverage auxiliary information from similar instances to improve the running time. We do this by identifying the key problem of Minimum Violation Permutations, and give algorithms with strong approximation guarantees as well as formal lower bounds. We complement the theoretical analysis with an empirical evaluation, showing that this approach can lead to a significant speed up in practice.

Given a graph with edges colored red or blue and an integer k, the exact perfect matching problem asks if there exists a perfect matching with exactly k red edges. There exists a randomized polylogarithmic-time parallel algorithm to solve this problem, dating back to the eighties, but no deterministic polynomial-time algorithm is known, even for bipartite graphs. In this paper we show that there is no sub-exponential sized linear program that can describe the convex hull of exact matchings in bipartite graphs. In fact, we prove something stronger, that there is no sub-exponential sized linear program to describe the convex hull of perfect matchings with an odd number of red edges.

We consider the classical problem of maximizing a monotone submodular function subject to a cardinality constraint, which, due to its numerous applications, has recently been studied in various computational models. We consider a clean multiplayer model that lies between the offline and streaming model, and study it under the aspect of one-way communication complexity. Our model captures the streaming setting (by considering a large number of players), and, in addition, two-player approximation results for it translate into the robust setting. We present tight one-way communication complexity results for our model, which, due to the connections mentioned previously, have multiple implications in the data stream and robust setting. Even for just two players, a prior information-theoretic hardness result implies that no approximation factor above 1/2 can be achieved in our model, if only queries to feasible sets (i.e., sets respecting the cardinality constraint) are allowed. We show that the possibility of querying infeasible sets can actually be exploited to beat this bound, by presenting a tight 2/3-approximation taking exponential time, and an efficient 0.514-approximation. To the best of our knowledge, this is the first example where querying a submodular function on infeasible sets leads to provably better results. Through the link to the (non-streaming) robust setting mentioned previously, both of these algorithms improve on the current state of the art for robust submodular maximization, showing that approximation factors beyond 1/2 are possible. Moreover, exploiting the link of our model to streaming, we settle the approximability for streaming algorithms by presenting a tight 1/2+ɛ hardness result, based on the construction of a new family of coverage functions. This improves on a prior 0.586 hardness and matches, up to an arbitrarily small margin, the best-known approximation algorithm.

Designing algorithms for machine learning problems targeting beyond worst-case analysis and, in particular, analyzing the effect of side-information on the complexity of such problems is a very important line of research with many practical applications. In this paper we study the classic k-means clustering problem in the presence of noisy labels. In this problem, in addition to a set of points and parameter \(k\), we receive cluster labels of each point generated by either an adversarial or a random perturbation of the optimal solution. Our main goal is to formally study the effect of this extra information on the complexity of the k-means problem. In particular, in the context of random perturbations, we give an efficient algorithm that finds a clustering of cost within a factor $1+o(1)$ of the optimum even when the label of each point is perturbed with a large probability (think 99%). In contrast, we show that the side-information with adversarial perturbations is as hard as the original problem even if only a small $\epsilon$ fraction of the labels are perturbed. We complement this negative result by giving a simple algorithm in the case when the adversary is only allowed to perturb an $\epsilon$ fraction of the labels per \emph{each cluster}.

We relate discrepancy theory with the classic scheduling problems of minimizing max flow time and total flow time on unrelated machines. Specifically, we give a general reduction that allows us to transfer discrepancy bounds in the prefix Beck-Fiala (bounded ℓ1-norm) setting to bounds on the flow time of an optimal schedule. Combining our reduction with a deep result proved by Banaszczyk via convex geometry, give guarantees of O(√logn) and O(√logn logP) for max flow time and total flow time, respectively, improving upon the previous best guarantees of O(logn) and O(logn logP). Apart from the improved guarantees, the reduction motivates seemingly easy versions of prefix discrepancy questions: any constant bound on prefix Beck-Fiala where vectors have sparsity two (sparsity one being trivial) would already yield tight guarantees for both max flow time and total flow time. While known techniques solve this case when the entries take values in {−1,0,1}, we show that they are unlikely to transfer to the more general 2-sparse case of bounded ℓ1-norm.

The secretary problem became one of the most prominent online selection problems due to its numerous applications in online mechanism design. The task is to select a maximum weight subset of elements subject to given constraints, where elements arrive one-by-one in random order, revealing a weight upon arrival. The decision whether to select an element has to be taken immediately after its arrival. The different applications that map to the secretary problem ask for different constraint families to be handled. The most prominent ones are matroid constraints, which both capture many relevant settings and admit strongly competitive secretary algorithms. However, dealing with more involved constraints proved to be much more difficult, and strong algorithms are known only for a few specific settings. In this paper, we present a general framework for dealing with the secretary problem over the intersection of several matroids. This framework allows us to combine and exploit the large set of matroid secretary algorithms known in the literature. As one consequence, we get constant-competitive secretary algorithms over the intersection of any constant number of matroids whose corresponding (single-)matroid secretary problems are currently known to have a constant-competitive algorithm. Moreover, we show that our results extend to submodular objectives.

As a fundamental unsupervised learning task, hierarchical clustering has been extensively studied in the past decade. In particular, standard metric formulations as hierarchical $k$-center, $k$-means, and $k$-median received a lot of attention and the problems have been studied extensively in different models of computation. Despite all this interest, not many efficient parallel algorithms are known for these problems. In this paper we introduce a new parallel algorithm for the Euclidean hierarchical $k$-median problem that, when using machines with memory $s$ (for $s\in \Omega(\log^2 (n+\Delta+d))$), outputs a hierarchical clustering such that for every fixed value of $k$ the cost of the solution is at most an $O(\min\{d, \log n\} \log \Delta)$ factor larger in expectation than that of an optimal solution. Furthermore, we also get that for all $k$ simultanuously the cost of the solution is at most an $O(\min\{d, \log n\} \log \Delta \log (\Delta d n))$ factor bigger that the corresponding optimal solution. The algorithm requires in $O\left(\log_{s} (nd\log(n+\Delta))\right)$ rounds. Here $d$ is the dimension of the data set and $\Delta$ is the ratio between the maximum and minimum distance of two points in the input dataset. To the best of our knowledge, this is the first \emph{parallel} algorithm for the hierarchical $k$-median problem with theoretical guarantees. We further complement our theoretical results with an empirical study of our algorithm that shows its effectiveness in practice.

We study the problem of explainable clustering in the setting first formalized by Dasgupta, Frost, Moshkovitz, and Rashtchian (ICML 2020). A $k$-clustering is said to be explainable if it is given by a decision tree where each internal node splits data points with a threshold cut in a single dimension (feature), and each of the $k$ leaves corresponds to a cluster. We give an algorithm that outputs an explainable clustering that loses at most a factor of $O(\log^2 k)$ compared to an optimal (not necessarily explainable) clustering for the $k$-medians objective, and a factor of $O(k \log^2 k)$ for the $k$-means objective. This improves over the previous best upper bounds of $O(k)$ and $O(k^2)$, respectively, and nearly matches the previous $\Omega(\log k)$ lower bound for $k$-medians and our new $\Omega(k)$ lower bound for $k$-means. The algorithm is remarkably simple. In particular, given an initial not necessarily explainable clustering in $\mathbb{R}^d$, it is oblivious to the data points and runs in time $O(dk \log^2 k)$, independent of the number of data points $n$. Our upper and lower bounds also generalize to objectives given by higher $\ell_p$-norms.

Given a stream of points in a metric space, is it possible to maintain a constant approximate clustering by changing the cluster centers only a small number of times during the entire execution of the algorithm? This question received attention in recent years in the machine learning literature and, before our work, the best known algorithm performs Õ(k2) center swaps (the Õ(·) notation hides polylogarithmic factors in the number of points n and the aspect ratio Δ of the input instance). This is a quadratic increase compared to the offline case — the whole stream is known in advance and one is interested in keeping a constant approximation at any point in time — for which Õ(k) swaps are known to be sufficient and simple examples show that Ω(k log(nΔ)) swaps are necessary. We close this gap by developing an algorithm that, perhaps surprisingly, matches the guarantees in the offline setting. Specifically, we show how to maintain a constant-factor approximation for the k-median problem by performing an optimal (up to polylogarithimic factors) number Õ(k) of center swaps. To obtain our result we leverage new structural properties of k-median clustering that may be of independent interest.

We consider the problem of scheduling jobs on unrelated machines so as to minimize the sum of weighted completion times. Our main result is a $(\nicefrac{3}{2}-c)$-approximation algorithm for some fixed $c>0$, improving upon the long-standing bound of $\nicefrac{3}{2}$. To do this, we first introduce a new lift-and-project-based SDP relaxation for the problem. This is necessary, as the previous convex programming relaxations have an integrality gap of $\nicefrac{3}{2}$. Second, we give a new general bipartite-rounding procedure that produces an assignment with certain strong negative correlation properties.

We introduce a new rounding technique designed for online optimization problems, which is related to contention resolution schemes, a technique initially introduced in the context of submodular function maximization. Our rounding technique, which we call online contention resolution schemes (OCRSs), is applicable to many online selection problems, including Bayesian online selection, oblivious posted pricing mechanisms, and stochastic probing models. It allows for handling a wide set of constraints and shares many strong properties of offline contention resolution schemes. In particular, OCRSs for different constraint families can be combined to obtain an OCRS for their intersection. Moreover, we can approximately maximize submodular functions in the online settings we consider. We thus get a broadly applicable framework for several online selection problems, which improves on previous approaches in terms of the types of constraints that can be handled, the objective functions that can be dealt with, and the assumptions on the strength of the adversary. Furthermore, we resolve two open problems from the literature; namely, we present the first constant-factor constrained oblivious posted price mechanism for matroid constraints and the first constant-factor algorithm for weighted stochastic probing with deadlines.

In this paper, we study streaming and online algorithms in the context of randomness in the input. For several problems, a random order of the input sequence---as opposed to the worst-case order---appears to be a necessary evil in order to prove satisfying guarantees. However, algorithmic techniques that work under this assumption tend to be vulnerable to even small changes in the distribution. For this reason, we propose a new \emph{adversarial injections} model, in which the input is ordered randomly, but an adversary may inject misleading elements at arbitrary positions. We believe that studying algorithms under this much weaker assumption can lead to new insights and, in particular, more robust algorithms. We investigate two classical combinatorial-optimization problems in this model: Maximum matching and cardinality constrained monotone submodular function maximization. Our main technical contribution is a novel streaming algorithm for the latter that computes a $0.55$-approximation. While the algorithm itself is clean and simple, an involved analysis shows that it emulates a subdivision of the input stream which can be used to greatly limit the power of the adversary.

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.

$k$-means++ \cite{arthur2007k} is a widely used clustering algorithm that is easy to implement, has nice theoretical guarantees and strong empirical performance. Despite its wide adoption, $k$-means++ sometimes suffers from being slow on large data-sets so a natural question has been to obtain more efficient algorithms with similar guarantees. In this paper, we present such a near linear time algorithm for $k$-means++ seeding. Interestingly our algorithm obtains the same theoretical guarantees as $k$-means++ and significantly improves earlier results on fast $k$-means++ seeding. Moreover, we show empirically that our algorithm is significantly faster than $k$-means++ and obtains solutions of equivalent quality.

We consider the classical problem of maximizing a monotone submodular function subject to a cardinality constraint, which, due to its numerous applications, has recently been studied in various computational models. We consider a clean multi-player model that lies between the offline and streaming model, and study it under the aspect of one-way communication complexity. Our model captures the streaming setting (by considering a large number of players), and, in addition, two player approximation results for it translate into the robust setting. We present tight one-way communication complexity results for our model, which, due to the above-mentioned connections, have multiple implications in the data stream and robust setting.

We give a constant-factor approximation algorithm for the asymmetric traveling salesman problem (ATSP). Our approximation guarantee is analyzed with respect to the standard LP relaxation, and thus our result confirms the conjectured constant integrality gap of that relaxation. The main idea of our approach is a reduction to Subtour Partition Cover, an easier problem obtained by significantly relaxing the general connectivity requirements into local connectivity conditions. We first show that any algorithm for Subtour Partition Cover can be turned into an algorithm for ATSP while only losing a small constant factor in the performance guarantee. Next, we present a reduction from general ATSP instances to structured instances, on which we then solve Subtour Partition Cover, yielding our constant-factor approximation algorithm for ATSP.

Clustering is a classic topic in optimization with $k$-means being one of the most fundamental such problems. In the absence of any restrictions on the input, the best-known algorithm for $k$-means in Euclidean space with a provable guarantee is a simple local search heuristic yielding an approximation guarantee of $9+\epsilon$, a ratio that is known to be tight with respect to such methods. We overcome this barrier by presenting a new primal-dual approach that allows us to (1) exploit the geometric structure of $k$-means and (2) satisfy the hard constraint that at most $k$ clusters are selected without deteriorating the approximation guarantee. Our main result is a 6.357-approximation algorithm with respect to the standard linear programming (LP) relaxation. Our techniques are quite general, and we also show improved guarantees for $k$-median in Euclidean metrics and for a generalization of $k$-means in which the underlying metric is not required to be Euclidean.

A sparsifier of a graph G (Benczúr and Karger; Spielman and Teng) is a sparse weighted subgraph G that approximately retains the same cut structure of G. For general graphs, non-trivial sparsification is possible only by using weighted graphs in which different edges have different weights. Even for graphs that admit unweighted sparsifiers (that is, sparsifiers in which all the edge weights are equal to the same scaling factor), there are no known polynomial time algorithms that find such unweighted sparsifiers. We study a weaker notion of sparsification suggested by Oveis Gharan, in which the number of cut edges in each cut (S, S) is not approximated within a multiplicative factor (1+ε), but is, instead, approximated up to an additive term bounded by ε times d·|S| + vol (S), where d is the average degree of the graph and vol (S) is the sum of the degrees of the vertices in S. We provide a probabilistic polynomial time construction of such sparsifiers for every graph, and our sparsifiers have a near-optimal number of edges O(ε-2npolylog (1/ε)). We also provide a deterministic polynomial time construction that constructs sparsifiers with a weaker property having the optimal number of edges O(ε-2n). Our constructions also satisfy a spectral version of the “additive sparsification'' property. Notions of sparsification have also been studied for hypergraphs. Our construction of “additive sparsifiers'' with Oε(n) edges also works for hypergraphs, and provides the first non-trivial notion of sparsification for hypergraphs achievable with O(n) hyperedges when ε and the rank r of the hyperedges are constant. Finally, we provide a new construction of spectral hypergraph sparsifiers, according to the standard definition, with poly (ε-1, r) · n log n hyperedges, improving over the previous spectral construction (Soma and Yoshida) that used Õ(n3) hyperedges even for constant r and ε.

The online matching problem was introduced by Karp, Vazirani and Vazirani nearly three decades ago. In that seminal work, they studied this problem in bipartite graphs with vertices arriving only on one side, and presented optimal deterministic and randomized algorithms for this setting. In comparison, more general arrival models, such as edge arrivals and general vertex arrivals, have proven more challenging and positive results are known only for various relaxations of the problem. In particular, even the basic question of whether randomization allows one to beat the trivially-optimal deterministic competitive ratio of 1/2 for either of these models was open. In this paper, we resolve this question for both these natural arrival models, and show the following.1) For edge arrivals, randomization does not help - no randomized algorithm is better than 1/2 competitive. 2)For general vertex arrivals, randomization helps - there exists a randomized (1/2+Ω(1)) -competitive online matching algorithm.

We design a generic method to reduce the task of finding weighted matchings to that of finding short augmenting paths in unweighted graphs. This method enables us to provide efficient implementations for approximating weighted matchings in the massively parallel computation (MPC) model and in the streaming model.

We consider the maximum bipartite matching problem in stochastic settings, namely the query-commit and price-of-information models. In the query-commit model, an edge e independently exists with probability pe. We can query whether an edge exists or not, but if it does exist, then we have to take it into our solution. In the unweighted case, one can query edges in the order given by the classical online algorithm of Karp, Vazirani, and Vazirani [20] to get a (1 – 1/e)-approximation. In contrast, the previously best known algorithm in the weighted case is the (1/2)-approximation achieved by the greedy algorithm that sorts the edges according to their weights and queries in that order. Improving upon the basic greedy, we give a (1 – 1/e)-approximation algorithm in the weighted query-commit model. We use a linear program (LP) to upper bound the optimum achieved by any strategy. The proposed LP admits several structural properties that play a crucial role in the design and analysis of our algorithm. We also extend these techniques to get a (1 – 1/e)-approximation algorithm for maximum bipartite matching in the price-of-information model introduced by Singla [25], who also used the basic greedy algorithm to give a (1/2)-approximation.

The traveling salesman problem is one of the most fundamental optimization problems. Given n cities and pairwise distances, it is the problem of finding a tour of minimum total distance that visits each city once. In spite of significant research efforts, current techniques seem insufficient for settling the approximability of the traveling salesman problem. The gap in our understanding is especially large in the general asymmetric setting where the distance from city i to j is not assumed to equal the distance from j to i. Indeed, until recently, it remained an open problem to design an algorithm with any constant approximation guarantee. This status is particularly intriguing as the standard linear programming relaxation is believed to give a constant-factor approximation algorithm, where the constant may in fact be as small as 2. In this talk, we will give an overview of old and new approaches for settling this question. We shall, in particular, talk about our new approach that gives the first constant-factor approximation algorithm for the asymmetric traveling salesman problem. Our approximation guarantee is analyzed with respect to the standard LP relaxation, and thus our result confirms the conjectured constant integrality gap of that relaxation. The main idea of our approach is to first give a generic reduction to structured instances and on those instances we then solve an easier problem (but equivalent in terms of constant-factor approximation) obtained by relaxing the general connectivity requirements into local connectivity conditions. This is based on joint work with Jakub Tarnawski and László A. Végh.

We study $k$-means clustering in a semi-supervised setting. Given an oracle that returns whether two given points belong to the same cluster in a fixed optimal clustering, we investigate the following question: how many oracle queries are sufficient to efficiently recover a clustering that, with probability at least $(1 - δ)$, simultaneously has a cost of at most $(1 + ε)$ times the optimal cost and an accuracy of at least $(1 - ε)$? We show how to achieve such a clustering on $n$ points with $O{((k^2 \log n) \cdot m{(Q, ε^4, δ/ (k\log n))})}$ oracle queries, when the $k$ clusters can be learned with an $ε'$ error and a failure probability $δ'$ using $m(Q, ε',δ')$ labeled samples in the supervised setting, where $Q$ is the set of candidate cluster centers. We show that $m(Q, ε', δ')$ is small both for $k$-means instances in Euclidean space and for those in finite metric spaces. We further show that, for the Euclidean $k$-means instances, we can avoid the dependency on $n$ in the query complexity at the expense of an increased dependency on $k$: specifically, we give a slightly more involved algorithm that uses $O(k^4/(ε^2 δ) + (k^{9}/ε^4) \log(1/δ) + k \cdot m(\mathbb{R}^r, ε^4/k, δ))$ oracle queries. We also show that the number of queries needed for $(1 - ε)$-accuracy in Euclidean $k$-means must linearly depend on the dimension of the underlying Euclidean space, and for finite metric space $k$-means, we show that it must at least be logarithmic in the number of candidate centers. This shows that our query complexities capture the right dependencies on the respective parameters.

Many tasks in machine learning and data mining, such as data diversification, non-parametric learning, kernel machines, clustering etc., require extracting a small but representative summary from a massive dataset. Often, such problems can be posed as maximizing a submodular set function subject to a cardinality constraint. We consider this question in the streaming setting, where elements arrive over time at a fast pace and thus we need to design an efficient, low-memory algorithm. One such method, proposed by Badanidiyuru et al. (2014), always finds a 0.5-approximate solution. Can this approximation factor be improved? We answer this question affirmatively by designing a new algorithm Salsa for streaming submodular maximization. It is the first low-memory, singlepass algorithm that improves the factor 0.5, under the natural assumption that elements arrive in a random order. We also show that this assumption is necessary, i.e., that there is no such algorithm with better than 0.5-approximation when elements arrive in arbitrary order. Our experiments demonstrate that Salsa significantly outperforms the state of the art in applications related to exemplar-based clustering, social graph analysis, and recommender systems.

The secretary problem became one of the most prominent online selection problems due to its numerous applications in online mechanism design. The task is to select a maximum weight subset of elements subject to given constraints, where elements arrive one-by-one in random order, revealing a weight upon arrival. The decision whether to select an element has to be taken immediately after its arrival. The different applications that map to the secretary problem ask for different constraint families to be handled. The most prominent ones are matroid constraints, which both capture many relevant settings and admit strongly competitive secretary algorithms. However, dealing with more involved constraints proved to be much more difficult, and strong algorithms are known only for a few specific settings. In this paper, we present a general framework for dealing with the secretary problem over the intersection of several matroids. This framework allows us to combine and exploit the large set of matroid secretary algorithms known in the literature. As one consequence, we get constant-competitive secretary algorithms over the intersection of any constant number of matroids whose corresponding (single-)matroid secretary problems are currently known to have a constant-competitive algorithm. Moreover, we show that our results extend to submodular objectives.

We give a constant-factor approximation algorithm for the asymmetric traveling salesman problem. Our approximation guarantee is analyzed with respect to the standard LP relaxation, and thus our result confirms the conjectured constant integrality gap of that relaxation.

Clustering is a classic topic in optimization with k-means being one of the most fundamental such problems. In the absence of any restrictions on the input, the best known algorithm for k-means with a provable guarantee is a simple local search heuristic yielding an approximation guarantee of 9 + ε, a ratio that is known to be tight with respect to such methods. We overcome this barrier by presenting a new primal-dual approach that allows us to (1) exploit the geometric structure of k-means and (2) to satisfy the hard constraint that at most k clusters are selected without deteriorating the approximation guarantee. Our main result is a 6.357-approximation algorithm with respect to the standard LP relaxation. Our techniques are quite general and we also show improved guarantees for the general version of k-means where the underlying metric is not required to be Euclidean and for k-median in Euclidean metrics.

We show that the perfect matching problem in general graphs is in Quasi-NC. That is, we give a deterministic parallel algorithm which runs in O(log3n) time on nO(log2 n)processors. The result is obtained by a derandomization of the Isolation Lemma for perfect matchings, which was introduced in the classic paper by Mulmuley, Vazirani and Vazirani [1987] to obtain a Randomized NC algorithm.Our proof extends the framework of Fenner, Gurjar and Thierauf [2016], who proved the analogous result in the special case of bipartite graphs. Compared to that setting, several new ingredients are needed due to the significantly more complex structure of perfect matchings in general graphs. In particular, our proof heavily relies on the laminar structure of the faces of the perfect matching polytope.

Initially developed for the min-knapsack problem, the knapsack cover inequalities are used in the current best relaxations for numerous combinatorial optimization problems of covering type. In spite of their widespread use, these inequalities yield linear programming (LP) relaxations of exponential size, over which it is not known how to optimize exactly in polynomial time. In this paper we address this issue and obtain LP relaxations of quasi-polynomial size that are at least as strong as that given by the knapsack cover inequalities. For the min-knapsack cover problem, our main result can be stated formally as follows: for any ∊ > 0, there is a (1/∊)O(1)nO(log n)-size LP relaxation with an integrality gap of at most 2 + ∊, where n is the number of items. Prior to this work, there was no known relaxation of subexponential size with a constant upper bound on the integrality gap. Our construction is inspired by a connection between extended formulations and monotone circuit complexity via Karchmer-Wigderson games. In particular, our LP is based on O (log2 n)-depth monotone circuits with fan-in 2 for evaluating weighted threshold functions with n inputs, as constructed by Beimel and Weinreb. We believe that a further understanding of this connection may lead to more positive results complementing the numerous lower bounds recently proved for extended formulations.

We consider the classic problem of scheduling jobs on unrelated machines so as to minimize the weighted sum of completion times. Recently, for a small constant ∊ > 0, Bansal et al. gave a (3/2 – ∊)-approximation algorithm improving upon the “natural” barrier of 3/2 which follows from independent randomized rounding. In simplified terms, their result is obtained by an enhancement of independent randomized rounding via strong negative correlation properties. In this work, we take a different approach and propose to use the same elegant rounding scheme for the weighted completion time objective as devised by Shmoys and Tardos for optimizing a linear function subject to makespan constraints. Our main result is a 1.21-approximation algorithm for the natural special case where the weight of a job is proportional to its processing time (specifically, all jobs have the same Smith ratio), which expresses the notion that each unit of work has the same weight. In addition, as a direct consequence of the rounding, our algorithm also achieves a bi-criteria 2-approximation for the makespan objective. Our technical contribution is a tight analysis of the expected cost of the solution compared to the one given by the Configuration-LP relaxation - we reduce this task to that of understanding certain worst-case instances which are simple to analyze.

Linear programming (LP) has played a key role in the study of algorithms for combinatorial optimization problems. In the field of approximation algorithms, this is well illustrated by the uncapacitated facility location problem. A variety of algorithmic methodologies, such as LP-rounding and the primal-dual method, have been applied to and evolved from algorithms for this problem. Unfortunately, this collection of powerful algorithmic techniques had not yet been applicable to the more general capacitated facility location problem. In fact, all of the known algorithms with good performance guarantees were based on a single technique, local search, and no LP relaxation was known to efficiently approximate the problem. In this paper, we present an LP relaxation with a constant integrality gap for the capacitated facility location. We demonstrate that the fundamental theories of multicommodity flows and matchings provide key insights that lead to the strong relaxation. Our algorithmic proof of integrality gap is obtained by finally accessing the rich toolbox of LP-based methodologies: we present a constant factor approximation algorithm based on LP-rounding.

The dynamic facility location problem is a generalization of the classic facility location problem proposed by Eisenstat, Mathieu, and Schabanel to model the dynamics of evolving social/infrastructure networks. The generalization lies in that the distance metric between clients and facilities changes over time. This leads to a trade-off between optimizing the classic objective function and the “stability” of the solution: There is a switching cost charged every time a client changes the facility to which it is connected. While the standard linear program (LP) relaxation for the classic problem naturally extends to this problem, traditional LP-rounding techniques do not, as they are often sensitive to small changes in the metric resulting in frequent switches. We present a new LP-rounding algorithm for facility location problems, which yields the first constant approximation algorithm for the dynamic facility location problem. Our algorithm installs competing exponential clocks on the clients and facilities and connects every client by the path that repeatedly follows the smallest clock in the neighborhood. The use of exponential clocks gives rise to several properties that distinguish our approach from previous LP roundings for facility location problems. In particular, we use no clustering and we allow clients to connect through paths of arbitrary lengths . In fact, the clustering-free nature of our algorithm is crucial for applying our LP-rounding approach to the dynamic problem.

We study the basic allocation problem of assigning resources to players to maximize fairness. This is one of the few natural problems that enjoys the intriguing status of having a better estimation algorithm than approximation algorithm. Indeed, a certain Configuration-LP can be used to estimate the value of the optimal allocation to within a factor of 4+ ε. In contrast, however, the best-known approximation algorithm for the problem has an unspecified large constant guarantee. In this article, we significantly narrow this gap by giving a 13-approximation algorithm for the problem. Our approach develops a local search technique introduced by Haxell [13] for hypergraph matchings and later used in this context by Asadpour, Feige, and Saberi [2]. For our local search procedure to terminate in polynomial time, we introduce several new ideas, such as lazy updates and greedy players . Besides the improved approximation guarantee, the highlight of our approach is that it is purely combinatorial and uses the Configuration-LP only in the analysis.

In this talk, we give an overview of the current best approximation algorithms for fundamental clustering problems, such as k-center, k-median, k-means, and facility location. We focus on recent progress and point out several important open problems. For the uncapacitated versions, a variety of algorithmic methodologies, such as LP-rounding and primal-dual method, have been applied to a standard linear programming relaxation. This has given a uniform way of addressing these problems resulting in small constant approximation guarantees. In spite of this impressive progress, it remains a challenging open problem to give tight guarantees. Moreover, this collection of powerful algorithmic techniques is not easily applicable to the capacitated setting. In fact, there is no simple strong convex relaxation known for the capacitated versions. As a result, our understanding of these problems is significantly weaker and several fundamental questions remain open.

Choosing a diverse subset of a large collection of points in a metric space is a fundamental problem, with applications in feature selection, recommender systems, web search, data summarization, etc. Various notions of diversity have been proposed, tailored to different applications. The general algorithmic goal is to find a subset of points that maximize diversity, while obeying a cardinality (or more generally, matroid) constraint. The goal of this paper is to develop a novel linear programming (LP) framework that allows us to design approximation algorithms for such problems. We study an objective known as {\em sum-min} diversity, which is known to be effective in many applications, and give the first constant factor approximation algorithm. Our LP framework allows us to easily incorporate additional constraints, as well as secondary objectives. We also prove a hardness result for two natural diversity objectives, under the so-called {\em planted clique} assumption. Finally, we study the empirical performance of our algorithm on several standard datasets. We first study the approximation quality of the algorithm by comparing with the LP objective. Then, we compare the quality of the solutions produced by our method with other popular diversity maximization algorithms.

We introduce a new rounding technique designed for online optimization problems, which is related to contention resolution schemes, a technique initially introduced in the context of submodular function maximization. Our rounding technique, which we call online contention resolution schemes (OCRSs), is applicable to many online selection problems, including Bayesian online selection, oblivious posted pricing mechanisms, and stochastic probing models. It allows for handling a wide set of constraints, and shares many strong properties of offline contention resolution schemes. In particular, OCRSs for different constraint families can be combined to obtain an OCRS for their intersection. Moreover, we can approximately maximize submodular functions in the online settings we consider. We, thus, get a broadly applicable framework for several online selection problems, which improves on previous approaches in terms of the types of constraints that can be handled, the objective functions that can be dealt with, and the assumptions on the strength of the adversary. Furthermore, we resolve two open problems from the literature; namely, we present the first constant-factor constrained oblivious posted price mechanism for matroid constraints, and the first constant-factor algorithm for weighted stochastic probing with deadlines.

We consider the problem of scheduling jobs on unrelated machines so as to minimize the sum of weighted completion times. Our main result is a (3/2-c)-approximation algorithm for some fixed c>0, improving upon the long-standing bound of 3/2. To do this, we first introduce a new lift-and-project based SDP relaxation for the problem. This is necessary as the previous convex programming relaxations have an integrality gap of 3/2. Second, we give a new general bipartite-rounding procedure that produces an assignment with certain strong negative correlation properties.

We present a framework for approximating the metric TSP based on a novel use of matchings. Traditionally, matchings have been used to add edges to make a given graph Eulerian, whereas our approach also allows for the removal of certain edges leading to a decreased cost. For the TSP on graphic metrics (graph-TSP), we show that the approach gives a 1.461-approximation algorithm with respect to the Held-Karp lower bound. For graph-TSP restricted either to half-integral solutions to the Held-Karp relaxation or to a class of graphs that contains subcubic and claw-free graphs, we show that the integrality gap of the Held-Karp relaxation matches the conjectured ratio 4/3. The framework also allows for generalizations in a natural way and leads to analogous results for the s , t -path traveling salesman problem on graphic metrics where the start and end vertices are prespecified.

We present a novel approximation algorithm for $k$-median that achieves an approximation guarantee of $1+\sqrt{3}+\epsilon$, improving upon the decade-old ratio of $3+\epsilon$. Our improved approximation ratio is achieved by exploiting the power of pseudo-approximation. More specifically, our approach is based on two components, each of which, we believe, is of independent interest. First, we show that in order to give an $\alpha$-approximation algorithm for $k$-median, it is sufficient to give a pseudo-approximation algorithm that finds an $\alpha$-approximate solution by opening $k+O(1)$ facilities. This is a rather surprising result as there exist instances for which opening $k+1$ facilities may lead to a significantly smaller cost than that of opening only $k$ facilities. Second, we give such a pseudo-approximation algorithm with $\alpha=1+\sqrt{3}+\epsilon$. Prior to our work, it was not even known whether opening $k+o(k)$ facilities would help improve the approximation ratio.

The restricted max-min fair allocation problem (also known as the restricted Santa Claus problem) is one of few problems that enjoys the intriguing status of having a better estimation algorithm than approximation algorithm. Indeed, Asadpour et al. [2012] proved that a certain configuration LP can be used to estimate the optimal value within a factor of 1/(4 + ϵ), for any ϵ > 0, but at the same time it is not known how to efficiently find a solution with a comparable performance guarantee. A natural question that arises from their work is if the difference between these guarantees is inherent or results from a lack of suitable techniques. We address this problem by giving a quasi-polynomial approximation algorithm with the mentioned performance guarantee. More specifically, we modify the local search of Asadpour et al. [2012] and provide a novel analysis that lets us significantly improve the bound on its running time: from 2 O ( n ) to n O (log n ) . Our techniques also have the interesting property that although we use the rather complex configuration LP in the analysis, we never actually solve it and therefore the resulting algorithm is purely combinatorial.

The vertex cover problem is one of the most important and intensively studied combinatorial optimization problems. Khot and Regev [30], [31] proved that the problem is NP-hard to approximate within a factor 2 - ε, assuming the Unique Games Conjecture (UGC). This is tight because the problem has an easy 2-approximation algorithm. Without resorting to the UGC, the best in approximability result for the problem is due to Dinur and Safra [16], [17]: vertex cover is NP-hard to approximate within a factor 1.3606. We prove the following unconditional result about linear programming (LP) relaxations of the problem: every LP relaxation that approximates vertex cover within a factor of 2 - ε has super-polynomially many inequalities. As a direct consequence of our methods, we also establish that LP relaxations (as well as SDP relaxations) that approximate the independent set problem within any constant factor have super-polynomially many inequalities.

The standard LP relaxation of the asymmetric traveling salesman problem has been conjectured to have a constant integrality gap in the metric case. We prove this conjecture when restricted to shortest path metrics of node-weighted digraphs. Our arguments are constructive and give a constant factor approximation algorithm for these metrics. We remark that the considered case is more general than the directed analog of the special case of the symmetric traveling salesman problem for which there were recent improvements on Christofides' algorithm. The main idea of our approach is to first consider an easier problem obtained by significantly relaxing the general connectivity requirements into local connectivity conditions. For this relaxed problem, it is quite easy to give an algorithm with a guarantee of 3 on node-weighted shortest path metrics. More surprisingly, we then show that any algorithm (irrespective of the metric) for the relaxed problem can be turned into an algorithm for the asymmetric traveling salesman problem by only losing a small constant factor in the performance guarantee. This leaves open the intriguing task of designing a "good" algorithm for the relaxed problem on general metrics.

The dynamic facility location problem is a generalization of the classic facility location problem proposed by Eisenstat, Mathieu, and Schabanel to model the dynamics of evolving social/infrastructure networks. The generalization lies in that the distance metric between clients and facilities changes over time. This leads to a trade-off between optimizing the classic objective function and the “stability” of the solution: there is a switching cost charged every time a client changes the facility to which it is connected. While the standard linear program (LP) relaxation for the classic problem naturally extends to this problem, traditional LP-rounding techniques do not, as they are often sensitive to small changes in the metric resulting in frequent switches. We present a new LP-rounding algorithm for facility location problems, which yields the first constant approximation algorithm for the dynamic facility location problem. Our algorithm installs competing exponential clocks on the clients and facilities, and connect every client by the path that repeatedly follows the smallest clock in the neighborhood. The use of exponential clocks gives rise to several properties that distinguish our approach from previous LP-roundings for facility location problems. In particular, we use no clustering and we allow clients to connect through paths of arbitrary lengths. In fact, the clustering-free nature of our algorithm is crucial for applying our LP-rounding approach to the dynamic problem.

We study the basic allocation problem of assigning resources to players so as to maximize fairness. This is one of the few natural problems that enjoys the intriguing status of having a better estimation algorithm than approximation algorithm. Indeed, a certain configuration-LP can be used to estimate the value of the optimal allocation to within a factor of 4 + ε. In contrast, however, the best known approximation algorithm for the problem has an unspecified large constant guarantee. In this paper we significantly narrow this gap by giving a 13-approximation algorithm for the problem. Our approach develops a local search technique introduced by Haxell [11] for hypergraph matchings, and later used in this context by Asadpour, Feige, and Saberi [1]. For our local search procedure to terminate in polynomial time, we introduce several new ideas such as lazy updates and greedy players. Besides the improved approximation guarantee, the highlight of our approach is that it is purely combinatorial and uses the configuration-LP only in the analysis.

Only recently progress has been made in obtaining o(log(rank))-competitive algorithms for the matroid secretary problem. More precisely, Chakraborty and Lachish (2012) presented a -competitive procedure, and Lachish (2014) recently presented a O(log log(rank))-competitive algorithm. Both algorithms are involved with complex analyses. Using different tools, we present a considerably simpler O(log log(rank))-competitive algorithm. Our algorithm can be interpreted as a distribution over a simple type of matroid secretary algorithms which are easy to analyze. We are also able to vastly improve on the hidden constant in the competitive ratio.

Linear programming has played a key role in the study of algorithms for combinatorial optimization problems. In the field of approximation algorithms, this is well illustrated by the uncapacitated facility location problem. A variety of algorithmic methodologies, such as LP-rounding and primal-dual method, have been applied to and evolved from algorithms for this problem. Unfortunately, this collection of powerful algorithmic techniques had not yet been applicable to the more general capacitated facility location problem. In fact, all of the known algorithms with good performance guarantees were based on a single technique, local search, and no linear programming relaxation was known to efficiently approximate the problem. In this paper, we present a linear programming relaxation with constant integrality gap for capacitated facility location. We demonstrate that the fundamental theories of multi-commodity flows and matchings provide key insights that lead to the strong relaxation. Our algorithmic proof of integrality gap is obtained by finally accessing the rich toolbox of LP-based methodologies: we present a constant factor approximation algorithm based on LP-rounding.

We present a novel approximation algorithm for k-median that achieves an approximation guarantee of 1+√3+ε, improving upon the decade-old ratio of 3+ε. Our approach is based on two components, each of which, we believe, is of independent interest. First, we show that in order to give an α-approximation algorithm for k-median, it is sufficient to give a pseudo-approximation algorithm that finds an α-approximate solution by opening k+O(1) facilities. This is a rather surprising result as there exist instances for which opening k+1 facilities may lead to a significant smaller cost than if only k facilities were opened.

One of the classic results in scheduling theory is the $2$-approximation algorithm by Lenstra, Shmoys, and Tardos for the problem of scheduling jobs to minimize makespan on unrelated machines; i.e., job $j$ requires time $p_{ij}$ if processed on machine $i$. More than two decades after its introduction it is still the algorithm of choice even in the restricted model where processing times are of the form $p_{ij} \in \{p_j, \infty\}$. This problem, also known as the restricted assignment problem, is NP-hard to approximate within a factor less than $1.5$, which is also the best known lower bound for the general version. Our main result is a polynomial time algorithm that estimates the optimal makespan of the restricted assignment problem within a factor $33/17 + \epsilon \approx 1.9412 + \epsilon$, where $\epsilon > 0$ is an arbitrarily small constant. The result is obtained by upper bounding the integrality gap of a certain strong linear program, known as the configuration LP, that was previously successfully used for the related Santa Claus problem. Similar to the strongest analysis for that problem our proof is based on a local search algorithm that will eventually find a schedule of the mentioned approximation guarantee but is not known to converge in polynomial time.

We study two closely related problems in nonpreemptive scheduling of jobs on identical parallel machines. In these two settings there are either fixed jobs or nonavailability intervals during which the machines are not available; in both cases, the objective is to minimize the makespan. Both formulations have different applications, for example, in turnaround scheduling or overlay computing. For both problems we contribute approximation algorithms with an improved ratio of 3/2. For scheduling with fixed jobs, a lower bound of 3/2 on the approximation ratio has been obtained by Scharbrodt et al. [1999]; for scheduling with nonavailability we provide the same lower bound. We use dual approximation, creation of a gap structure, and a PTAS for the multiple subset sum problem, combined with a postprocessing step to assign large jobs.

We present a framework for approximating the metric TSP based on a novel use of matchings. Traditionally, matchings have been used to add edges in order to make a given graph Eulerian, whereas our approach also allows for the removal of certain edges leading to a decreased cost. For the TSP on graphic metrics (graph-TSP), the approach yields a 1.461-approximation algorithm with respect to the Held-Karp lower bound. For graph-TSP restricted to a class of graphs that contains degree three bounded and claw-free graphs, we show that the integrality gap of the Held-Karp relaxation matches the conjectured ratio 4/3. The framework allows for generalizations in a natural way and also leads to a 1.586-approximation algorithm for the traveling salesman path problem on graphic metrics where the start and end vertices are prespecified.

One of the classic results in scheduling theory is the 2-approximation algorithm by Lenstra, Shmoys, and Tardos for the problem of scheduling jobs to minimize makespan on unrelated machines, i.e., job j requires time pij if processed on machine i. More than two decades after its introduction it is still the algorithm of choice even in the restricted model where processing times are of the form pij ∈ pj, ∞. This problem, also known as the restricted assignment problem, is NP-hard to approximate within a factor less than 1.5 which is also the best known lower bound for the general version.

We consider several variants of the job shop problem that is a fundamental and classical problem in scheduling. The currently best approximation algorithms have worse than logarithmic performance guarantee, but the only previously known inapproximability result says that it is NP-hard to approximate job shops within a factor less than 5/4. Closing this big approximability gap is a well-known and long-standing open problem. This article closes many gaps in our understanding of the hardness of this problem and answers several open questions in the literature. It is shown the first nonconstant inapproximability result that almost matches the best-known approximation algorithm for acyclic job shops. The same bounds hold for the general version of flow shops, where jobs are not required to be processed on each machine. Similar inapproximability results are obtained when the objective is to minimize the sum of completion times. It is also shown that the problem with two machines and the preemptive variant with three machines have no PTAS.

We consider the Minimum Linear Arrangement problem and the (Uniform) Sparsest Cut problem. So far, these two notorious NP-hard graph problems have resisted all attempts to prove inapproximability results. We show that they have no polynomial time approximation scheme, unless NP-complete problems can be solved in randomized subexponential time. Furthermore, we show that the same techniques can be used for the Maximum Edge Biclique problem, for which we obtain a hardness factor similar to previous results but under a more standard assumption.

In 1966, Graham showed that a simple procedure called list scheduling yields a 2-approximation algorithm for the central problem of scheduling precedence constrained jobs on identical machines to minimize makespan. To date this has remained the best algorithm, and whether it can or cannot be improved has become a major open problem in scheduling theory. We address this problem by establishing a quite surprising relation between the approximability of the considered problem and that of scheduling precedence constrained jobs on a single machine to minimize weighted completion time. More specifically, we prove that if the single machine problem is hard to approximate within a factor of $2-\epsilon$, then the considered parallel machine problem, even in the case of unit processing times, is hard to approximate within a factor of $2-\zeta$, where $\zeta$ tends to 0 as $\epsilon$ tends to 0. Combining this with Bansal and Khot's recent hardness result for the single machine problem gives that it is NP-hard to improve upon the approximation ratio obtained by Graham, assuming a new variant of the unique games conjecture.

Already in 1966, Graham showed that a simple procedure called list scheduling yields a 2-approximation algorithm for the central problem of scheduling precedence constrained jobs on identical machines to minimize makespan. Till this date it has remained the best algorithm and whether it can or cannot be improved has become a major open problem in scheduling theory.

For every euro > 0, we show that the (acyclic) job shop problem cannot be approximated within ratio O(log1+eurolb), unless NP has quasi-polynomial Las-Vegas algorithms, and where lb denotes a trivial lower bound on the optimal value. This almost matches the best known results for acyclic job shops, since an O(log1+eurolb)-approximate solution can be obtained in polynomial time for every euro > 0. Recently, a PTAS was given for the job shop problem, where the number of machines and the number of operations per job are assumed to be constant. Under P ne NP, and when the number mu of operations per job is a constant, we provide an inapproximability result whose value grows with mu to infinity. Moreover, we show that the problem with two machines and the preemptive variant with three machines have no PTAS, unless NP has quasi-polynomial algorithms. These results show that the restrictions on the number of machines and operations per job are necessary to obtain a PTAS.In summary, the presented results close many gaps in our understanding of the hardness of the job shop problem and resolve (negatively) several open problems in the literature.

We consider (uniform) sparsest cut, optimal linear arrangement and the precedence constrained scheduling problem 1 |prec| SigmawjCj-So far, these three notorious NP-hard problems have resisted all attempts to prove inapproximability results. We show that they have no polynomial time approximation scheme (PTAS), unless NP-complete pmblems can be solved in randomized subexponential time. Furthermore, we prove that the scheduling problem is as-hard to approximate as vertex cover when the so-called fixed cost, that is present in all feasible solutions, is subtracted from the objective function.