Our main contribution is a general framework to design efficient polynomial time approximation schemes (EPTAS) for fundamental stochastic combinatorial optimization problems. Given an error parameter ε>0, such algorithmic schemes attain a (1-ε)-approximation in t(ε)· poly(n) time, where t(·) is some function that depends only on ε. Technically speaking, our approach relies on presenting tailor-made reductions to a newly-introduced multi-dimensional load balancing problem. Even though the single-dimensional problem is already known to be APX-Hard, we prove that an EPTAS can be designed under certain structural assumptions, which hold for each of our applications. To demonstrate the versatility of our framework, we first study selection-stopping settings to derive an EPTAS for the Free-Order Prophets problem [Agrawal et al., EC'20] and for its cost-driven generalization, Pandora's Box with Commitment [Fu et al., ICALP'18]. These results constitute the first approximation schemes in the non-adaptive setting and improve on known inefficient polynomial time approximation schemes (PTAS) for their adaptive variants. Next, turning our attention to stochastic probing problems, we obtain an EPTAS for the adaptive ProbeMax problem as well as for its non-adaptive counterpart; in both cases, state-of-the-art approximability results have been inefficient PTASes [Chen et al., NIPS'16; Fu et al., ICALP'18].

In the adversarial edge arrival model for maximum cardinality matching, edges of an unknown graph are revealed one-by-one in arbitrary order, and should be irrevocably accepted or rejected. Here, the goal of an online algorithm is to maximize the number of accepted edges while maintaining a feasible matching at any point in time. For this model, the standard greedy heuristic is 1/2-competitive, and on the other hand, no algorithm that outperforms this ratio is currently known, even for very simple graphs. We present a clean Min-Index framework for devising a family of randomized algorithms, and provide a number of positive and negative results in this context. Among these results, we present a 5/9-competitive algorithm when the underlying graph is a forest, and prove that this ratio is best possible within the Min-Index framework. In addition, we prove a new general upper bound of 2/(3+1/phi^2) ~ 0.5914 on the competitiveness of any algorithm in the edge arrival model. Interestingly, this bound holds even for an easier model in which vertices (along with their adjacent edges) arrive online, and when the underlying graph is a tree of maximum degree at most 3.

We consider the assortment optimization problem when customer preferences follow a mixture of Mallows distributions. The assortment optimization problem focuses on determining the revenue/profit maximizing subset of products from a large universe of products; it is an important decision that is commonly faced by retailers in determining what to offer their customers. There are two key challenges: (a) the Mallows distribution lacks a closed-form expression (and requires summing an exponential number of terms) to compute the choice probability and, hence, the expected revenue/profit per customer; and (b) finding the best subset may require an exhaustive search. Our key contributions are an efficiently computable closed-form expression for the choice probability under the Mallows model and a compact mixed integer linear program (MIP) formulation for the assortment problem.

Assortment planning is an important problem that arises in many industries such as retailing and airlines where one of the key challenges is to identify the "right model" for the consumer preferences and substitution behavior. Distribution over preference lists or permutations is the most general framework for modeling preferences but is intractable in general. In this paper, we present a parsimonious distribution over permutations model that is induced by random swaps from a central preference list (which we also refer to as the prototype list. In particular, a random list from this distribution can be sampled as follows: sample N from a given distribution on the number of swaps and starting from the initial prototype list, perform N random swaps. A random swap operation consists of selecting a random pair of items in the current list and swapping their positions. We consider two types of random swaps: i) swapping an arbitrary pair of items, and $ii)$ swapping an adjacent pair of items. More precisely, a pair is picked uniformly at random out of all n+1\choose 2 possible pairs in i), and out all all $n$ pairs of adjacent items in ii). If the distribution over the number of swaps has sufficient large support, the distribution over permutations has non-zero probability on all preference lists. This model is motivated by practical applications where consumers preferences generally have many common items appear according to the same relative order and differ only in a small number of items. The prototype list used to generate a random preference list can intuitively be thought of as the mode of the distribution implied by the random swap model. This model also captures the well known Mallows distribution over permutation that is specified by a model permutation and a concentration parameter that determines how the probability of permutations decrease as a function of the distance from the modal permutation. Therefore, this is a fairly general model for consumer preferences. This model is motivated by practical applications where consumer preference are more or less similar over most items and differ in the relative order of only a few items.

When designing a preemptive online algorithm for the maximum matching problem, we wish to maintain a valid matching M while edges of the underlying graph are presented one after the other. When presented with an edge e, the algorithm should decide whether to augment the matching M by adding e (in which case e may be removed later on) or to keep M in its current form without adding e (in which case e is lost for good). The objective is to eventually hold a matching M with maximum weight. The main contribution of this paper is to establish new lower and upper bounds on the competitive ratio achievable by preemptive online algorithms: - We provide a lower bound of 1 + ln 2 \approx 1.693 on the competitive ratio of any randomized algorithm for the maximum cardinality matching problem, thus improving on the currently best known bound of e / (e-1) \approx 1.581 due to Karp, Vazirani, and Vazirani [STOC'90]. - We devise a randomized algorithm that achieves an expected competitive ratio of 5.356 for maximum weight matching. This finding demonstrates the power of randomization in this context, showing how to beat the tight bound of 3 + 2\sqrt{2} \approx 5.828 for deterministic algorithms, obtained by combining the 5.828 upper bound of McGregor [APPROX'05] and the recent 5.828 lower bound of Varadaraja [ICALP'11].

We study the maximum weight matching problem in the semi-streaming model, and improve on the currently best one-pass algorithm due to Zelke [Proceedings of the 25th Annual Symposium on Theoretical Aspects of Computer Science, 2008, pp. 669–680] by devising a deterministic approach whose performance guarantee is [Formula: see text]. In addition, we study preemptive online algorithms, a class of algorithms related to one-pass semi-streaming algorithms, where we are allowed to maintain only a feasible matching in memory at any point in time. We provide a lower bound of 4.967 on the competitive ratio of any such deterministic algorithm, and hence show that future improvements will have to store in memory a set of edges that is not necessarily a feasible matching. We conclude by presenting an empirical study, conducted in order to compare the practical performance of our approach to that of previously suggested algorithms.

In the generalized connectivity problem, we are given an edge-weighted graph G = ( V , E ) and a collection D = {( S 1 , T 1 ), …, ( S k , T k )} of distinct demands each demand ( S i , T i ) is a pair of disjoint vertex subsets. We say that a subgraph F of G connects a demand ( S i , T i ) when it contains a path with one endpoint in S i and the other in T i . The goal is to identify a minimum weight subgraph that connects all demands in D . Alon et al. (SODA '04) introduced this problem to study online network formation settings and showed that it captures some well-studied problems such as Steiner forest, facility location with nonmetric costs, tree multicast, and group Steiner tree. Obtaining a nontrivial approximation ratio for generalized connectivity was left as an open problem. We describe the first poly-logarithmic approximation algorithm for generalized connectivity that has a performance guarantee of O (log 2 n log 2 k ). Here, n is the number of vertices in G and k is the number of demands. We also prove that the cut-covering relaxation of this problem has an O (log 3 n log 2 k ) integrality gap. Building upon the results for generalized connectivity, we obtain improved approximation algorithms for two problems that contain generalized connectivity as a special case. For the directed Steiner network problem, we obtain an O ( k 1/2 + ϵ ) approximation which improves on the currently best performance guarantee of Õ ( k 2/3 ) due to Charikar et al. (SODA '98). For the set connector problem, recently introduced by Fukunaga and Nagamochi (IPCO '07), we present a poly-logarithmic approximation; this result improves on the previously known ratio which can be Ω( n ) in the worst case.

We study the maximum weight matching problem in the semi-streaming model, and improve on the currently best one-pass algorithm due to Zelke (Proc.\ STACS~'08, pages 669--680) by devising a deterministic approach whose performance guarantee is $4.91 + \eps$. In addition, we study {\em preemptive} online algorithms, a sub-class of one-pass algorithms where we are only allowed to maintain a feasible matching in memory at any point in time. All known results prior to Zelke's belong to this sub-class. We provide a lower bound of $4.967$ on the competitive ratio of any such deterministic algorithm, and hence show that future improvements will have to store in memory a set of edges which is not necessarily a feasible matching. We conclude by presenting an empirical study, conducted in order to compare the practical performance of our approach to that of previously suggested algorithms.

An instance of the sorting buffer problem consists of a metric space and a server, equipped with a finite-capacity buffer capable of holding a limited number of requests. An additional ingredient of the input is an online sequence of requests, each of which is characterized by a destination in the given metric space; whenever a request arrives, it must be stored in the sorting buffer. At any point in time, a currently pending request can be served by drawing it out of the buffer and moving the server to its corresponding destination. The objective is to serve all input requests in a way that minimizes the total distance traveled by the server. In this article, we focus our attention on instances of the problem in which the underlying metric is either an evenly-spaced line metric or a continuous line metric . Our main findings can be briefly summarized as follows. (1) We present a deterministic O (log n )-competitive algorithm for n -point evenly-spaced line metrics. This result improves on a randomized O (log 2 n )-competitive algorithm due to Khandekar and Pandit [2006b]. It also refutes their conjecture, stating that a deterministic strategy is unlikely to obtain a nontrivial competitive ratio. (2) We devise a deterministic O (log N log log N )-competitive algorithm for continuous line metrics, where N denotes the length of the input sequence. In this context, we introduce a novel discretization technique of independent interest. (3) We establish the first nontrivial lower bound for the evenly-spaced case, by proving that the competitive ratio of any deterministic algorithm is at least 2 + √3/√3 ≈ 2.154. This result settles, to some extent, an open question due to Khandekar and Pandit [2006b], who posed the task of attaining lower bounds on the achievable competitive ratio as a foundational objective for future research.

We consider the problem of approximating a set P of n points in Rd by a collection of j-dimensional flats, andextensions thereof, under the standard median / mean / centermeasures, in which we wish to minimize, respectively, the sum of thedistances from each point of P to its nearest flat, the sum of thesquares of these distances, or the maximal such distance.Such problems cannot be approximated unless P=NP but do allowbi-criteria approximations where one allows some leeway in both the numberof flats and the quality of the objective function.We give a very simple bi-criteria approximation algorithm, which producesat most α(k,j,n) = (k j log n)O(j) flats, which exceeds the optimalobjective value for any k j-dimensional flats by a factor of nomore than β(j)= 2O(j). Given this bi-criteria approximation, wecan use it to reduce the approximation factor arbitrarily, at the costof increasing the number of flats. Our algorithm hasmany advantages over previous work, in that it is muchmore widely applicable (wider set of objective functions and classes ofclusters) and much more efficient -- reducing the running time bound from O(n Poly(k,j)) to nd · (jk)O(j).Our algorithm is randomized and successful with probability 1/2(easily boosted to probabilities arbitrary close to 1).

Consider a graph problem which is associated with a parameter, for example, that of finding a longest tour spanning k vertices. The following question is natural: Is there a small subgraph that contains an optimal or near optimal solution for every possible value of the given parameter? Such a subgraph is said to be robust . In this article we consider the problems of finding heavy paths and heavy trees of k edges. In these two cases, we prove surprising bounds on the size of a robust subgraph for a variety of approximation ratios. For both problems, we show that in every complete weighted graph on n vertices there exists a subgraph with approximately α/1−α 2 n edges that contains an α-approximate solution for every k = 1,…, n − 1. In the analysis of the tree problem, we also describe a new result regarding balanced decomposition of trees. In addition, we consider variants in which the subgraph itself is restricted to be a path or a tree. For these problems, we describe polynomial time algorithms and corresponding proofs of negative results.