Concentration inequalities quantify the deviation of a random variable from a fixed value. In spite of numerous applications, such as opinion surveys or ecological counting procedures , few concentration results are known for the setting of sampling without replacement from a finite population. Until now, the best general concentration inequality has been a Hoeffding inequality due to ?. In this paper, we first improve on the fundamental result of ?, and further extend it to obtain a Bernstein concentration bound for sampling without replacement. We then derive an empirical version of our bound that does not require the variance to be known to the user.

We study the problem of maximizing a monotone submodular function subject to a matroid independence constraint. For more than a decade, a rich body of work has studied this problem. Initially, a tight approximation of (1−1e) was given using the continuous greedy algorithm [Calinescu-Chekuri-Pal-Vondrák STOC‘2008] and later non-oblivious local search techniques were able to match this tight approximation guarantee [Filmus-Ward FOCS‘2012] and [Buchbinder-Feldman FOCS‘2024].

We study graph ordering problems with a min-max objective. A classical problem of this type is cutwidth, where given a graph we want to order its vertices such that the number of edges crossing any point is minimized. We give a$\log^{1+o(1)}(n)$approximation for the problem, substantially improving upon the previous poly-logarithmic guarantees based on the standard recursive balanced partitioning approach of Leighton and Rao (FOCS'88). Our key idea is a new metric decomposition procedure that is suitable for handling min-max objectives, which could be of independent interest. We also use this to show other results, including an improved$\log^{1+o(1)}(n)$approximation for computing the pathwidth of a graph.

In this paper we consider the online Submodular Welfare (SW) problem. In this problem we are given $n$ bidders each equipped with a general (not necessarily monotone) submodular utility and $m$ items that arrive online. The goal is to assign each item, once it arrives, to a bidder or discard it, while maximizing the sum of utilities. When an adversary determines the items' arrival order we present a simple randomized algorithm that achieves a tight competitive ratio of $\nicefrac{1}{4}$. The algorithm is a specialization of an algorithm due to [Harshaw-Kazemi-Feldman-Karbasi MOR`22], who presented the previously best known competitive ratio of $3-2\sqrt{2}\approx 0.171573 $ to the problem. When the items' arrival order is uniformly random, we present a competitive ratio of $\approx 0.27493$, improving the previously known $\nicefrac{1}{4}$ guarantee. Our approach for the latter result is based on a better analysis of the (offline) Residual Random Greedy (RRG) algorithm of [Buchbinder-Feldman-Naor-Schwartz SODA`14], which we believe might be of independent interest.

We consider the Max--Section problem, where we are given an undirected graph G=(V,E)equipped with non-negative edge weights w: E → R_+ and the goal is to find a partition of V into three equisized parts while maximizing the total weight of edges crossing between different parts. Max-3-Section is closely related to other well-studied graph partitioning problems, e.g., Max-Cut, Max-3-Cut, and Max-Bisection. We present a polynomial time algorithm achieving an approximation of 0.795, that improves upon the previous best known approximation of 0.673. The requirement of multiple parts that have equal sizes renders Max-3-Section much harder to cope with compared to, e.g., Max-Bisection. We show a new algorithm that combines the existing approach of Lassere hierarchy along with a random cut strategy that suffices to give our result.

We consider the family of Correlation Clustering optimization problems under fairness constraints. In Correlation Clustering we are given a graph whose every edge is labeled either with a + or a -, and the goal is to find a clustering that agrees the most with the labels: + edges within clusters and - edges across clusters. The notion of fairness implies that there is no over, or under, representation of vertices in the clustering: every vertex has a color and the distribution of colors within each cluster is required to be the same as the distribution of colors in the input graph. Previously, approximation algorithms were known only for fair disagreement minimization in complete unweighted graphs. We prove the following: (1) there is no finite approximation for fair disagreement minimization in general graphs unless P = NP (this hardness holds also for bicriteria algorithms); and (2) fair agreement maximization in general graphs admits a bicriteria approximation of ≈ 0.591 (an improved ≈ 0.609 true approximation is given for the special case of two uniformly distributed colors). Our algorithm is based on proving that the sticky Brownian motion rounding of [Abbasi Zadeh-Bansal-Guruganesh-Nikolov-Schwartz-Singh SODA'20] copes well with uncut edges.

Semidefinite programming is a powerful tool in the design and analysis of approximation algorithms for combinatorial optimization problems. In particular, the random hyperplane rounding method of Goemans and Williamson [ 31 ] has been extensively studied for more than two decades, resulting in various extensions to the original technique and beautiful algorithms for a wide range of applications. Despite the fact that this approach yields tight approximation guarantees for some problems, e.g., Max-Cut , for many others, e.g., Max-SAT and Max-DiCut , the tight approximation ratio is still unknown. One of the main reasons for this is the fact that very few techniques for rounding semi-definite relaxations are known. In this work, we present a new general and simple method for rounding semi-definite programs, based on Brownian motion. Our approach is inspired by recent results in algorithmic discrepancy theory. We develop and present tools for analyzing our new rounding algorithms, utilizing mathematical machinery from the theory of Brownian motion, complex analysis, and partial differential equations. Focusing on constraint satisfaction problems, we apply our method to several classical problems, including Max-Cut , Max-2SAT , and Max-DiCut , and derive new algorithms that are competitive with the best known results. To illustrate the versatility and general applicability of our approach, we give new approximation algorithms for the Max-Cut problem with side constraints that crucially utilizes measure concentration results for the Sticky Brownian Motion, a feature missing from hyperplane rounding and its generalizations.

In this work, we initiate the study of fault tolerant Max Cut, where given an edge-weighted undirected graph $G=(V,E)$, the goal is to find a cut $S\subseteq V$ that maximizes the total weight of edges that cross $S$ even after an adversary removes $k$ vertices from $G$. We consider two types of adversaries: an adaptive adversary that sees the outcome of the random coin tosses used by the algorithm, and an oblivious adversary that does not. For any constant number of failures $k$ we present an approximation of $(0.878-ε)$ against an adaptive adversary and of $α_{GW}\approx 0.8786$ against an oblivious adversary (here $α_{GW}$ is the approximation achieved by the random hyperplane algorithm of [Goemans-Williamson J. ACM `95]). Additionally, we present a hardness of approximation of $α_{GW}$ against both types of adversaries, rendering our results (virtually) tight. The non-linear nature of the fault tolerant objective makes the design and analysis of algorithms harder when compared to the classic Max Cut. Hence, we employ approaches ranging from multi-objective optimization to LP duality and the ellipsoid algorithm to obtain our results.

We consider the 0-Extension problem, where we are given an undirected graph G=(V,E) equipped with non-negative edge weights w:E→ ℝ+, a collection T={ t1,…,tk}⊆ V of k special vertices called terminals, and a semi-metric D over T. The goal is to assign every non-terminal vertex to a terminal while minimizing the sum over all edges of the weight of the edge multiplied by the distance in D between the terminals to which the endpoints of the edge are assigned. 0-Extension admits two known algorithms, achieving approximations of O(logk) [Călinescu-Karloff-Rabani SICOMP ’05] and O(logk/loglogk) [Fakcharoenphol-Harrelson-Rao-Talwar SODA ’03]. Both known algorithms are based on rounding a natural linear programming relaxation called the metric relaxation, in which D is extended from T to the entire of V. The current best known integrality gap for the metric relaxation is Ω (√logk). In this work we present an improved integrality gap of Ω(log2/3k) for the metric relaxation. Our construction is based on the randomized extension of one graph by another, a notion that captures lifts of graphs as a special case and might be of independent interest. Inspired by algebraic topology, our analysis of the gap instance is based on proving no continuous section (in the topological sense) exists in the randomized extension.

Correlation Clustering is an elegant model where given a graph with edges labeled + or -, the goal is to produce a clustering that agrees the most with the labels: + edges should reside within clusters and - edges should cross between clusters. In this work we study the MaxCorr objective, aiming to find a clustering that maximizes the difference between edges classified correctly and incorrectly. We focus on the case of bipartite graphs and present an improved approximation of 0.254, improving upon the known approximation of 0.219 given by Charikar and Wirth [FOCS`2004] and going beyond the 0.2296 barrier imposed by their technique. Our algorithm is inspired by Krivine’s method for bounding Grothendieck’s constant, and we extend this method to allow for more than two clusters in the output. Moreover, our algorithm leads to two additional results: (1) the first known approximation guarantees for MaxCorr where the output is constrained to have a bounded number of clusters; and (2) a natural extension of Grothendieck’s inequality to large domains.

We consider the {Requirement Cut} problem, where given an undirected graph G = (V,E) equipped with non-negative edge weights c:E → R_{+}, and g groups of vertices X₁,…,X_{g} ⊆ V each equipped with a requirement r_i, the goal is to find a collection of edges F ⊆ E, with total minimum weight, such that once F is removed from G in the resulting graph every X_{i} is broken into at least r_{i} connected components. {Requirement Cut} captures multiple classic cut problems in graphs, e.g., {Multicut}, {Multiway Cut}, {Min k-Cut}, {Steiner k-Cut}, {Steiner Multicut}, and {Multi-Multiway Cut}. Nagarajan and Ravi [Algoritmica`10] presented an approximation of O(log{n}log{R}) for the problem, which was subsequently improved to O(log{g} log{k}) by Gupta, Nagarajan and Ravi [Operations Research Letters`10] (here R = ∑ _{i = 1}^g r_i and k = |∪ _{i = 1}^g X_i |). We present an approximation of O(Xlog{R} √{log{k}}log{log{k}}) for {Requirement Cut} (here X = max _{i = 1,…,g} {|X_i|}). Our approximation in general is incomparable to the above mentioned previous results, however when all groups are not too large, i.e., X = o((√{log{k}}log{g})/(log{R}log{log{k}})), it is better. Our algorithm is based on a new configuration linear programming relaxation for the problem, which is accompanied by a remarkably simple randomized rounding procedure.

Semi-definite programming is a powerful tool in the design and analysis of approximation algorithms for combinatorial optimization problems. In particular, the random hyperplane rounding method of Goemans and Williamson [23] has been extensively studied for more than two decades, resulting in various extensions to the original technique and beautiful algorithms for a wide range of applications. Despite the fact that this approach yields tight approximation guarantees for some problems, e.g., Max-Cut, for many others, e.g., Max-SAT and Max-DiCut, the tight approximation ratio is still unknown. One of the main reasons for this is the fact that very few techniques for rounding semi-definite relaxations are known. In this work, we present a new general and simple method for rounding semi-definite programs, based on Brownian motion. Our approach is inspired by recent results in algorithmic discrepancy theory. We develop and present tools for analyzing our new rounding algorithms, utilizing mathematical machinery from the theory of Brownian motion, complex analysis, and partial differential equations. Focusing on constraint satisfaction problems, we apply our method to several classical problems, including Max-Cut, Max-2SAT, and Max-DiCut, and derive new algorithms that are competitive with the best known results. To illustrate the versatility and general applicability of our approach, we give new approximation algorithms for the Max-Cut problem with side constraints that crucially utilizes measure concentration results for the Sticky Brownian Motion, a feature missing from hyperplane rounding and its generalizations.

In the graph balancing problem the goal is to orient a weighted undirected graph to minimize the maximum weighted in-degree. This special case of makespan minimization is NP-hard to approximate to a factor better than 3/2 even when there are only two types of edge weights. In this note we describe a simple 3/2 approximation for the graph balancing problem with two-edge types, settling this very special case of makespan minimization.

Motivated by the use of high speed circuit switches in large scale data centers, we consider the problem of circuit switch scheduling. In this problem we are given demands between pairs of servers and the goal is to schedule at every time step a matching between the servers while maximizing the total satisfied demand over time. The crux of this scheduling problem is that once one shifts from one matching to a different one a fixed delay delta is incurred during which no data can be transmitted. For the offline version of the problem we present a (1-(1/e)-epsilon) approximation ratio (for any constant epsilon >0). Since the natural linear programming relaxation for the problem has an unbounded integrality gap, we adopt a hybrid approach that combines the combinatorial greedy with randomized rounding of a different suitable linear program. For the online version of the problem we present a (bi-criteria) ((e-1)/(2e-1)-epsilon)-competitive ratio (for any constant epsilon >0 ) that exceeds time by an additive factor of O(delta/epsilon). We note that no uni-criteria online algorithm is possible. Surprisingly, we obtain the result by reducing the online version to the offline one.

Motivated by applications in machine learning, such as subset selection and data summarization, we consider the problem of maximizing a monotone submodular function subject to mixed packing and covering constraints. We present a tight approximation algorithm that for any constant $ε>0$ achieves a guarantee of $1-\frac{1}{\mathrm{e}}-ε$ while violating only the covering constraints by a multiplicative factor of $1-ε$. Our algorithm is based on a novel enumeration method, which unlike previous known enumeration techniques, can handle both packing and covering constraints. We extend the above main result by additionally handling a matroid independence constraints as well as finding (approximate) pareto set optimal solutions when multiple submodular objectives are present. Finally, we propose a novel and purely combinatorial dynamic programming approach that can be applied to several special cases of the problem yielding not only {\em deterministic} but also considerably faster algorithms. For example, for the well studied special case of only packing constraints (Kulik {\em et. al.} [Math. Oper. Res. `13] and Chekuri {\em et. al.} [FOCS `10]), we are able to present the first deterministic non-trivial approximation algorithm. We believe our new combinatorial approach might be of independent interest.

Submodular function maximization has been studied extensively in recent years under various constraints and models. The problem plays a major role in various disciplines. We study a natural online variant of this problem in which elements arrive one by one and the algorithm has to maintain a solution obeying certain constraints at all times. Upon arrival of an element, the algorithm has to decide whether to accept the element into its solution and may preempt previously chosen elements. The goal is to maximize a submodular function over the set of elements in the solution. We study two special cases of this general problem and derive upper and lower bounds on the competitive ratio. Specifically, we design a 1/ e -competitive algorithm for the unconstrained case in which the algorithm may hold any subset of the elements, and constant competitive ratio algorithms for the case where the algorithm may hold at most k elements in its solution.

In the Minimum Hypergraph Bisection problem, the vertex set of a hypergraph has to be partitioned into two parts of equal size so that the number of hyperedges intersecting both parts is minimized. This problem is a natural generalization of the well-studied Minimum Bisection problem in graphs. In this paper we present a sharp distinction between Minimum Bisection in hypergraphs and graphs. Whereas it is well-known that all bi-criteria approximation algorithms for Minimum Bisection in graphs can be extended to hypergraphs with the exact same guarantees, in this paper we prove that this is not the case when considering true (i.e., non bi-criteria) approximation algorithms. Specifically, we show that Minimum Bisection in Hypergraphs admits an $\tilde\mathcalO (\sqrtn )$ approximation algorithm (and highlight several special cases where a better approximation ratio is possible). Additionally, we show that the problem is at least as hard as the Densest k -Subgraph problem. Assuming the Dense vs. Random Conjecture~\citeCDK12, no approximation ratio better than $\bigO(n^1/4-\varepsilon )$ is possible. In particular, Minimum Hypergraph Bisection is much harder to approximate than Minimum Bisection in graphs, for which a logarithmic approximation algorithms exist~\citeRae08. We also consider the problem of constructing trees that are cut sparsifiers for hypergraph and vertex cuts. While similar trees lie at the heart of powerful algorithms for Minimum Bisection in graphs, we prove that this is not the case for hypergraphs. We give upper and lower bounds to the quality of such trees. Our bounds show that this tree cut sparsifying approach cannot improve the general approximation ratio of Minimum Hypergraph Bisection and Minimum Vertex Bisection.

The \sf Multiway-Cut problem is a fundamental graph partitioning problem in which the objective is to find a minimum weight set of edges disconnecting a given set of special vertices called terminals. This problem is NP-hard and there is a well-known geometric relaxation in which the graph is embedded into a high dimensional simplex. Rounding a solution to the geometric relaxation is equivalent to partitioning the simplex. We present a novel simplex partitioning algorithm which is based on two ingredients: competing exponential clocks and distortion. Unlike previous methods, it utilizes cuts that are not parallel to the faces of the simplex. Applying this partitioning algorithm to the multiway cut problem, we obtain a simple (4/3)-approximation algorithm, thus, improving upon the current best-known result. This bound is further pushed to obtain an approximation factor of 1.32388. It is known that under the assumption of the unique games conjecture, the best possible approximation for the \sf Multiway-Cut problem can be attained via the geometric relaxation.

We introduce correlated randomized dependent rounding where, given multiple points y^1,...,y^n in some polytope P\subseteq [0,1]^k, the goal is to simultaneously round each y^i to some integral z^i in P while preserving both marginal values and expected distances between the points. In addition to being a natural question in its own right, the correlated randomized dependent rounding problem is motivated by multi-label classification applications that arise in machine learning, e.g., classification of web pages, semantic tagging of images, and functional genomics. The results of this work can be summarized as follows: (1) we present an algorithm for solving the correlated randomized dependent rounding problem in uniform matroids while losing only a factor of O(log{k}) in the distances (k is the size of the ground set); (2) we introduce a novel multi-label classification problem, the metric multi-labeling problem, which captures the above applications. We present a (true) O(log{k})-approximation for the general case of metric multi-labeling and a tight 2-approximation for the special case where there is no limit on the number of labels that can be assigned to an object.

We consider Multiway Cut, a basic graph partitioning problem in which the goal is to find the minimum weight collection of edges disconnecting a given set of special vertices called terminals. Multiway Cut admits a well known simplex embedding relaxation, where rounding this embedding is equivalent to partitioning the simplex. Current best known solutions to the problem are comprised of a mix of several different ingredients, resulting in intricate algorithms. Moreover, the best of these algorithms is too complex to fully analyze analytically and its approximation factor was verified using a computer. We propose a new approach to simplex partitioning and the Multiway Cut problem based on general transformations of the simplex that allow dependencies between the different variables. Our approach admits much simpler algorithms, and in addition yields an approximation guarantee for the Multiway Cut problem that (roughly) matches the current best computer verified approximation factor.

We study a variant of the generalized assignment problem ( gap ), which we label all-or-nothing gap ( agap ). We are given a set of items, partitioned into n groups, and a set of m bins. Each item ℓ has size s ℓ > 0, and utility a ℓ j ⩾ 0 if packed in bin j . Each bin can accommodate at most one item from each group; the total size of the items in a bin cannot exceed its capacity. A group of items is satisfied if all of its items are packed. The goal is to find a feasible packing of a subset of the items in the bins such that the total utility from satisfied groups is maximized. We motivate the study of agap by pointing out a central application in scheduling advertising campaigns. Our main result is an O (1)-approximation algorithm for agap instances arising in practice, in which each group consists of at most m /2 items. Our algorithm uses a novel reduction of agap to maximizing submodular function subject to a matroid constraint. For agap instances with a fixed number of bins, we develop a randomized polynomial time approximation scheme (PTAS) , relying on a nontrivial LP relaxation of the problem. We present a (3 + ε)-approximation as well as PTASs for other special cases of agap , where the utility of any item does not depend on the bin in which it is packed. Finally, we derive hardness results for the different variants of agap studied in this paper.

Fast algorithms for submodular maximization problems have a vast potential use in applicative settings, such as machine learning, social networks, and economics. Though fast algorithms were known for some special cases, only recently Badanidiyuru and Vondrák [4] were the first to explicitly look for such algorithms in the general case of maximizing a monotone submodular function subject to a matroid independence constraint. The algorithm of Badanidiyuru and Vondrák matches the best possible approximation guarantee, while trying to reduce the number of value oracle queries the algorithm performs. Our main result is a new algorithm for this general case which establishes a surprising tradeoff between two seemingly unrelated quantities: the number of value oracle queries and the number of matroid independence queries performed by the algorithm. Specifically, one can decrease the former by increasing the latter and vice versa, while maintaining the best possible approximation guarantee. Such a tradeoff is very useful since various applications might incur significantly different costs in querying the value and matroid independence oracles. Furthermore, in case the rank of the matroid is O(nc), where n is the size of the ground set and c is an absolute constant smaller than 1, the total number of oracle queries our algorithm uses can be made to have a smaller magnitude compared to that needed by [4]. We also provide even faster algorithms for the well studied special cases of a cardinality constraint and a partition matroid independence constraint, both of which capture many real-world applications and have been widely studied both theorically and in practice.

Submodular function maximization has been studied extensively in recent years under various constraints and models. The problem plays a major role in various disciplines. We study a natural online variant of this problem in which elements arrive one-by-one and the algorithm has to maintain a solution obeying certain constraints at all times. Upon arrival of an element, the algorithm has to decide whether to accept the element into its solution and may preempt previously chosen elements. The goal is to maximize a submodular function over the set of elements in the solution. We study two special cases of this general problem and derive upper and lower bounds on the competitive ratio. Specifically, we design a 1/e-competitive algorithm for the unconstrained case in which the algorithm may hold any subset of the elements, and constant competitive ratio algorithms for the case where the algorithm may hold at most k elements in its solution.

We consider the \sf Unconstrained Submodular Maximization problem in which we are given a nonnegative submodular function $f:2^{\mathcal{N}}\rightarrow \mathbb{R}^+$, and the objective is to find a subset $S\subseteq \mathcal{N}$ maximizing $f(S)$. This is one of the most basic submodular optimization problems, having a wide range of applications. Some well-known problems captured by \sf Unconstrained Submodular Maximization include \sf Max-Cut, \sf Max-DiCut, and variants of \sf Max-SAT and maximum facility location. We present a simple randomized linear time algorithm achieving a tight approximation guarantee of 1/2, thus matching the known hardness result of Feige, Mirrokni, and Vondrák [SIAM J. Comput., 40 (2011), pp. 1133--1153]. Our algorithm is based on an adaptation of the greedy approach which exploits certain symmetry properties of the problem.

Spencer's theorem asserts that, for any family of n subsets of ground set of size n, the elements of the ground set can be "colored" by the values +1 or -1 such that the sum of every set is O(sqrt(n)) in absolute value. All existing proofs of this result recursively construct "partial colorings", which assign +1 or -1 values to half of the ground set. We devise the first algorithm for Spencer's theorem that directly computes a coloring, without recursively computing partial colorings.

Datacenter WAN traffic consists of high priority transfers that have to be carried as soon as they arrive alongside large transfers with pre-assigned deadlines on their completion (ranging from minutes to hours). The ability to offer guarantees to large transfers is crucial for business needs and impacts overall cost-of-business. State-of-the-art traffic engineering solutions only consider the current time epoch and hence cannot provide pre-facto promises for long-lived transfers. We present Tempus, an online traffic engineering scheme that exploits information on transfer size and deadlines to appropriately pack long-running transfers across network paths and time, thereby leaving enough capacity slack for future high-priority requests. Tempus builds on a tailored approximate solution to a mixed packing-covering linear program, which is parallelizable and scales well in both running time and memory usage. Consequently, Tempus is able to quickly and effectively update its solution when new transfers arrive or unexpected changes happen. These updates involve only small edits to existing transfers. Therefore, as experiments on traces from a large production WAN show, Tempus can offer and keep promises to long-lived transfers well in advance of their actual deadline; the promise on minimal transfer size is comparable with an offline optimal solution and outperforms state-of-the-art solutions by 2-3X.

We consider the problem of maximizing a (non-monotone) submodular function subject to a cardinality constraint. In addition to capturing well-known combinatorial optimization problems, e.g., Max-k-Coverage and Max-Bisection, this problem has applications in other more practical settings such as natural language processing, information retrieval, and machine learning. In this work we present improved approximations for two variants of the cardinality constraint for non-monotone functions. When at most k elements can be chosen, we improve the current best approximation to a factor that is in the range [ ], achieving a tight approximation of for and breaking the barrier for all values of k. When exactly k elements must be chosen, our algorithms improve the current best approximation to a factor that is in the range [0.356, ], again achieving a tight approximation of for . Additionally, some of the algorithms we provide are very fast with time complexities of O(nk), as opposed to previous known algorithms which are continuous in nature, and thus, too slow for applications in the practical settings mentioned above. Our algorithms are based on two new techniques. First, we present a simple randomized greedy approach where in each step a random element is chosen from a set of “reasonably good” elements. This approach might be considered a natural substitute for the greedy algorithm of Nemhauser, Wolsey and Fisher [45], as it retains the same tight guarantee of for monotone objectives and the same time complexity of O(nk), while giving an approximation of for general non-monotone objectives (while the greedy algorithm of Nemhauser et. al. fails to provide any constant guarantee). Second, we extend the double greedy technique, which achieves a tight approximation for unconstrained submodular maximization, to the continuous setting. This allows us to manipulate the natural rates by which elements change, thus bounding the total number of elements chosen.

We consider the problem of Non-Uniform Graph Partitioning, where the input is an edge-weighted undirected graph G = (V, E) and k capacities n1, …, nk, and the goal is to find a partition {S1, S2, …, Sk} of V satisfying |Sj| ≤ nj for all 1 ≤ j ≤ k, that minimizes the total weight of edges crossing between different parts. This natural graph partitioning problem arises in practical scenarios, and generalizes well-studied balanced partitioning problems such as Minimum Bisection, Minimum Balanced Cut, and Minimum k-Partitioning. Unlike these problems, Non-Uniform Graph Partitioning seems to be resistant to many of the known partitioning techniques, such as spreading metrics, recursive partitioning, and Räcke's tree decomposition, because k can be a function of n and the capacities could be of different magnitudes. We present a bicriteria approximation algorithm for Non-Uniform Graph Partitioning that approximates the objective within O(logn) factor while deviating from the required capacities by at most a constant factor. Our approach is to apply stopping-time based concentration results to a simple randomized rounding of a configuration LP. These concentration bounds are needed as the commonly used techniques of bounded differences and bounded conditioned variances do not suffice.

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.

The Multiway-Cut problem is a fundamental graph partitioning problem in which the objective is to find a minimum weight set of edges disconnecting a given set of special vertices called terminals. This problem is NP-hard and there is a well known geometric relaxation in which the graph is embedded into a high dimensional simplex. Rounding a solution to the geometric relaxation is equivalent to partitioning the simplex. We present a novel simplex partitioning algorithm which is based on em competing exponential clocks and distortion. Unlike previous methods, it utilizes cuts that are not parallel to the faces of the simplex. Applying this partitioning algorithm to the multiway cut problem, we obtain a simple (4/3)-approximation algorithm, thus, improving upon the current best known result. This bound is further pushed to obtain an approximation factor of 1.32388. It is known that under the assumption of the unique games conjecture, the best possible approximation for the Multiway-Cut problem can be attained via the geometric relaxation.

We study the problem of aggregating and summarizing partial orders, on a large scale. Our motivation is two-fold: to discover elements at similar preference levels and to reduce the number of bits needed to store an element's position in a full ranking.We proceed in two steps: first, we find a total order by linearizing the rankings induced by the multiple partial orders and removing potentially inconsistent pairwise preferences. Next, given a total order, we introduce and formalize the rank quantization problem, which intuitively aims to bucketize the total order in a manner that mostly preserves the relations appearing in the partial orders. We show an exact quadratic-time quantization algorithm, as well as a greedy 2/3-approximation algorithm whose running is substantially faster on sparse instances. As an application, we aggregate rankings of top-10 search results over millions of search engine queries, approximately reproducing and then efficiently encoding the underlying static ranks used by the engine. We evaluate the performance of our algorithms on a web dataset of 12 million(2^{23.5}) unique pages and show that we can quantize the pages' static ranks using as few as eight bits, with only a minor degradation in search quality.

We consider the Unconstrained Submodular Maximization problem in which we are given a non-negative submodular function f : 2N→ ℝ+, and the objective is to find a subset S ⊆ N maximizing f(S). This is one of the most basic submodular optimization problems, having a wide range of applications. Some well known problems captured by Unconstrained Submodular Maximization include MaxCut, Max-DiCut, and variants of Max-SAT and maximum facility location. We present a simple randomized linear time algorithm achieving a tight approximation guarantee of 1/2, thus matching the known hardness result of Feige et al. [11]. Our algorithm is based on an adaptation of the greedy approach which exploits certain symmetry properties of the problem. Our method might seem counterintuitive, since it is known that the greedy algorithm fails to achieve any bounded approximation factor for the problem.

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 study of combinatorial problems with a submodular objective function has attracted much attention in recent years, and is partly motivated by the importance of such problems to economics, algorithmic game theory and combinatorial optimization. Classical works on these problems are mostly combinatorial in nature. Recently, however, many results based on continuous algorithmic tools have emerged. The main bottleneck of such continuous techniques is how to approximately solve a non-convex relaxation for the sub- modular problem at hand. Thus, the efficient computation of better fractional solutions immediately implies improved approximations for numerous applications. A simple and elegant method, called "continuous greedy", successfully tackles this issue for monotone submodular objective functions, however, only much more complex tools are known to work for general non-monotone submodular objectives. In this work we present a new unified continuous greedy algorithm which finds approximate fractional solutions for both the non-monotone and monotone cases, and improves on the approximation ratio for many applications. For general non-monotone submodular objective functions, our algorithm achieves an improved approximation ratio of about 1/e. For monotone submodular objective functions, our algorithm achieves an approximation ratio that depends on the density of the polytope defined by the problem at hand, which is always at least as good as the previously known best approximation ratio of 1-1/e. Some notable immediate implications are an improved 1/e-approximation for maximizing a non-monotone submodular function subject to a matroid or O(1)-knapsack constraints, and information-theoretic tight approximations for Submodular Max-SAT and Submodular Welfare with k players, for any number of players k. A framework for submodular optimization problems, called the contention resolution framework, was introduced recently by Chekuri et al. [11]. The improved approximation ratio of the unified continuous greedy algorithm implies improved ap- proximation ratios for many problems through this framework. Moreover, via a parameter called stopping time, our algorithm merges the relaxation solving and re-normalization steps of the framework, and achieves, for some applications, further improvements. We also describe new monotone balanced con- tention resolution schemes for various matching, scheduling and packing problems, thus, improving the approximations achieved for these problems via the framework.

We consider the problem of embedding a directed graph onto evenly spaced points on a circle while minimizing the total weighted edge length. We present the first poly-logarithmic approximation factor algorithm for this problem which yields an approximation factor of O (log n log log n ), thus improving the previous Õ (√ n ) approximation factor. In order to achieve this, we introduce a new problem which we call the directed penalized linear arrangement . This problem generalizes both the directed feedback edge set problem and the directed linear arrangement problem. We present an O (log n log log n )-approximation factor algorithm for this newly defined problem. Our solution uses two distinct directed metrics (“right” and “left”) which together yield a lower bound on the value of an optimal solution. In addition, we define a sequence of new directed spreading metrics that are used for applying the algorithm recursively on smaller subgraphs. The new spreading metrics allow us to define an asymmetric region growing procedure that accounts simultaneously for both incoming and outgoing edges. To the best of our knowledge, this is the first time that a region growing procedure is defined in directed graphs that allows for such an accounting.

We consider the k-balanced partitioning problem, where the goal is to partition the vertices of an input graph G into k equally sized components, while minimizing the total weight of the edges connecting different components. We allow k to be part of the input and denote the cardinality of the vertex set by n. This problem is a natural and important generalization of well-known graph partitioning problems, including minimum bisection and minimum balanced cut. We present a (bi-criteria) approximation algorithm achieving an approximation of , which matches or improves over previous algorithms for all relevant values of k. Our algorithm uses a semidefinite relaxation which combines metrics with spreading metrics. Surprisingly, we show that the integrality gap of the semidefinite relaxation is Ω(log k) even for large values of k (e.g., k = nΩ(1)), implying that the dependence on k of the approximation factor is necessary. This is in contrast to previous approximation algorithms for k-balanced partitioning, which are based on linear programming relaxations and their approximation factor is independent of k.

The connection between integrality gaps and computational hardness of discrete optimization problems is an intriguing question. In recent years, this connection has prominently figured in several tight UGC-based hardness results. We show in this paper a direct way of turning integrality gaps into hardness results for several fundamental classification problems. Specifically, we convert linear programming integrality gaps for the Multiway Cut, 0-Extension, and and Metric Labeling problems into UGC-based hardness results. Qualitatively, our result suggests that if the unique games conjecture is true then a linear relaxation of the latter problems studied in several papers (so-called earthmover linear program) yields the best possible approximation. Taking this a step further, we also obtain integrality gaps for a semi-definite programming relaxation matching the integrality gaps of the earthmover linear program. Prior to this work, there was an intriguing possibility of obtaining better approximation factors for labeling problems via semi-definite programming.

We define the balanced metric labeling problem, a generalization of the metric labeling problem, in which each label has a capacity, i.e., at most l vertices can be assigned to it. The balanced metric labeling problem is a generalization of fundamental problems in the area of approximation algorithms, e.g., arrangements and balanced partitions of graphs. It is also motivated by resource limitations in certain practical scenarios. We focus on the case where the given metric is uniform and note that this case alone encompasses various well-known graph partitioning problems. We present the first (pseudo) approximation algorithm for this problem, achieving for any ε, 0 < ε < 1, an approximation factor of O((ln n)/ε), while assigning at most min {O(ln k)/1 - ε, l + 1| ( 1 + ε) l vertices to each label (k is the number of labels). Our approximation algorithm is based on a novel randomized rounding of a linear programming formulation that combines an embedding of the graph in a simplex together with spreading metrics and additional constraints that strengthen the formulation. Our randomized rounding technique uses both a randomized metric decomposition technique and a randomized label assignment technique. At the heart of our approach is the fact that only limited dependency is created between the labels assigned to different vertices, allowing us to bound the expected cost of the solution and the number of vertices assigned to each label, simultaneously. We note that the number of vertices assigned to each label is bounded via a new inequality of Janson[15] for tail bounds of (partly) dependent random variables.