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This report documents the program and the outcomes of Dagstuhl Seminar 23061 "Scheduling". The seminar focused on the emerging models for beyond-worst case algorithm design, in particular, recent approaches that incorporate learning. This includes models for the integration of learning into algorithm design that have been proposed recently and that have already demonstrated advances in the state-of-art for various scheduling applications: (i) scheduling with error-prone learned predictions, (ii) data-driven algorithm design, and (iii) stochastic and Bayesian learning in scheduling.

The hitting set problem is a well-known NP-hard optimization problem in which, given a set of elements and a collection of subsets, the goal is to find the smallest selection of elements, such that each subset contains at least one element in the selection. Many geometric set systems enjoy improved approximation ratios, which have recently been shown to be tight with respect to the shallow cell complexity of the set system. The algorithms that exploit the cell complexity, however, tend to be involved and computationally intensive. This paper shows that a slightly improved asymptotic approximation ratio for the hitting set problem can be attained using a much simpler algorithm: solve the linear programming relaxation, take one initial random sample from the set of elements with probabilities proportional to the LP-solution, and, while there is an unhit set, take an additional sample from it proportional to the LP-solution. Our algorithm is a simple generalization of the elegant net-finder algorithm by Nabil Mustafa. To analyze this algorithm for the hitting set problem, we generalize the classic Packing Lemma, and the more recent Shallow Packing Lemma, to the setting of weighted epsilon-nets., Comment: Accepted by WAOA2023

Understanding how subsets of items are chosen from offered sets is critical to assortment planning, wireless network planning, and many other applications. There are two seemingly unrelated subset choice models that capture dependencies between items: intuitive and interpretable random utility models; and tractable determinantal point processes (DPPs). This paper connects the two. First, all DPPs are shown to be random utility models. Next, a determinantal choice model that enjoys the best of both worlds is specified; the model is shown to subsume logistic regression when dependence is minimal, and MNL when dependence is maximally negative. This makes the model interpretable, while retaining the tractability of DPPs. A simulation study verifies that the model can learn a continuum of negative dependencies from data, and an applied study using original experimental data produces novel insights on wireless interference in LoRa networks., Comment: submitted to ICML2023

We introduce the technique of decomposing an undirected graph by finding a maximal set of edge-disjoint cycles. We give a parallel algorithm to find this decomposition in O(log n) time on (m+ n)/log n processors. We then use this decomposition to to give the first efficient parallel algorithm for finding an approximation to a minimum cycle cover.

Electric vehicle (EV) adoption in long-distance logistics faces challenges such as range anxiety and uneven distribution of charging stations. Two pivotal questions emerge: How can EVs be efficiently routed in a charging network considering range limits, charging speeds and prices? And, can the existing charging infrastructure sustain the increasing demand for EVs in long-distance logistics? This paper addresses these questions by introducing a novel theoretical and computational framework to study the EV network flow problems. We present an EV network flow model that incorporates range constraints and nonlinear charging rates, and identify conditions under which polynomial-time solutions can be obtained for optimal single EV routing, maximum flow, and minimum-cost flow problems. Our findings provide insights for optimizing EV routing in logistics, ensuring an efficient and sustainable future.

Sharing infrastructure between many users is often advantageous, however finding a fair and reasonable way to allocate its cost between its users can be challenging. This is particularly true for LPWANs, a popular Internet of Things solution for wirelessly connecting devices to the internet. We study cost-allocation of LPWANS using a covering integer program. Standard cost-allocation methods are inapplicable in this model, because the integrality gap of its natural LP-relaxation is unbounded. We overcome this challenge by strengthening the natural LP with knapsack-cover inequalities. Our main result is proving that all dual-feasible solutions to the strengthened LP produce cost-allocations that satisfy the core property. This reduces the problem of finding a cost-allocation to that of finding a strengthened-LP-relative approximation algorithm. Existing algorithms imply improved cost-recovery ratios for families of sparse CIP instances. Finally, we show that the strengthened formulation simplifies and improves the analysis of a cross-monotone cost-allocation mechanism as well., Comment: Preprint

The joint replenishment problem (JRP) is a classical inventory management problem. We consider a natural generalization with outliers, where we are allowed to reject (that is, not service) a subset of demand points. In this paper, we are motivated by issues of fairness - if we do not serve all of the demands, we wish to ``spread out the pain'' in a balanced way among customers, communities, or any specified market segmentation. One approach is to constrain the rejections allowed, and to have separate bounds for each given customer. In our most general setting, we consider a set of C features, where each demand point has an associated rejection cost for each feature, and we have a given bound on the allowed rejection cost incurred in total for each feature. This generalizes a model of fairness introduced in earlier work on the Colorful k-Center problem in which (analogously) each demand point has a given color, and we bound the number of rejections of each color class. We give the first constant approximation algorithms for the fairness-constrained JRP with a constant number of features; specifically, we give a 2.86-approximation algorithm in this case. Even for the special case in which we bound the total (weighted) number of outliers, this performance guarantee improves upon bounds previously known for this case. Our approach is an LP-based algorithm that splits the instance into two subinstances. One is solved by a novel iterative rounding approach and the other by pipage-based rounding. The standard LP relaxation has an unbounded integrality gap, and hence another key element of our algorithm is to strengthen the relaxation by correctly guessing key attributes of the optimal solution, which are sufficiently concise, so that we can enumerate over all possible guesses in polynomial time - albeit exponential in C, the number of features., Comment: 43 pages, 6 figures

There is significant interest in deploying machine learning algorithms for diagnostic radiology, as modern learning techniques have made it possible to detect abnormalities in medical images within minutes. While machine-assisted diagnoses cannot yet reliably replace human reviews of images by a radiologist, they could inform prioritization rules for determining the order by which to review patient cases so that patients with time-sensitive conditions could benefit from early intervention. We study this scenario by formulating it as a learning-augmented online scheduling problem. We are given information about each arriving patient's urgency level in advance, but these predictions are inevitably error-prone. In this formulation, we face the challenges of decision making under imperfect information, and of responding dynamically to prediction error as we observe better data in real-time. We propose a simple online policy and show that this policy is in fact the best possible in certain stylized settings. We also demonstrate that our policy achieves the two desiderata of online algorithms with predictions: consistency (performance improvement with prediction accuracy) and robustness (protection against the worst case). We complement our theoretical findings with empirical evaluations of the policy under settings that more accurately reflect clinical scenarios in the real world.

A retrospective fleet-sizing problem can be solved via bipartite matching, where a maximum cardinality matching corresponds to the minimum number of vehicles needed to cover all trips. We prove a min-max theorem on this minimum fleet-size problem: the maximum number of pairwise incompatible trips is equal to the minimum fleet size needed.

The Simplex algorithm for solving linear programs-one of Computing in Science & Engineering's top 10 most influential algorithms of the 20th century-is an important topic in many algorithms courses. While the Simplex algorithm relies on intuitive geometric ideas, the computationally-involved mechanics of the algorithm can obfuscate a geometric understanding. In this paper, we present gilp, an easy-to-use Simplex algorithm visualization tool designed to explicitly connect the mechanical steps of the algorithm with their geometric interpretation. We provide an extensive library with example visualizations, and our tool allows an instructor to quickly produce custom interactive HTML files for students to experiment with the algorithm (without requiring students to install anything!). The tool can also be used for interactive assignments in Jupyter notebooks, and has been incorporated into a forthcoming Data Science and Decision Making interactive textbook. In this paper, we first describe how the tool fits into the existing literature on algorithm visualizations: how it was designed to facilitate student engagement and instructor adoption, and how it substantially extends existing algorithm visualization tools for Simplex. We then describe the development and usage of the tool, and report feedback from its use in a course with roughly 100 students. Student feedback was overwhelmingly positive, with students finding the tool easy to use: it effectively helped them link the algebraic and geometrical views of the Simplex algorithm and understand its nuances. Finally, gilp is open-source, includes an extension to visualizing linear programming-based branch and bound, and is readily amenable to further extensions., Comment: ACM SIGCSE 2023 Manuscript, 12 pages, 6 figures

The recently introduced online Minimum Peak Appointment Scheduling (MPAS) problem is a variant of the online bin-packing problem that allows for deferred decision making. Specifically, it allows for the problem to be split into an online phase where a stream of appointment requests arrive requiring a scheduled time, followed by an offline phase where those appointments are scheduled into rooms. Similar to the bin-packing problem, the aim is to use the minimum number of rooms in the final configuration. This model more accurately captures scheduling appointments than bin packing. For example, a dialysis patient needs to know what time to arrive for an appointment, but does not need to know the assigned station ahead of time. Previous work developed a randomized algorithm for this problem which achieved an asymptotic competitive ratio of at most 1.5, proving that online MPAS was fundamentally different from the online bin-packing problem. Our main contribution is to develop a new randomized algorithm for the problem that achieves an asymptotic competitive ratio under 1.455, indicating the potential for further progress. This improvement is attained by modifying the process for scheduling appointments to increase the density of the packing in the worst case, along with utilizing the dual of the bin-packing linear programming relaxation to perform the analysis. We also present the first known lower bound of 1.2 on the asymptotic competitive ratio of both deterministic and randomized online MPAS algorithm. These results demonstrate how deferred decision-making can be leveraged to yield improved worst-case performance, a phenomenon which should be investigated in a broader class of settings.

Every representative democracy must specify a mechanism under which voters choose their representatives. The most common mechanism in the United States -- Winner takes all single-member districts -- both enables substantial partisan gerrymandering and constrains `fair' redistricting, preventing proportional representation in legislatures. We study the design of multi-member districts (MMDs), in which each district elects multiple representatives, potentially through a non-Winner takes all voting rule. We carry out large-scale empirical analyses for the U.S. House of Representatives under MMDs with different social choice functions, under algorithmically generated maps optimized for either partisan benefit or proportionality. Doing so requires efficiently incorporating predicted partisan outcomes -- under various multi-winner social choice functions -- into an algorithm that optimizes over an ensemble of maps. We find that with three-member districts using Single Transferable Vote, fairness-minded independent commissions would be able to achieve proportional outcomes in every state up to rounding, and advantage-seeking partisans would have their power to gerrymander significantly curtailed. Simultaneously, such districts would preserve geographic cohesion, an arguably important aspect of representative democracies. In the process, we advance a rich research agenda at the intersection of social choice and computational gerrymandering., Comment: 34 pages

The American winner-take-all congressional district system empowers politicians to engineer electoral outcomes by manipulating district boundaries. Existing computational solutions mostly focus on drawing unbiased maps by ignoring political and demographic input, and instead simply optimize for compactness. We claim that this is a flawed approach because compactness and fairness are orthogonal qualities, and introduce a scalable two-stage method to explicitly optimize for arbitrary piecewise-linear definitions of fairness. The first stage is a randomized divide-and-conquer column generation heuristic which produces an exponential number of distinct district plans by exploiting the compositional structure of graph partitioning problems. This district ensemble forms the input to a master selection problem to choose the districts to include in the final plan. Our decoupled design allows for unprecedented flexibility in defining fairness-aligned objective functions. The pipeline is arbitrarily parallelizable, is flexible to support additional redistricting constraints, and can be applied to a wide array of other regionalization problems. In the largest ever ensemble study of congressional districts, we use our method to understand the range of possible expected outcomes and the implications of this range on potential definitions of fairness.

This report documents the program and the outcomes of Dagstuhl Seminar 20081 "Scheduling". The seminar focused on the interplay between scheduling problems and problems that arise in the management of transportation and traffic. Important aspects at the intersection of these two research directions include data-driven approaches in dynamic decision-making, scheduling in combination with routing, shared mobility, and coordination versus competition.

This report documents the program and the outcomes of Dagstuhl Seminar 20081 "Scheduling". The seminar focused on the interplay between scheduling problems and problems that arise in the management of transportation and traffic. Important aspects at the intersection of these two research directions include data-driven approaches in dynamic decision-making, scheduling in combination with routing, shared mobility, and coordination versus competition.

In the winter of 1976, Alexander Rinnooy Kan and Jan Karel Lenstra defended their PhD theses at the University of Amsterdam. Gene Lawler was on their committees. It was a natural idea to turn the theses into a textbook on scheduling. They set out to compile a survey with Ron Graham (1979), but progress on the book was hampered by the many research opportunities offered by the field. After David Shmoys joined the team in the mid 1980's, several chapters were drafted, and the survey was rewritten (1993). Gene passed away in 1994. Colleagues were asked to contribute chapters or to complete existing drafts. However, by the turn of the century the project was losing its momentum, and finite convergence to completion fell beyond our reach. Over the years, several chapters have been used in the classroom. We continue to receive requests from colleagues who look for a text on the elements of scheduling at an advanced undergraduate or early graduate level. This document is a compilation of what currently exists. We have made a marginal effort in patching it up at some places but is essentially what was written long ago. We did make an attempt to include most of the citations in the bibliography.

We present the first nontrivial approximation algorithm for the bottleneck asymmetric traveling salesman problem. Given an asymmetric metric cost between n vertices, the problem is to find a Hamiltonian cycle that minimizes its bottleneck (or maximum-length edge) cost. We achieve an O(log n / log log n) approximation performance guarantee by giving a novel algorithmic technique to shortcut Eulerian circuits while bounding the lengths of the shortcuts needed. This allows us to build on a related result of Asadpour, Goemans, M\k{a}dry, Oveis Gharan, and Saberi to obtain this guarantee. Furthermore, we show how our technique yields stronger approximation bounds in some cases, such as the bounded orientable genus case studied by Oveis Gharan and Saberi. We also explore the possibility of further improvement upon our main result through a comparison to the symmetric counterpart of the problem., Comment: 16 pages, 3 figures

We consider constrained versions of the prize-collecting traveling salesman and the minimum spanning tree problems. The goal is to maximize the number of vertices in the returned tour/tree subject to a bound on the tour/tree cost. We present a 2-approximation algorithm for these problems based on a primal-dual approach. The algorithm relies on finding a threshold value for the dual variable corresponding to the budget constraint in the primal and then carefully constructing a tour/tree that is just within budget. Thereby, we improve the best-known guarantees from 3+epsilon and 2+epsilon for the tree and the tour version, respectively. Our analysis extends to the setting with weighted vertices, in which we want to maximize the total weight of vertices in the tour/tree subject to the same budget constraint.

We consider constrained versions of the prize-collecting traveling salesman and the minimum spanning tree problems. The goal is to maximize the number of vertices in the returned tour/tree subject to a bound on the tour/tree cost. We present a 2-approximation algorithm for these problems based on a primal-dual approach. The algorithm relies on finding a threshold value for the dual variable corresponding to the budget constraint in the primal and then carefully constructing a tour/tree that is just within budget. Thereby, we improve the best-known guarantees from 3+epsilon and 2+epsilon for the tree and the tour version, respectively. Our analysis extends to the setting with weighted vertices, in which we want to maximize the total weight of vertices in the tour/tree subject to the same budget constraint.

We consider the following single-machine scheduling problem, which is often denoted $1||\sum f_{j}$: we are given $n$ jobs to be scheduled on a single machine, where each job $j$ has an integral processing time $p_j$, and there is a nondecreasing, nonnegative cost function $f_j(C_{j})$ that specifies the cost of finishing $j$ at time $C_{j}$; the objective is to minimize $\sum_{j=1}^n f_j(C_j)$. Bansal \& Pruhs recently gave the first constant approximation algorithm with a performance guarantee of 16. We improve on this result by giving a primal-dual pseudo-polynomial-time algorithm based on the recently introduced knapsack-cover inequalities. The algorithm finds a schedule of cost at most four times the constructed dual solution. Although we show that this bound is tight for our algorithm, we leave open the question of whether the integrality gap of the LP is less than 4. Finally, we show how the technique can be adapted to yield, for any $\epsilon >0$, a $(4+\epsilon )$-approximation algorithm for this problem., Comment: 26 pages. A preliminary version appeared in APPROX 2011. arXiv admin note: text overlap with arXiv:1403.0298

The growing popularity of bike-sharing systems around the world has motivated recent attention to models and algorithms for their effective operation. Most of this literature focuses on their daily operation for managing asymmetric demand. In this work, we consider the more strategic question of how to (re-)allocate dock-capacity in such systems. We develop mathematical formulations for variations of this problem (either for service performance over the course of one day or for a long-run-average) and exhibit discrete convex properties in associated optimization problems. This allows us to design a practically fast polynomial-time allocation algorithm to compute an optimal solution for this problem, which can also handle practically motivated constraints, such as a limit on the number of docks moved in the system. We apply our algorithm to data sets from Boston, New York City, and Chicago to investigate how different dock allocations can yield better service in these systems. Recommendations based on our analysis have led to changes in the system design in Chicago and New York City. Beyond optimizing for improved quality of service through better allocations, our results also provide a metric to compare the impact of strategically reallocating docks and the rebalancing of bikes.

The joint replenishment problem is a fundamental model in supply chain management theory that has applications in inventory management, logistics, and maintenance scheduling. In this problem, there are multiple item types, each having a given time-dependent sequence of demands that need to be satisfied. In order to satisfy demand, orders of the item types must be placed in advance of the due dates for each demand. Every time an order of item types is placed, there is an associated joint setup cost depending on the subset of item types ordered. This ordering cost can be due to machine, transportation, or labor costs, for example. In addition, there is a cost to holding inventory for demand that has yet to be served. The overall goal is to minimize the total ordering costs plus inventory holding costs. In this paper, the cost of an order, also known as a joint setup cost, is a monotonically increasing, submodular function over the item types. For this general problem, we show that a greedy approach provides an approximation guarantee that is logarithmic in the number of demands. Then we consider three special cases of submodular functions which we call the laminar, tree, and cardinality cases, each of which can model real world scenarios that previously have not been captured. For each of these cases, we provide a constant factor approximation algorithm. Specifically, we show that the laminar case can be solved optimally in polynomial time via a dynamic programming approach. For the tree and cardinality cases, we provide two different linear programming based approximation algorithms that provide guarantees of three and five, respectively., National Science Foundation (U.S.) (CAREER Grant CMMI-0846554), United States. Air Force Office of Scientific Research (Award FA9550-11-1-0150), SMA Grant, Solomon Buchsbaum AT&T Research Fund

Computing the group of units in a field of algebraic numbers is one of the central tasks of computational algebraic number theory. It is believed to be hard classically, which is of interest for cryptography. In the quantum setting, efficient algorithms were previously known for fields of constant degree. We give a quantum algorithm that is polynomial in the degree of the field and the logarithm of its discriminant. This is achieved by combining three new results. The first is a classical algorithm for computing a basis for certain ideal lattices with doubly exponentially large generators. The second shows that a Gaussian-weighted superposition of lattice points, with an appropriate encoding, can be used to provide a unique representation of a real-valued lattice. The third is an extension of the hidden subgroup problem to continuous groups and a quantum algorithm for solving the HSP over the group ℝ^n.

We introduce a new optimization framework to maximize the expected spread of cascades in networks. Our model allows a rich set of actions that directly manipulate cascade dynamics by adding nodes or edges to the network. Our motivating application is one in spatial conservation planning, where a cascade models the dispersal of wild animals through a fragmented landscape. We propose a mixed integer programming (MIP) formulation that combines elements from network design and stochastic optimization. Our approach results in solutions with stochastic optimality guarantees and points to conservation strategies that are fundamentally different from naive approaches., Comment: Appears in Proceedings of the Twenty-Sixth Conference on Uncertainty in Artificial Intelligence (UAI2010)

This paper has been merged into 1110.4604., Comment: This paper has been merged into 1110.4604

We present a deterministic (1+sqrt(5))/2-approximation algorithm for the s-t path TSP for an arbitrary metric. Given a symmetric metric cost on n vertices including two prespecified endpoints, the problem is to find a shortest Hamiltonian path between the two endpoints; Hoogeveen showed that the natural variant of Christofides' algorithm is a 5/3-approximation algorithm for this problem, and this asymptotically tight bound in fact has been the best approximation ratio known until now. We modify this algorithm so that it chooses the initial spanning tree based on an optimal solution to the Held-Karp relaxation rather than a minimum spanning tree; we prove this simple but crucial modification leads to an improved approximation ratio, surpassing the 20-year-old barrier set by the natural Christofides' algorithm variant. Our algorithm also proves an upper bound of (1+sqrt(5))/2 on the integrality gap of the path-variant Held-Karp relaxation. The techniques devised in this paper can be applied to other optimization problems as well: these applications include improved approximation algorithms and improved LP integrality gap upper bounds for the prize-collecting s-t path problem and the unit-weight graphical metric s-t path TSP., Comment: 31 pages, 5 figures

Stochastic optimization problems provide a means to model uncertainty in the input data where the uncertainty is modeled by a probability distribution over the possible realizations of the data. We consider a broad class of these problems, called {it multi-stage stochastic programming problems with recourse}, where the uncertainty evolves through a series of stages and one take decisions in each stage in response to the new information learned. These problems are often computationally quite difficult with even very specialized (sub)problems being $#P$-complete. We obtain the first fully polynomial randomized approximation scheme (FPRAS) for a broad class of multi-stage stochastic linear programming problems with any constant number of stages, without placing any restrictions on the underlying probability distribution or on the cost structure of the input. For any fixed $k$, for a rich class of $k$-stage stochastic linear programs (LPs), we show that, for any probability distribution, for any $epsilon>0$, one can compute, with high probability, a solution with expected cost at most $(1+e)$ times the optimal expected cost, in time polynomial in the input size, $frac{1}{epsilon}$, and a parameter $lambda$ that is an upper bound on the cost-inflation over successive stages. Moreover, the algorithm analyzed is a simple and intuitive algorithm that is often used in practice, the {it sample average approximation} (SAA) method. In this method, one draws certain samples from the underlying distribution, constructs an approximate distribution from these samples, and solves the stochastic problem given by this approximate distribution. This is the first result establishing that the SAA method yields near-optimal solutions for (a class of) multi-stage programs with a polynomial number of samples. As a corollary of this FPRAS, by adapting a generic rounding technique of Shmoys and Swamy, we also obtain the first approximation algorithms for the analogous class of multi-stage sto

Stochastic optimization problems provide a means to model uncertainty in the input data where the uncertainty is modeled by a probability distribution over the possible realizations of the data. We consider a broad class of these problems, called {it multi-stage stochastic programming problems with recourse}, where the uncertainty evolves through a series of stages and one take decisions in each stage in response to the new information learned. These problems are often computationally quite difficult with even very specialized (sub)problems being $#P$-complete. We obtain the first fully polynomial randomized approximation scheme (FPRAS) for a broad class of multi-stage stochastic linear programming problems with any constant number of stages, without placing any restrictions on the underlying probability distribution or on the cost structure of the input. For any fixed $k$, for a rich class of $k$-stage stochastic linear programs (LPs), we show that, for any probability distribution, for any $epsilon>0$, one can compute, with high probability, a solution with expected cost at most $(1+e)$ times the optimal expected cost, in time polynomial in the input size, $frac{1}{epsilon}$, and a parameter $lambda$ that is an upper bound on the cost-inflation over successive stages. Moreover, the algorithm analyzed is a simple and intuitive algorithm that is often used in practice, the {it sample average approximation} (SAA) method. In this method, one draws certain samples from the underlying distribution, constructs an approximate distribution from these samples, and solves the stochastic problem given by this approximate distribution. This is the first result establishing that the SAA method yields near-optimal solutions for (a class of) multi-stage programs with a polynomial number of samples. As a corollary of this FPRAS, by adapting a generic rounding technique of Shmoys and Swamy, we also obtain the first approximation algorithms for the analogous class of multi-stage sto

Stochastic optimization problems attempt to model uncertainty in the data by assuming that (part of) the input is specified by a probability distribution. We consider the well-studied paradigm of stochastic recourse models, where the uncertainty evolves through a series of stages and one can take decisions in each stage in response to the new information learned. We obtain the first approximation algorithms for a variety of 2-stage and k-stage stochastic integer optimization problems where the underlying random data is given by a "black box" and no restrictions are placed on the costs of the two stages: one can merely sample data from this distribution, but no direct information about the distributions is given. Our results are based on two principal components. First, we show that for a broad class of 2-stage and k-stage linear programs, where k is not part of the input, given only a "black box" to draw independent samples from the distribution, one can, for any epsilon>0, compute a solution of cost guaranteed to be within a (1+epsilon) factor of the optimum, in time polynomial in 1/epsilon, the size of the input, and a parameter lambda that is the ratio of the cost of the same action in successive stages which is a lower bound on the sample complexity in the "black-box" model. This is based on reformulating the stochastic linear program, which has both an exponential number of variables and an exponential number of constraints, as a compact convex program, and adapting tools from convex optimization to solve the resulting program to near optimality. In doing so, a significant difficulty that we must overcome is that even evaluating the objective function of this convex program at a given point may be quite difficult and provably hard. To the best of our knowledge, this is the first such result for multi-stage stochastic programs. Second, we give a rounding approach for stochastic integer programs that shows that approximation algorithms for a deterministic analogu

Stochastic optimization problems provide a means to model uncertainty in the input data where the uncertainty is modeled by a probability distribution over the possible realizations of the actual data. We consider a broad class of these problems in which the realized input is revealed through a series of stages, and hence are called multi-stage stochastic programming problems. Our main result is to give the first fully polynomial approximation scheme for a broad class of multi-stage stochastic linear programming problems with any constant number of stages. The algorithm analyzed, known as the sample average approximation (SAA) method, is quite simple, and is the one most commonly used in practice. The algorithm accesses the input by means of a "black box" that can generate, given a series of outcomes for the initial stages, a sample of the input according to the conditional probability distribution (given those outcomes). We use this to obtain the first polynomial-time approximation algorithms for a variety of k-stage generalizations of basic combinatorial optimization problems.

Stochastic optimization problems attempt to model uncertainty in the data by assuming that (part of) the input is specified by a probability distribution. We consider the well-studied paradigm of stochastic recourse models, where the uncertainty evolves through a series of stages and one can take decisions in each stage in response to the new information learned. We obtain the first approximation algorithms for a variety of 2-stage and k-stage stochastic integer optimization problems where the underlying random data is given by a "black box" and no restrictions are placed on the costs of the two stages: one can merely sample data from this distribution, but no direct information about the distributions is given. Our results are based on two principal components. First, we show that for a broad class of 2-stage and k-stage linear programs, where k is not part of the input, given only a "black box" to draw independent samples from the distribution, one can, for any \epsilon>0, compute a solution of cost guaranteed to be within a (1+\epsilon) factor of the optimum, in time polynomial in 1/\epsilon, the size of the input, and a parameter \lambda that is the ratio of the cost of the same action in successive stages which is a lower bound on the sample complexity in the "black-box" model. This is based on reformulating the stochastic linear program, which has both an exponential number of variables and an exponential number of constraints, as a compact convex program, and adapting tools from convex optimization to solve the resulting program to near optimality. In doing so, a significant difficulty that we must overcome is that even evaluating the objective function of this convex program at a given point may be quite difficult and provably hard. To the best of our knowledge, this is the first such result for multi-stage stochastic programs. Second, we give a rounding approach for stochastic integer programs that shows that approximation algorithms for a deterministic ana

The research efforts reported here have centered in the areas of parallel and distributed computing, network architecture, combinatorial algorithms, and complexity theory. Significant progress has been made on the development of efficient sorting circuits, network management protocols for high-speed networks, distributed graph algorithms and data structures, improved algorithms for packet routing and sorting in parallel machines, algorithms for reconfiguring networks around faults, improved approximation algorithms for a variety of NP-hard optimization problems, VLSI design, and algorithms for combinatorial problems such as multicommodity flow.

We show that if there exists a deterministic oracle that can determine the sum of the bits in the binary representation of x when presented with the RSA encryption of x, then there exists a probabilistic algorithm using this oracle to recover x when presented with the RSA encryption of x. We present a similar result for Rabin encryption., Sponsored in part by the Defense Advanced Research Projects Agency.

In this paper we use interior-point methods for linear programming, developed in the context of sequential computation, to obtain a parallel algorithm for the bipartite matching problem. Our algorithm runs in O(square root of m) time. Our results extend to the weighted bipartite matching problem and to the zero-one minimum-cost flow problem, yielding O((square root of m) logC) algorithms. This improves previous bounds on these problems and illustrate the importance of interior-point methods in the context of parallel algorithm design.