We present a new Correlation Clustering algorithm for a dynamic setting where nodes are added one at a time. In this model, proposed by Cohen-Addad, Lattanzi, Maggiori, and Parotsidis (ICML 2024), the algorithm uses database queries to access the input graph and updates the clustering as each new node is added. Our algorithm has the amortized update time of $\log^{O(1)}(n)$. Its approximation factor is $20+\varepsilon$, which is a substantial improvement over the approximation factor of the algorithm by Cohen-Addad et al. We complement our theoretical findings by empirically evaluating the approximation guarantee of our algorithm. The results show that it outperforms the algorithm by Cohen-Addad et al.~in practice.

We initiate the study of algorithms for constraint satisfaction problems with ML oracle advice. We introduce two models of advice and then design approximation algorithms for Max Cut, Max 2-Lin, and Max 3-Lin in these models. In particular, we show the following.

Motivated by applications to classification problems on metric data, we study Weighted Metric Clustering problem: given a metric d over n points and a k x k symmetric matrix A with non-negative entries, the goal is to find a k-partition of these points into clusters C1,...,Ck, while minimizing the sum of A[i,j] * d(u,v) over all pairs of clusters Ci and Cj and all pairs of points u from Ci and v from Cj. Specific choices of A lead to Weighted Metric Clustering capturing well-studied graph partitioning problems in metric spaces, such as Min-Uncut, Min-k-Sum, Min-k-Cut, and more. Our main result is that Weighted Metric Clustering admits a polynomial-time approximation scheme (PTAS). Our algorithm handles all the above problems using the Sherali-Adams linear programming relaxation. This subsumes several prior works, unifies many of the techniques for various metric clustering objectives, and yields a PTAS for several new problems, including metric clustering on manifolds and a new family of hierarchical clustering objectives. Our experiments on the hierarchical clustering objective show that it better captures the ground-truth structural information compared to the popular Dasgupta's objective.

Given a graph with positive and negative edge labels, the correlation clustering problem aims to cluster the nodes so to minimize the total number of between-cluster positive and within-cluster negative edges. This problem has many applications in data mining, particularly in unsupervised learning. Inspired by the prevalence of large graphs and constantly changing data in modern applications, we study correlation clustering in dynamic, parallel (MPC), and local computation (LCA) settings. We design an approach that improves state-of-the-art runtime complexities in all these settings. In particular, we provide the first fully dynamic algorithm that runs in an expected amortized constant time, without any dependence on the graph size. Moreover, our algorithm essentially matches the approximation guarantee of the celebrated Pivot algorithm.

We prove a new generalization of the higher-order Cheeger inequality for partitioning with buffers. Consider a graph G = (V, E). The buffered expansion of a set S ⊆ V with a buffer B ⊆ V \ S is the edge expansion of S after removing all the edges from set S to its buffer B. An ɛ-buffered k-partitioning is a partitioning of a graph into disjoint components Pi and buffers Bi, in which the size of buffer Bi for Pi is small relative to the size of Pi: |Bi| ≤ ɛ|Pi|. The buffered expansion of a buffered partition is the maximum of buffered expansions of the k sets Pi with buffers Bi. Let be the buffered expansion of the optimal ɛ-buffered k-partitioning, then for every δ > 0,

We consider variants of the classic Multiway Cut problem. Multiway Cut asks to partition a graph $G$ into $k$ parts so as to separate $k$ given terminals. Recently, Chandrasekaran and Wang (ESA 2021) introduced $\ell_p$-norm Multiway, a generalization of the problem, in which the goal is to minimize the $\ell_p$ norm of the edge boundaries of $k$ parts. We provide an $O(\log^{1/2} n\log^{1/2+1/p} k)$ approximation algorithm for this problem, improving upon the approximation guarantee of $O(\log^{3/2} n \log^{1/2} k)$ due to Chandrasekaran and Wang. We also introduce and study Norm Multiway Cut, a further generalization of Multiway Cut. We assume that we are given access to an oracle, which answers certain queries about the norm. We present an $O(\log^{1/2} n \log^{7/2} k)$ approximation algorithm with a weaker oracle and an $O(\log^{1/2} n \log^{5/2} k)$ approximation algorithm with a stronger oracle. Additionally, we show that without any oracle access, there is no $n^{1/4-\varepsilon}$ approximation algorithm for every $\varepsilon > 0$ assuming the Hypergraph Dense-vs-Random Conjecture.

We study the natural problem of Triplet Reconstruction (also known as Rooted Triplets Consistency or Triplet Clustering), originally motivated by applications in computational biology and relational databases (Aho, Sagiv, Szymanski, and Ullman, 1981) [2]: given n datapoints, we want to embed them onto the n leaves of a rooted binary tree (also known as a hierarchical clustering, or ultrametric embedding) such that a given set of m triplet constraints is satisfied. A triplet constraint $i j \mid k$ for points $i, j, k$ indicates that “$i, j$ are more closely related to each other than to k,” (in terms of distances $d(i, j) \leq d(i, k)$ and $d(i, j) \leq d(j, k)$) and we say that a tree satisfies the triplet $i j \mid k$ if the distance in the tree between $i, j$ is smaller than the distance between $i, k$ (or $j, k$). Among all possible trees with n leaves, can we efficiently find one that satisfies a large fraction of the m given triplets?Aho et al. (1981) [2] studied the decision version and gave an elegant polynomial-time algorithm that determines whether or not there exists a tree that satisfies all of the m constraints. Moreover, it is straightforward to see that a random binary tree achieves a constant $\frac{1}{3}$-approximation, since there are only 3 distinct triplets $i j|k, i k| j, j k \mid i$ (each will be satisfied w.p. $\frac{1}{3}$). Unfortunately, despite more than four decades of research by various communities, there is no better approximation algorithm for this basic Triplet Reconstruction problem.Our main theorem-which captures Triplet Reconstruction as a special case-is a general hardness of approximation result about Constraint Satisfaction Problems (CSPs) over infinite domains (CSPs where instead of boolean values $\{0,1\}$ or a fixed-size domain, the variables can be mapped to any of the n leaves of a tree). Specifically, we prove that assuming the Unique Games Conjecture [57], Triplet Reconstruction and more generally, every Constraint Satisfaction Problem (CSP) over hierarchies is approximation resistant, i.e., there is no polynomial-time algorithm that does asymptotically better than a biased random assignment.Our result settles the approximability not only for Triplet Reconstruction, but for many interesting problems that have been studied by various scientific communities such as the popular Quartet Reconstruction and Subtree/Supertree Aggregation Problems. More broadly, our result significantly extends the list of approximation resistant predicates by pointing to a large new family of hard problems over hierarchies. Our main theorem is a generalization of Guruswami, Håstad, Manokaran, Raghavendra, and Charikar (2011) [36], who showed that ordering CSPs (CSPs over permutations of n elements, e.g., Max Acyclic Subgraph, Betweenness, Non-Betweenness) are approximation resistant. The main challenge in our analyses stems from the fact that trees have topology (in contrast to permutations and ordering CSPs) and it is the tree topology that determines whether a given constraint on the variables is satisfied or not. As a byproduct, we also present some of the first CSPs where their approximation resistance is proved against biased random assignments, instead of uniformly random assignments.

We show that a simple single-pass semi-streaming variant of the Pivot algorithm for Correlation Clustering gives a (3+eps)-approximation using O(n/eps) words of memory. This is a slight improvement over the recent results of Cambus, Kuhn, Lindy, Pai, and Uitto, who gave a (3+eps)-approximation using O(n log n) words of memory, and Behnezhad, Charikar, Ma, and Tan, who gave a 5-approximation using O(n) words of memory. One of the main contributions of our paper is that the algorithm and its analysis are simple and easy to understand.

We show that the RandomCoordinateCut algorithm gives the optimal competitive ratio for explainable $k$-medians in $\ell_1$. The problem of explainable $k$-medians was introduced by Dasgupta, Frost, Moshkovitz, and Rashtchian in 2020. Several groups of authors independently proposed a simple polynomial-time randomized algorithm for the problem and showed that this algorithm is $O(\log k \log\log k)$ competitive. We provide a tight analysis of the algorithm and prove that its competitive ratio is upper bounded by $2\ln k+2$. This bound matches the $\Omega(\log k)$ lower bound by Dasgupta et al (2020).

We provide a new bi-criteria Õ(log2 k) competitive algorithm for explainable k-means clustering. Explainable k-means was recently introduced by Dasgupta, Frost, Moshkovitz, and Rashtchian (ICML 2020). It is described by an easy to interpret and understand (threshold) decision tree or diagram. The cost of the explainable k-means clustering equals to the sum of costs of its clusters; and the cost of each cluster equals the sum of squared distances from the points in the cluster to the center of that cluster. The best non bi-criteria algorithm for explainable clustering Õ(k) competitive, and this bound is tight.

Large pools of synthetic DNA molecules have been recently used to reliably store significant volumes of digital data. While DNA as a storage medium has enormous potential because of its high storage density, its practical use is currently severely limited because of the high cost and low throughput of available DNA synthesis technologies. We study the role of batch optimization in reducing the cost of large scale DNA synthesis, which translates to the following algorithmic task. Given a large pool$\mathcal {S}$of random quaternary strings of fixed length, partition$\mathcal {S}$into batches in a way that minimizes the sum of the lengths of the shortest common supersequences across batches. We introduce two ideas for batch optimization that both improve (in different ways) upon a naive baseline: (1) using both$(ACGT)^{\ast}$and its reverse$(TGCA)^{\ast}$as reference strands, and batching appropriately, and (2) batching via the quantiles of an appropriate ordering of the strands. We also prove asymptotically matching lower bounds on the cost of DNA synthesis, showing that one cannot improve upon these two ideas. Our results uncover a surprising separation between two cases that naturally arise in the context of DNA data storage: the asymptotic cost savings of batch optimization are significantly greater in the case where strings in$\mathcal {S}$do not contain repeats of the same character (homopolymers), as compared to the case where strings in$\mathcal {S}$are unconstrained.

In this paper, we prove a two-sided variant of the Kirszbraun theorem. Consider an arbitrary subset X of Euclidean space and its superset Y. Let f be a 1-Lipschitz map from X to ℝ^m. The Kirszbraun theorem states that the map f can be extended to a 1-Lipschitz map ̃ f from Y to ℝ^m. While the extension ̃ f does not increase distances between points, there is no guarantee that it does not decrease distances significantly. In fact, ̃ f may even map distinct points to the same point (that is, it can infinitely decrease some distances). However, we prove that there exists a (1 + ε)-Lipschitz outer extension f̃:Y → ℝ^{m'} that does not decrease distances more than "necessary". Namely, ‖f̃(x) - f̃(y)‖ ≥ c √{ε} min(‖x-y‖, inf_{a,b ∈ X} (‖x - a‖ + ‖f(a) - f(b)‖ + ‖b-y‖)) for some absolutely constant c > 0. This bound is asymptotically optimal, since no L-Lipschitz extension g can have ‖g(x) - g(y)‖ > L min(‖x-y‖, inf_{a,b ∈ X} (‖x - a‖ + ‖f(a) - f(b)‖ + ‖b-y‖)) even for a single pair of points x and y. In some applications, one is interested in the distances ‖f̃(x) - f̃(y)‖ between images of points x,y ∈ Y rather than in the map f̃ itself. The standard Kirszbraun theorem does not provide any method of computing these distances without computing the entire map ̃ f first. In contrast, our theorem provides a simple approximate formula for distances ‖f̃(x) - f̃(y)‖.

In the Correlation Clustering problem, we are given a complete weighted graph $G$ with its edges labeled as “similar" and “dissimilar" by a noisy binary classifier. For a clustering $\mathcal{C}$ of graph $G$, a similar edge is in disagreement with $\mathcal{C}$, if its endpoints belong to distinct clusters; and a dissimilar edge is in disagreement with $\mathcal{C}$ if its endpoints belong to the same cluster. The disagreements vector, $\disagree$, is a vector indexed by the vertices of $G$ such that the $v$-th coordinate $\disagree_v$ equals the weight of all disagreeing edges incident on $v$. The goal is to produce a clustering that minimizes the $\ell_p$ norm of the disagreements vector for $p\geq 1$. We study the $\ell_p$ objective in Correlation Clustering under the following assumption: Every similar edge has weight in $[\alpha\mathbf{w},\mathbf{w}]$ and every dissimilar edge has weight at least $\alpha\mathbf{w}$ (where $\alpha \leq 1$ and $\mathbf{w}>0$ is a scaling parameter). We give an $O\left((\nicefrac{1}{\alpha})^{\nicefrac{1}{2}-\nicefrac{1}{2p}}\cdot \log\nicefrac{1}{\alpha}\right)$ approximation algorithm for this problem. Furthermore, we show an almost matching convex programming integrality gap.

We consider the problem of explainable $k$-medians and $k$-means introduced by Dasgupta, Frost, Moshkovitz, and Rashtchian (ICML 2020). In this problem, our goal is to find a \emph{threshold decision tree} that partitions data into $k$ clusters and minimizes the $k$-medians or $k$-means objective. The obtained clustering is easy to interpret because every decision node of a threshold tree splits data based on a single feature into two groups. We propose a new algorithm for this problem which is $\tilde O(\log k)$ competitive with $k$-medians with $\ell_1$ norm and $\tilde O(k)$ competitive with $k$-means. This is an improvement over the previous guarantees of $O(k)$ and $O(k^2)$ by Dasgupta et al (2020). We also provide a new algorithm which is $O(\log^{\nicefrac{3}{2}} k)$ competitive for $k$-medians with $\ell_2$ norm. Our first algorithm is near-optimal: Dasgupta et al (2020) showed a lower bound of $\Omega(\log k)$ for $k$-medians; in this work, we prove a lower bound of $\tilde\Omega(k)$ for $k$-means. We also provide a lower bound of $\Omega(\log k)$ for $k$-medians with $\ell_2$ norm.

Large pools of synthetic DNA molecules have been recently used to reliably store significant volumes of digital data. While DNA as a storage medium has enormous potential because of its high storage density, its practical use is currently severely limited because of the high cost and low throughput of available DNA synthesis technologies. We study the role of batch optimization in reducing the cost of large scale DNA synthesis, which translates to the following algorithmic task. Given a large pool$S$of random quaternary strings of fixed length, partition$S$into batches in a way that minimizes the sum of the lengths of the shortest common supersequences across batches. We introduce two ideas for batch optimization that both improve (in different ways) upon a naive baseline: (1) using both (ACGT)* and its reverse (TGCA)* as reference strands, and batching appropriately, and (2) batching via the quantiles of an appropriate ordering of the strands. We also prove asymptotically matching lower bounds on the cost of DNA synthesis, showing that one cannot improve upon these two ideas. Our results uncover a surprising separation between two cases that naturally arise in the context of DNA data storage: the asymptotic cost savings of batch optimization are significantly greater in the case where strings in$S$do not contain repeats of the same character (homopolymers), as compared to the case where strings in$S$are unconstrained. A full version of this paper is accessible at: https://arxiv.org/abs/2011.14532

Hierarchical Clustering is an unsupervised data analysis method which has been widely used for decades. Despite its popularity, it had an underdeveloped analytical foundation and to address this, Dasgupta recently introduced an optimization viewpoint of hierarchical clustering with pairwise similarity information that spurred a line of work shedding light on old algorithms (e.g., Average-Linkage), but also designing new algorithms. Here, for the maximization dual of Dasgupta’s objective (introduced by Moseley-Wang), we present polynomial-time 42.46% approximation algorithms that use Max-Uncut Bisection as a subroutine. The previous best worst-case approximation factor in polynomial time was 33.6%, improving only slightly over Average-Linkage which achieves 33.3%. Finally, we complement our positive results by providing APX-hardness (even for 0-1 similarities), under the Small Set Expansion hypothesis.

In the Correlation Clustering problem, we are given a weighted graph $G$ with its edges labelled as "similar" or "dissimilar" by a binary classifier. The goal is to produce a clustering that minimizes the weight of "disagreements": the sum of the weights of "similar" edges across clusters and "dissimilar" edges within clusters. We study the correlation clustering problem under the following assumption: Every "similar" edge $e$ has weight $w_e \in [ \alpha w, w ]$ and every "dissimilar" edge $e$ has weight $w_e \geq \alpha w$ (where $\alpha \leq 1$ and $w > 0$ is a scaling parameter). We give a $(3 + 2 \log_e (1/\alpha))$ approximation algorithm for this problem. This assumption captures well the scenario when classification errors are asymmetric. Additionally, we show an asymptotically matching Linear Programming integrality gap of $\Omega(\log 1/\alpha)$.

In this paper, we introduce the notion of a certified algorithm. Certified algorithms provide worst-case and beyond-worst-case performance guarantees. First, a γ-certified algorithm is also a γ-approximation algorithm - it finds a γ-approximation no matter what the input is. Second, it exactly solves γ-perturbation-resilient instances (γ-perturbation-resilient instances model real-life instances). Additionally, certified algorithms have a number of other desirable properties: they solve both maximization and minimization versions of a problem (e.g. Max Cut and Min Uncut), solve weakly perturbation-resilient instances, and solve optimization problems with hard constraints. In the paper, we define certified algorithms, describe their properties, present a framework for designing certified algorithms, provide examples of certified algorithms for Max Cut/Min Uncut, Minimum Multiway Cut, k-medians and k-means. We also present some negative results.

In this paper, we study k-means++ and k-means||, the two most popular algorithms for the classic k-means clustering problem. We provide novel analyses and show improved approximation and bi-criteria approximation guarantees for k-means++ and k-means||. Our results give a better theoretical justification for why these algorithms perform extremely well in practice.

Correlation Clustering is a powerful graph partitioning model that aims to cluster items based on the notion of similarity between items. An instance of the Correlation Clustering problem consists of a graph G (not necessarily complete) whose edges are labeled by a binary classifier as similar and dissimilar. Classically, we are tasked with producing a clustering that minimizes the number of disagreements: an edge is in disagreement if it is a similar edge and is present across clusters or if it is a dissimilar edge and is present within a cluster. Define the disagreements vector to be an n dimensional vector indexed by the vertices, where the v-th index is the number of disagreements at vertex v. Recently, Puleo and Milenkovic (ICML '16) initiated the study of the Correlation Clustering framework in which the objectives were more general functions of the disagreements vector. In this paper, we study algorithms for minimizing \ellq norms (q >= 1) of the disagreements vector for both arbitrary and complete graphs. We present the first known algorithm for minimizing the \ellq norm of the disagreements vector on arbitrary graphs and also provide an improved algorithm for minimizing the \ell_q norm (q >= 1) of the disagreements vector on complete graphs. We also study an alternate cluster-wise local objective introduced by Ahmadi, Khuller and Saha (IPCO '19), which aims to minimize the maximum number of disagreements associated with a cluster. We present an improved (2 + \eps) approximation algorithm for this objective.

Consider an instance of Euclidean k-means or k-medians clustering. We show that the cost of the optimal solution is preserved up to a factor of (1+ε) under a projection onto a random O(log(k /ε) / ε2)-dimensional subspace. Further, the cost of every clustering is preserved within (1+ε). More generally, our result applies to any dimension reduction map satisfying a mild sub-Gaussian-tail condition. Our bound on the dimension is nearly optimal. Additionally, our result applies to Euclidean k-clustering with the distances raised to the p-th power for any constant p.

An instance of the constraint satisfaction problem (CSP) is given by a family of constraints on overlapping sets of variables, and the goal is to assign values from a fixed domain to the variables so that all constraints are satisfied. In the optimization version, the goal is to maximize the number of satisfied constraints. An approximation algorithm for a CSP is called robust if it outputs an assignment satisfying an $(1-g(\varepsilon))$-fraction of constraints on any $(1-\varepsilon)$-satisfiable instance, where the loss function $g$ is such that $g(\varepsilon)\rightarrow 0$ as $\varepsilon\rightarrow 0$. We study how the robust approximability of CSPs depends on the set of constraint relations allowed in instances, the so-called constraint language. All constraint languages admitting a robust polynomial-time algorithm (with some $g$) have been characterized by Barto and Kozik, with the general bound on the loss $g$ being doubly exponential, specifically $g(\varepsilon)=O((\log\log(1/\varepsilon))/\log(1/\varepsilon))$. It is natural to ask when a better loss can be achieved, in particular polynomial loss $g(\varepsilon)=O(\varepsilon^{1/k})$ for some constant $k$. In this paper, we consider CSPs with a constraint language having a near-unanimity polymorphism. This general condition almost matches a known necessary condition for having a robust algorithm with polynomial loss. We give two randomized robust algorithms with polynomial loss for such CSPs: one works for any near-unanimity polymorphism and the parameter $k$ in the loss depends on the size of the domain and the arity of the relations in $\Gamma$, while the other works for a special ternary near-unanimity operation called the dual discriminator with $k=2$ for any domain size. In the latter case, the CSP is a common generalization of Unique Games with a fixed domain and 2-Sat. In the former case, we use the algebraic approach to the CSP. Both cases use the standard semidefinite programming relaxation for the CSP.

Consider an instance of Euclidean $k$-means or $k$-medians clustering. We show that the cost of the optimal solution is preserved up to a factor of $(1+\varepsilon)$ under a projection onto a random $O(\log(k / \varepsilon) / \varepsilon^2)$-dimensional subspace. Further, the cost of every clustering is preserved within $(1+\varepsilon)$. More generally, our result applies to any dimension reduction map satisfying a mild sub-Gaussian-tail condition. Our bound on the dimension is nearly optimal. Additionally, our result applies to Euclidean $k$-clustering with the distances raised to the $p$th power for any constant $p$. For $k$-means, our result resolves an open problem posed by Cohen et al. [ STOC 2015, ACM, New York, 2015, pp. 163--172] for $k$-medians, it answers a question raised by Kannan.

We introduce and study the notion of *an outer bi-Lipschitz extension* of a map between Euclidean spaces. The notion is a natural analogue of the notion of *a Lipschitz extension* of a Lipschitz map. We show that for every map f there exists an outer bi-Lipschitz extension f′ whose distortion is greater than that of f by at most a constant factor. This result can be seen as a counterpart of the classic Kirszbraun theorem for outer bi-Lipschitz extensions. We also study outer bi-Lipschitz extensions of near-isometric maps and show upper and lower bounds for them. Then, we present applications of our results to prioritized and terminal dimension reduction problems, described next.

We present a new framework for solving optimization problems with a diseconomy of scale. In such problems, our goal is to minimize the cost of resources used to perform a certain task. The cost of resources grows superlinearly, as x q , q ≥ 1, with the amount x of resources used. We define a novel linear programming relaxation for such problems and then show that the integrality gap of the relaxation is A q , where A q is the q -th moment of the Poisson random variable with parameter 1. Using our framework, we obtain approximation algorithms for the Minimum Energy Efficient Routing, Minimum Degree Balanced Spanning Tree, Load Balancing on Unrelated Parallel Machines, and Unrelated Parallel Machine Scheduling with Nonlinear Functions of Completion Times problems. Our analysis relies on the decoupling inequality for nonnegative random variables. The inequality states that ║∑ i =1 n X i ║ q ≤ C q ║∑ i =1 n Y i ║ q , where X i are independent nonnegative random variables, Y i are possibly dependent nonnegative random variables, and each Y i has the same distribution as X i . The inequality was proved by de la Peña in 1990. De la Peña, Ibragimov, and Sharakhmetov showed that C q ≤ 2 for q ∈(1,2) and C q ≤ A q 1/ q for q ≥ 2. We show that the optimal constant is C q = A q 1/ q for any q ≥ 1. We then prove a more general inequality: For every convex function φ, E[φ(∑ i =1 n X i )] ≤ E[φ ( P ∑ i =1 n Y i )], and, for every concave function ψ, E[ψ (∑ i =1 n X i )] ≥ E[ψ(P∑ i =1 n Y i )], where P is a Poisson random variable with parameter 1 independent of the random variables Y i .

Storing data in synthetic DNA offers the possibility of improving information density and durability by several orders of magnitude compared to current storage technologies. However, DNA data storage requires a computationally intensive process to retrieve the data. In particular, a crucial step in the data retrieval pipeline involves clustering billions of strings with respect to edit distance. Datasets in this domain have many notable properties, such as containing a very large number of small clusters that are well-separated in the edit distance metric space. In this regime, existing algorithms are unsuitable because of either their long running time or low accuracy. To address this issue, we present a novel distributed algorithm for approximately computing the underlying clusters. Our algorithm converges efficiently on any dataset that satisfies certain separability properties, such as those coming from DNA data storage systems. We also prove that, under these assumptions, our algorithm is robust to outliers and high levels of noise. We provide empirical justification of the accuracy, scalability, and convergence of our algorithm on real and synthetic data. Compared to the state-of-the-art algorithm for clustering DNA sequences, our algorithm simultaneously achieves higher accuracy and a 1000x speedup on three real datasets.

An instance of the Constraint Satisfaction Problem (CSP) is given by a family of constraints on overlapping sets of variables, and the goal is to assign values from a fixed domain to the variables so that all constraints are satisfied. In the optimization version, the goal is to maximize the number of satisfied constraints. An approximation algorithm for CSP is called robust if it outputs an assignment satisfying a (1 — g(∊))-fraction of constraints on any (1 — ∊)-satisfiable instance, where the loss function g is such that g(∊) → 0 as ∊ → 0. We study how the robust approximability of CSPs depends on the set of constraint relations allowed in instances, the so-called constraint language. All constraint languages admitting a robust polynomial-time algorithm (with some g) have been characterised by Barto and Kozik, with the general bound on the loss g being doubly exponential, specifically g(∊) = O((loglog(1/ ∊))/log(1/ ∊)). It is natural to ask when a better loss can be achieved: in particular, polynomial loss g(∊) = O(∊1/k) for some constant k. In this paper, we consider CSPs with a constraint language having a near- unanimity polymorphism. We give two randomized robust algorithms with polynomial loss for such CSPs: one works for any near-unanimity polymorphism and the parameter k in the loss depends on the size of the domain and the arity of the relations in Γ, while the other works for a special ternary near-unanimity operation called dual discriminator with k = 2 for any domain size. In the latter case, the CSP is a common generalisation of Unique Games with a fixed domain and 2-Sat. In the former case, we use the algebraic approach to the CSP. Both cases use the standard semidefinite programming relaxation for CSP.

We study the notion of stability and perturbation resilience introduced by Bilu and Linial (2010) and Awasthi, Blum, and Sheffet (2012). A combinatorial optimization problem is α-stable or α-perturbation-resilient if the optimal solution does not change when we perturb all parameters of the problem by a factor of at most α. In this paper, we give improved algorithms for stable instances of various clustering and combinatorial optimization problems. We also prove several hardness results.

We study the q -adic assignment problem. We first give an O ( n (q−1)/2) )-approximation algorithm for the Koopmans--Beckman version of the problem, improving upon the result of Barvinok. Then, we introduce a new family of instances satisfying “tensor triangle inequalities” and give a constant factor approximation algorithm for them. We show that many classical optimization problems can be modeled by q -adic assignment problems from this family. Finally, we give several integrality gap examples for the natural LP relaxations of the problem.

We consider the classical k-means clustering problem in the setting of bi-criteria approximation, in which an algorithm is allowed to output beta*k > k clusters, and must produce a clustering with cost at most alpha times the to the cost of the optimal set of k clusters. We argue that this approach is natural in many settings, for which the exact number of clusters is a priori unknown, or unimportant up to a constant factor. We give new bi-criteria approximation algorithms, based on linear programming and local search, respectively, which attain a guarantee alpha(beta) depending on the number beta*k of clusters that may be opened. Our guarantee alpha(beta) is always at most 9 + epsilon and improves rapidly with beta (for example: alpha(2) < 2.59, and alpha(3) < 1.4). Moreover, our algorithms have only polynomial dependence on the dimension of the input data, and so are applicable in high-dimensional settings.

We study the problem of learning communities in the presence of modeling errors and give robust recovery algorithms for the Stochastic Block Model (SBM). This model, which is also known as the Planted Partition Model, is widely used for community detection and graph partitioning in various fields, including machine learning, statistics, and social sciences. Many algorithms exist for learning communities in the Stochastic Block Model, but they do not work well in the presence of errors. In this paper, we initiate the study of robust algorithms for partial recovery in SBM with modeling errors or noise. We consider graphs generated according to the Stochastic Block Model and then modified by an adversary. We allow two types of adversarial errors, Feige—Kilian or monotone errors, and edge outlier errors. Mossel, Neeman and Sly (STOC 2015) posed an open question about whether an almost exact recovery is possible when the adversary is allowed to add o(n) edges. Our work answers this question affirmatively even in the case of k>2 communities. We then show that our algorithms work not only when the instances come from SBM, but also work when the instances come from any distribution of graphs that is \varepsilon m close to SBM in the Kullback—Leibler divergence. This result also works in the presence of adversarial errors. Finally, we present almost tight lower bounds for two communities.

In this paper, we propose and study a semi-random model for the Correlation Clustering problem on arbitrary graphs G. We give two approximation algorithms for Correlation Clustering instances from this model. The first algorithm finds a solution of value (1+ δ)\mathrmopt-cost + O_δ(n\log^3 n) with high probability, where \mathrmopt-cost is the value of the optimal solution (for every δ> 0). The second algorithm finds the ground truth clustering with an arbitrarily small classification error η(under some additional assumptions on the instance).

We study the satisfiability of ordering constraint satisfaction problems (CSPs) above average. We prove the conjecture of Gutin, van Iersel, Mnich, and Yeo that the satisfiability above average of ordering CSPs of arity k is fixed-parameter tractable for every k. Previously, this was only known for k=2 and k=3. We also generalize this result to more general classes of CSPs, including CSPs with predicates defined by linear equations. To obtain our results, we prove a new Bonami-type inequality for the Efron -- Stein decomposition. The inequality applies to functions defined on arbitrary product probability spaces. In contrast to other variants of the Bonami Inequality, it does not depend on the mass of the smallest atom in the probability space. We believe that this inequality is of independent interest.

To reduce the impact of network congestion on big data jobs, cluster management frameworks use various heuristics to schedule compute tasks and/or network flows. Most of these schedulers consider the job input data fixed and greedily schedule the tasks and flows that are ready to run. However, a large fraction of production jobs are recurring with predictable characteristics, which allows us to plan ahead for them. Coordinating the placement of data and tasks of these jobs allows for significantly improving their network locality and freeing up bandwidth, which can be used by other jobs running on the cluster. With this intuition, we develop Corral, a scheduling framework that uses characteristics of future workloads to determine an offline schedule which (i) jointly places data and compute to achieve better data locality, and (ii) isolates jobs both spatially (by scheduling them in different parts of the cluster) and temporally, improving their performance. We implement Corral on Apache Yarn, and evaluate it on a 210 machine cluster using production workloads. Compared to Yarn's capacity scheduler, Corral reduces the makespan of these workloads up to 33% and the median completion time up to 56%, with 20-90% reduction in data transferred across racks.

We give new rounding schemes for the standard linear programming relaxation of the correlation clustering problem, achieving approximation factors almost matching the integrality gaps: For complete graphs our approximation is 2.06 - ε, which almost matches the previously known integrality gap of 2. For complete k-partite graphs our approximation is 3. We also show a matching integrality gap. For complete graphs with edge weights satisfying triangle inequalities and probability constraints, our approximation is 1.5, and we show an integrality gap of 1.2.

We present a new framework for solving optimization problems with a diseconomy of scale. In such problems, our goal is to minimize the cost of resources used to perform a certain task. The cost of resources grows superlinearly, as xq, q ≥ 1, with the amount x of resources used. We define a novel linear programming relaxation for such problems, and then show that the integrality gap of the relaxation is Aq, where Aqis the q-th moment of the Poisson random variable with parameter 1. Using our framework, we obtain approximation algorithms for the Minimum Energy Efficient Routing, Minimum Degree Balanced Spanning Tree, Load Balancing on Unrelated Parallel Machines, and Unrelated Parallel Machine Scheduling with Nonlinear Functions of Completion Times problems. Our analysis relies on the decoupling inequality for nonnegative random variables. The inequality states that ||Σi=1nXi||q≤ Cq ||Σi=1nYi||q, where Xi are independent nonnegative random variables, Yi are possibly dependent nonnegative random variable, and each Yihas the same distribution as Xi. The inequality was proved by de la Peña in 1990. However, the optimal constant Cq was not known. We show that the optimal constant is Cq= Aq1/q.

Given a graph G, the sparsest-cut problem asks to find the set of vertices S which has the least expansion defined as where w is the total edge weight of a subset. Here we study the natural generalization of this problem: given an integer k, compute a k-partition {P1, …, Pk} of the vertex set so as to minimize Our main result is a polynomial time bi-criteria approximation algorithm which outputs a (1 – ∊)k-partition of the vertex set such that each piece has expansion at most times OPT. We also study balanced versions of this problem.

We investigate the notion of stability proposed by Bilu and Linial. We obtain an exact polynomial-time algorithm for γ-stable Max Cut instances with for some absolute constant c > 0. Our algorithm is robust: it never returns an incorrect answer; if the instance is γ-stable, it finds the maximum cut, otherwise, it either finds the maximum cut or certifies that the instance is not γ-stable. We prove that there is no robust polynomial-time algorithm for γ-stable instances of Max Cut when γ < αSC(n/2), where αSC is the best approximation factor for Sparsest Cut with non-uniform demands. That suggests that solving γ-stable instances with might be difficult or even impossible. Our algorithm is based on semidefinite programming. We show that the standard SDP relaxation for Max Cut (with triangle inequalities) is integral if , where is the least distortion with which every n point metric space of negative type embeds into ℓ1. On the negative side, we show that the SDP relaxation is not integral when . Moreover, there is no tractable convex relaxation for γ-stable instances of Max Cut when γ < αSC(n/2). Our results significantly improve previously known results. The best previously known algorithm for γ-stable instances of Max Cut required that (for some c > 0) [Bilu, Daniely, Linial, and Saks]. No hardness results were known for the problem. Additionally, we present an exact robust polynomial-time algorithm for 4-stable instances of Minimum Multiway Cut. We also study a relaxed notion of weak stability and present algorithms for weakly stable instances of Max Cut and Minimum Multiway Cut.

We propose and study a new semi-random semi-adversarial model for Balanced Cut, a planted model with permutation invariant random edges (PIE). Our model is much more general than planted models considered previously. Consider a set of vertices V partitioned into two clusters L and R of equal size. Let G be an arbitrary graph on V with no edges between L and R. Let Erandom be a set of edges sampled from an arbitrary permutation-invariant distribution (a distribution that is invariant under permutation of vertices in L and in R). Then we say that G+Erandom is a graph with permutation-invariant random edges.

We study graph partitioning problems from a min-max perspective, in which an input graph on $n$ vertices should be partitioned into $k$ parts, and the objective is to minimize the maximum number of edges leaving a single part. The two main versions we consider are where the $k$ parts need to be of equal size, and where they must separate a set of $k$ given terminals. We consider a common generalization of these two problems, and design for it an $O(\sqrt{\log n\log k})$ approximation algorithm. This improves over an $O(\log^2 n)$ approximation for the second version due to Svitkina and Tardos [Min-max multiway cut, in APPROX-RANDOM, 2004, Springer, Berlin, 2004], and roughly $O(k\log n)$ approximation for the first version that follows from other previous work. We also give an $O(1)$ approximation algorithm for graphs that exclude any fixed minor. Our algorithm uses a new procedure for solving the small-set expansion problem. In this problem, we are given a graph $G$ and the goal is to find a nonempty set $S\subseteq V$ of size $|S| \leq \rho n$ with minimum edge expansion. We give an $O(\sqrt{\log{n}\log{(1/\rho)}})$ bicriteria approximation algorithm for small-set expansion in general graphs, and an improved factor of $O(1)$ for graphs that exclude any fixed minor.

We show that for every positive ε > 0, unless NP ⊂ BPQP, it is impossible to approximate the maximum quadratic assignment problem within a factor better than 2 log 1-ε n by a reduction from the maximum label cover problem. Our result also implies that Approximate Graph Isomorphism is not robust and is, in fact, 1 - ε versus ε hard assuming the Unique Games Conjecture. Then, we present an O (√n)-approximation algorithm for the problem based on rounding of the linear programming relaxation often used in state-of-the-art exact algorithms.

Word-classing has been used in language modeling for two distinct purposes: to improve the likelihood of the language model, and to improve the runtime speed. In particular, frequency-based heuristics have been proposed to improve the speed of recurrent neural network language models (RNN-LMs). In this paper, we present a dynamic programming algorithm for determining classes in a way that provably minimizes the runtime of the resulting class-based language models. However, we also find that the speed-based methods degrade the perplexity of the language models by 5-10% over traditional likelihood-based classing. We remedy this via the introduction of a speed-based regularization term in the likelihood objective function. This achieves a runtime close to that of the speed based methods without loss in perplexity performance. We demonstrate these improvements with both an RNN-LM and the Model M exponential language model, for three different tasks involving two different languages.

In this paper, we propose two semi-random models for the Minimum Feedback Arc Set Problem and present approximation algorithms for them. In the first model, which we call the Random Edge Flipping model, an instance is generated as follows. We start with an arbitrary acyclic directed graph and then randomly flip its edges (the adversary may later un-flip some of them). In the second model, which we call the Random Backward Edge model, again we start with an arbitrary acyclic graph but now add new random backward edges (the adversary may delete some of them). For the first model, we give an approximation algorithm that finds a solution of cost (1+ δ) OPT + n polylog n, where OPT is the cost of the optimal solution. For the second model, we give an approximation algorithm that finds a solution of cost O(planted) + n polylog n, where planted is the cost of the planted solution. Additionally, we present an approximation algorithm for semi-random instances of Minimum Directed Balanced Cut.

We prove that the Bounded Occurrence Ordering k-CSP Problem is not approximation resistant. We give a very simple local search algorithm that always performs better than the random assignment algorithm (unless, the number of satisfied constraints does not depend on the ordering). Specifically, the expected value of the solution returned by the algorithm is at least ALG >= AVG + alpha(B,k)(OPT-AVG), where OPT is the value of the optimal solution; AVG is the expected value of the random solution; and alpha(B,k) = Omega_k(B^{-(k+O(1))}) is a parameter depending only on k (the arity of the CSP) and B (the maximum number of times each variable is used in constraints). The question whether bounded occurrence ordering k-CSPs are approximation resistant was raised by Guruswami and Zhou (2012), who recently showed that bounded occurrence 3-CSPs and "monotone" k-CSPs admit a non-trivial approximation.

The erratic memory access pattern of graph algorithms makes it hard to optimize on cache-based architectures. While multithreading hides memory latency, it is unclear how hardware threads combined with caches impact the performance of typical graph workload. As modern architectures strike different balances between caching and multithreading, it remains an open question whether the benefit of optimizing locality behavior outweighs the cost. We study parallel graph algorithms on two different multi-threaded, multi-core platforms, that is, IBM Power7 and Sun Niagara2. Our experiments first demonstrate their performance advantage over prior architectures. We find nonetheless the number of hardware threads in either platform is not sufficient to fully mask memory latency. Our cache-friendly scheduling of memory accesses improves performance by up to 2.6 times on Power7 and prior cache-based architectures, yet the same technique significantly degrades performance on Niagara2. Software prefetching and manipulating the storage of the input to improve spatial locality improve performance by up to 2.1 times and 1.3 times on both platforms. Our study reveals interesting interplay between architecture and algorithm.

In this paper, we propose and study a new semi-random model for graph partitioning problems. We believe that it captures many properties of real-world instances. The model is more flexible than the semi-random model of Feige and Kilian and planted random model of Bui, Chaudhuri, Leighton and Sipser.

We present a new proof of a celebrated result of Gács and Körner that the common information is far less than the mutual information. Consider two sequences$\alpha_{1},\ldots\alpha_{n}$and$\beta_{1},\ldots,\beta_{n}$of random variables, where pairs$(\alpha_{1},\beta_{1}),\ldots, (\alpha_{n},\beta_{n})$are independent and identically distributed. Gács and Körner proved that it is not possible to extract “common information” from these two sequences unless the joint distribution matrix of random variables$(\alpha_{i},\beta_{i})$is a block matrix.

We study graph partitioning problems from a min-max perspective, in which an input graph on n vertices should be partitioned into k parts, and the objective is to minimize the maximum number of edges leaving a single part. The two main versions we consider are: (i) the k parts need to be of equal size, and (ii) the parts must separate a set of k given terminals. We consider a common generalization of these two problems, and design for it an O(√log n log k)-approximation algorithm. This improves over an O(log2n) approximation for the second version due to Svitkina and Tardos, and roughly O(k log n) approximation for the first version that follows from other previous work. We also give an improved O(1)-approximation algorithm for graphs that exclude any fixed minor. Our algorithm uses a new procedure for solving the Small Set Expansion problem. In this problem, we are given a graph G and the goal is to find a non-empty subset S of V of size at most pn with minimum edge-expansion. We give an O(√log n log (1/p)) bicriteria approximation algorithm for the general case of Small Set Expansion and O(1) approximation algorithm for graphs that exclude any fixed minor.

The classical Grothendieck constant, denoted KG, is equal to the integrality gap of the natural semidefinite relaxation of the problem of computing max {Σi-1mΣj=1naijεiδj: {εi}i=1m, {δj}j=1n⊆{-1,1} } a generic and well-studied optimization problem with many applications. Krivine proved in 1977 that KG ≤ 2log (1+√2)/π and conjectured that his estimate is sharp. We obtain a sharper Grothendieck inequality, showing that KGo>; 0. Our main contribution is conceptual: despite dealing with a binary rounding problem, random 2-dimensional projections combined with a careful partition of ℝ2in order to round the projected vectors, beat the random hyperplane technique, contrary to Krivine's long-standing conjecture.

In this paper, we study the average case complexity of the Unique Games problem. We propose a semi-random model, in which a unique game instance is generated in several steps. First an adversary selects a completely satisfiable instance of Unique Games, then she chooses an ε-fraction of all edges, and finally replaces ("corrupts") the constraints corresponding to these edges with new constraints. If all steps are adversarial, the adversary can obtain any (1 - ε)-satisfiable instance, so then the problem is as hard as in the worst case. We show however that we can find a solution satisfying a (1 - δ) fraction of all constraints in polynomial-time if at least one step is random (we require that the average degree of the graph is Ω̃(log k)). Our result holds only for ε less than some absolute constant. We prove that if ε ≥ 1/2, then the problem is hard in one of the models, that is, no polynomial-time algorithm can distinguish between the following two cases: (i) the instance is a (1 - ε)-satisfiable semi-random instance and (ii) the instance is at most δ-satisfiable (for every δ >; 0); the result assumes the 2-to-2 conjecture. Finally, we study semi-random instances of Unique Games that are at most (1 - ε)-satisfiable. We present an algorithm that distinguishes between the case when the instance is a semi-random instance and the case when the instance is an (arbitrary) (1 - δ)-satisfiable instances if ε >; cδ (for some absolute constant c).

The erratic memory access pattern makes it hard to implement fast large-scale graph analysis. Although algorithms of fine-grain parallelism seem to benefit from multithreading, it is unclear whether the long memory latency of such workload is fully masked on current systems, and if not, whether improving locality brings any performance benefit, especially when the cache is simple. We optimize several fundamental graph algorithms on a multi-threaded, multi-core platform, with simple caches. Although the naive implementation scales, we show nonetheless the number of hardware threads is insufficient to fully mask the memory latency for typical graph analysis workload and the processor is unlikely to be fully utilized. In optimizing for cache performance, we show that known cache-friendly designs that prove effective on traditional architectures do not perform well on this platform. We explore low-cost measures such as software prefetching and manipulating the storage of the input to improve performance. Our results show that compared with the original implementation speedups between 10% and 200% are achieved at different number of threads with our optimization.

This paper studies the ``explanation problem'' for tree- and linearly-ordered array data, a problem motivated by database applications and recently solved for the one-dimensional tree-ordered case. In this paper, one is given a matrix A=(a_{ij}) whose rows and columns have semantics: special subsets of the rows and special subsets of the columns are meaningful, others are not. A submatrix in A is said to be meaningful if and only if it is the cross product of a meaningful row subset and a meaningful column subset, in which case we call it an ``allowed rectangle.'' The goal is to ``explain'' A as a sparse sum of weighted allowed rectangles. Specifically, we wish to find as few weighted allowed rectangles as possible such that, for all i,j, a_ij equals the sum of the weights of all rectangles which include cell (i,j). In this paper we consider the natural cases in which the matrix dimensions are tree-ordered or linearly-ordered. In the tree-ordered case, we are given a rooted tree $T_1$ whose leaves are the rows of $A$ and another, $T_2$, whose leaves are the columns. Nodes of the trees correspond in an obvious way to the sets of their leaf descendants. In the linearly-ordered case, a set of rows or columns is meaningful if and only if it is contiguous. For tree-ordered data, we prove the explanation problem NP-Hard and give a randomized $2$-approximation algorithm for it. For linearly-ordered data, we prove the explanation problem NP-Har and give a $2.56$-approximation algorithm. To our knowledge, these are the first results for the problem of sparsely and exactly representing matrices by weighted rectangles.

We study vertex cut and flow sparsifiers that were recently introduced by Moitra, and Leighton and Moitra. We improve and generalize their results. We give a new polynomial-time algorithm for constructing O(log k/log log k) cut and flow sparsifiers, matching the best known existential upper bound on the quality of a sparsifier, and improving the previous algorithmic upper bound of O(log2k/log log k). We show that flow sparsifiers can be obtained from linear operators approximating minimum metric extensions. We introduce the notion of (linear) metric extension operators, prove that they exist, and give an exact polynomialtime algorithm for finding optimal operators. We then establish a direct connection between flow and cut sparsifiers and Lipschitz extendability of maps in Banach spaces, a notion studied in functional analysis since 1950s. Using this connection, we obtain a lower bound of Ω (√log k/ log log k) for flow sparsifiers and a lower bound of Ω( √g k/ log log k) for cut sparsifiers. We show that if a certain open question posed by Ball in 1992 has a positive answer, then there exist Õ(√log k) cut sparsifiers. On the other hand, any lower bound on cut sparsifiers better than Ω̃(√log k) would imply a negative answer to this question.

Suppose that every k points in a n point metric space X are D-distortion embeddable into $\ell_1$. We give upper and lower bounds on the distortion required to embed the entire space X into $\ell_1$. This is a natural mathematical question and is also motivated by the study of relaxations obtained by lift-and-project methods for graph partitioning problems. In this setting, we show that X can be embedded into $\ell_1$ with distortion $O(D\times\log(n/k))$. Moreover, we give a lower bound showing that this result is tight if D is bounded away from 1. For $D=1+\delta$ we give a lower bound of $\Omega(\log(n/k)/\log(1/\delta))$; and for $D=1$, we give a lower bound of $\Omega(\log n/(\log k+\log\log n))$. Our bounds significantly improve on the results of Arora et al. who initiated a study of these questions.

Over the past three decades, the suffix tree has served as a fundamental data structure in string processing. However, its widespread applicability has been hindered due to the fact that suffix tree construction does not scale well with the size of the input string. With advances in data collection and storage technologies, large strings have become ubiquitous, especially across emerging applications involving text, time series, and biological sequence data. To benefit from these advances, it is imperative that we have a scalable suffix tree construction algorithm. The past few years have seen the emergence of several disk-based suffix tree construction algorithms. However, construction times continue to be daunting—for example, indexing the entire human genome still takes over 30 hours on a system with 2 gigabytes of physical memory. In this article, we will empirically demonstrate and argue that all existing suffix tree construction algorithms have a severe limitation—to glean reasonable disk I/O efficiency, the input string being indexed must fit in main memory. This limitation is attributed to the poor locality exhibited by existing suffix tree construction algorithms and inhibits both sequential and parallel scalability. To deal with this limitation, we will show that through careful algorithm design, one of the simplest suffix tree construction algorithms can be rearchitected to build a suffix tree in a tiled manner, allowing the execution to operate within a fixed main memory budget when indexing strings of any size. We will also present a parallel extension of our algorithm that is designed for massively parallel systems like the IBM Blue Gene. An experimental evaluation will show that the proposed approach affords an improvement of several orders of magnitude in serial performance when indexing large strings. Furthermore, the proposed parallel extension is shown to be scalable—it is now possible to index the entire human genome on a 1024 processor IBM Blue Gene system in under 15 minutes.

With advances in sequencing technology and through aggressive sequencing efforts, DNA sequence data sets have been growing at a rapid pace. To gain from these advances, it is important to provide life science researchers with the ability to process and query large sequence data sets. For the past three decades, the suffix tree has served as a fundamental data structure in processing sequential data sets. However, tree construction times on large data sets have been excessive. While parallel suffix tree construction is an obvious solution to reduce execution times, poor locality of reference has limited parallel performance. In this paper, we show that through careful parallel algorithm design, this limitation can be removed, allowing tree construction to scale to massively parallel systems like the IBM Blue Gene. We demonstrate that the entire Human genome can be indexed on 1024 processors in under 15 minutes.

Over the past three decades, the suffix tree has served as a fundamental data structure in string processing. However, its widespread applicability has been hindered due to the fact that suffix tree construction does not scale well with the size of the input string. With advances in data collection and storage technologies, large strings have become ubiquitous, especially across emerging applications involving text, time series, and biological sequence data. To benefit from these advances, it is imperative that we realize a scalable suffix tree construction algorithm.

We prove strong lower bounds on integrality gaps of Sherali-Adams relaxations for MAX CUT, Vertex Cover, Sparsest Cut and other problems. Our constructions show gaps for Sherali-Adams relaxations that survive nδ rounds of lift and project. For MAX CUT and Vertex Cover, these show that even nδ rounds of Sherali-Adams do not yield a better than 2-ε approximation. The main combinatorial challenge in constructing these gap examples is the construction of a fractional solution that is far from an integer solution, but yet admits consistent distributions of local solutions for all small subsets of variables. Satisfying this consistency requirement is one of the major hurdles to constructing Sherali-Adams gap examples. We present a modular recipe for achieving this, building on previous work on metrics with a local-global structure. We develop a conceptually simple geometric approach to constructing Sherali-Adams gap examples via constructions of consistent local SDP solutions. This geometric approach is surprisingly versatile. We construct Sherali-Adams gap examples for Unique Games based on our construction for MAX CUT together with a parallel repetition like procedure. This in turn allows us to obtain Sherali-Adams gap examples for any problem that has a Unique Games based hardness result (with some additional conditions on the reduction from Unique Games). Using this, we construct 2-ε gap examples for Maximum Acyclic Subgraph that rules out any family of linear constraints with support at most nδ.

In this article, we present two approximation algorithms for the maximum constraint satisfaction problem with k variables in each constraint (MAX k -CSP). Given a (1 − ε) satisfiable 2CSP our first algorithm finds an assignment of variables satisfying a 1 − O (√ε) fraction of all constraints. The best previously known result, due to Zwick, was 1 − O (ε 1/3 ). The second algorithm finds a ck /2 k approximation for the MAX k -CSP problem (where c > 0.44 is an absolute constant). This result improves the previously best known algorithm by Hast, which had an approximation guarantee of Ω( k /(2 k log k )). Both results are optimal assuming the unique games conjecture and are based on rounding natural semidefinite programming relaxations. We also believe that our algorithms and their analysis are simpler than those previously known.

In this paper we present a new approximation algorithm for the Max Acyclic Subgraph problem. Given an instance where the maximum acyclic subgraph contains 1/2 + delta fraction of all edges, our algorithm finds an acyclic subgraph with 1/2 + Omega(delta/ log n) fraction of all edges.

Suppose that every k points in a metric space X are D-distortion embeddable intolscr1. We give upper and lower bounds on the distortion required to embed the entire space X intolscr1. This is a natural mathematical question and is also motivated by the study of relaxations obtained by lift-and-project methods for graph partitioning problems. In this setting, we show that X can be embedded intolscr1with distortion O(D times log(|X|/k)). Moreover, we give a lower bound showing that this result is tight if D is bounded away from I. For D = 1 + delta we give a lower bound of Omega(log(|X|/k/ log( 1/delta)); and for D = 1, we give a lower bound of Omega( log |X|/(log k +log log | X|)). Our bounds significantly improve on the results of Arora, Jjovdsz, Newman, Rabani, Rabinovich and Vempala, who initiated a study of these questions.

In this paper we present a new approximation algorithm for unique games. For a unique game with n vertices and k states (labels), if a (1 - epsiv) fraction of all constraints is satisfiable, the algorithm finds an assignment satisfying a 1 - O(epsiv radic(log n log k)) fraction of all constraints. To this end, we introduce new embedding techniques for rounding semidefinite relaxations of problems with large domain size

Unique games are constraint satisfaction problems that can be viewed as a generalization of Max-Cut to a larger domain size. The Unique Games Conjecture states that it is hard to distinguish between instances of unique games where almost all constraints are satisfiable and those where almost none are satisfiable. It has been shown to imply a number of inapproximability results for fundamental problems that seem difficult to obtain by more standard complexity assumptions. Thus, proving or refuting this conjecture is an important goal. We present significantly improved approximation algorithms for unique games. For instances with domain size k where the optimal solution satisfies 1-ε fraction of all constraints, our algorithms satisfy roughly k-ε/(2-ε) and 1- O(√εlog k) fraction of all constraints. Our algorithms are based on rounding a natural semidefinite programming relaxation for the problem and their performance almost matches the integrality gap of this relaxation. Our results are near optimal if the Unique Games Conjecture is true, i.e. any improvement (beyond low order terms) would refute the conjecture.

We give O(√log n)-approximation algorithms for the MIN UNCUT, MIN 2CNF DELETION, DIRECTED BALANCED SEPERATOR, and DIRECTED SPARSEST CUT problems. The previously best known algorithms give an O(log n)-approximation for MIN UNCUT [9], DIRECTED BALANCED SEPERATOR [17], DIRECTED SPARSEST CUT [17], and an O(log n log log n)-approximation for MIN 2CNF DELETION [14].We also show that the integrality gap of an SDP relaxation of the MINIMUM MULTICUT problem is Ω(log n).

We introduce a new graph parameter, called the Grothendieck constant of a graph G=(V,E), which is defined as the least constant K such that for every A:E→R,supf:V→S|V|-1 Σ(u,v) ∈ E A(u,v) · ‹f(u),f(v)› ≤ K supf:V→(-1,+1) Σ(u,v)∈ E A(u,v) · f(u)f(v).The classical Grothendieck inequality corresponds to the case of bipartite graphs, but the case of general graphs is shown to have various algorithmic applications. Indeed, our work is motivated by the algorithmic problem of maximizing the quadratic form ∑u,v∈EA(u,v)f(vover all f: V →-1,1, which arises in the study of correlation clustering and in the investigation of the spin glass model. We give upper and lower estimates for the integrality gap of this program. We show that the integrality gap is O(log θḠ)) where θ(Ḡ) is the Lovasz Theta Function of the complement of G, which is always smaller than the chromatic number of G. This yields an efficient constant factor approximation algorithm for the above maximization problem for a wide range of graphs G. We also show that the maximum possible integrality gap is always at least Ω(log ω(G)), where Ω(G) is the clique number of G. In particular it follows that the maximum possible integrality gap for the complete graph on n Θ vertices with no loops is ⏷(log n ). More generally, the maximum possible integrality gap for any perfect graph with chromatic number n is ⏷(log n). The lower bound for the complete graph improves a result of Kashin and Szarek on Gram matrices of uniformly bounded functions, and settles a problem of Megretski and of Charikar and Wirth.