Worst-case Guarantees Research Articles

A Parallelizable Acceleration Framework for Packing Linear Programs

This paper presents an acceleration framework for packing linear programming problems where the amount of data available is limited, i.e., where the number of constraints m is small compared to the variable dimension n. The framework can be used as a black box to speed up linear programming solvers dramatically, by two orders of magnitude in our experiments. We present worst-case guarantees on the quality of the solution and the speedup provided by the algorithm, showing that the framework provides an approximately optimal solution while running the original solver on a much smaller problem. The framework can be used to accelerate exact solvers, approximate solvers, and parallel/distributed solvers. Further, it can be used for both linear programs and integer linear programs.

Imperfect public monitoring with a fear of signal distortion

This paper proposes a model of signal distortion in a two-player game with imperfect public monitoring. We construct a tractable theoretical framework where each player has the opportunity to distort the true public signal and each player is uncertain about the distortion technologies available to the other player. We show that when players evaluate strategies according to their worst-case guarantees—i.e., are ambiguity averse over certain distributions in the environment—perceived continuation payoffs endogenously lie on a positively sloped line. We then provide examples showing that, counterintuitively, identifying deviators can be harmful in enforcing a strategy profile; moreover, we illustrate how the presence of such signal distortion can sustain cooperation when it is impossible in standard settings. We show that the main result and examples are robust to a number of natural modifications to our setting. Finally, we extend our model to a repeated game where our concept is a natural generalization of strongly symmetric equilibria. In this setting, we prove an anti-folk theorem, showing that payoffs under our equilibrium concept are under general conditions bounded away from efficiency.

Prioritized Metric Structures and Embedding

Metric data structures (distance oracles, distance labeling schemes, routing schemes) and low-distortion embeddings provide a powerful algorithmic methodology, which has been successfully applied for approximation algorithms [N. Linial, E. London, and Y. Rabinovich, Combinatorica, 15 (1995), pp. 215--245], online algorithms [N. Bansal et al., Proceedings of the 52th Annual IEEE Symposium on Foundations of Computer Science, FOCS '08, IEEE Computer Society, Washington, DC, 2011, pp. 267--276], distributed algorithms [M. Khan et al., Distrib. Comput., 25 (2012), pp. 189--205], and for computing sparsifiers [Y. Shavitt and T. Tankel, IEEE/ACM Trans. Netw., 12 (2004), pp. 993--1006]. However, this methodology appears to have a limitation: the worst-case performance inherently depends on the cardinality of the metric, and one could not specify in advance which vertices/points should enjoy a better service (i.e., stretch/distortion, label size/dimension) than that given by the worst-case guarantee. In this paper we alleviate this limitation by devising a suite of prioritized metric data structures and embeddings. We show that given a priority ranking $(x_1,x_2,\ldots,x_n)$ of the graph vertices (resp., metric points) one can devise a metric data structure (resp., embedding) in which the stretch (resp., distortion) incurred by any pair containing a vertex $x_j$ will depend on the rank $j$ of the vertex. We also show that other important parameters, such as the label size and (in some sense) the dimension, may depend only on $j$. In some of our metric data structures (resp., embeddings) we achieve both prioritized stretch (resp., distortion) and label size (resp., dimension) simultaneously. The worst-case performance of our metric data structures and embeddings is typically asymptotically no worse than of their nonprioritized counterparts.

Multiobjective $\mathcal{H}_2$/$\mathcal{H}_\infty$ Control Design Subject to Multiplicative Input Dependent Noises

This paper addresses the mixed $\mathcal{H}_2$/$\mathcal{H}_\infty$ control problem for linear discrete-time systems with structured multiplicative input noises. The structured random noises are modelled by a diagonal matrix where individual multiplicative noise is allowed to be different. The Nash game methodology is adopted to deal with such a multiobjective control problem, and a necessary and sufficient condition is analytically given in terms of two cross-coupled modified algebraic Riccati equations (MAREs). Based on the stabilizing solutions to MAREs, bounded causal equilibrium strategies are thus conducted and the resulting control law optimizes some specific performance together with a guaranteed worst case performance. Some relevant issues that can be regarded as special cases of the problem under consideration are also discussed. Moreover, a numerical example is included to show the validity of the present results.

Greedy Sampling of Graph Signals

Sampling is a fundamental topic in graph signal processing, having found applications in estimation, clustering, and video compression. In contrast to traditional signal processing, the irregularity of the signal domain makes selecting a sampling set non-trivial and hard to analyze. Indeed, though conditions for graph signal interpolation from noiseless samples exist, they do not lead to a unique sampling set. The presence of noise makes choosing among these sampling sets a hard combinatorial problem. Although greedy sampling schemes are commonly used in practice, they have no performance guarantee. This work takes a twofold approach to address this issue. First, universal performance bounds are derived for the Bayesian estimation of graph signals from noisy samples. In contrast to currently available bounds, they are not restricted to specific sampling schemes and hold for any sampling sets. Second, this paper provides near-optimal guarantees for greedy sampling by introducing the concept of approximate submodularity and updating the classical greedy bound. It then provides explicit bounds on the approximate supermodularity of the interpolation mean-square error showing that it can be optimized with worst-case guarantees using greedy search even though it is not supermodular. Simulations illustrate the derived bound for different graph models and show an application of graph signal sampling to reduce the complexity of kernel principal component analysis.

Social Influence Spectrum at Scale

Given a social network, the Influence Maximization (InfMax) problem seeks a seed set of k people that maximizes the expected influence for a viral marketing campaign. However, a solution for a particular seed size k is often not enough to make an informed choice regarding budget and cost-effectiveness. In this article, we propose the computation of Influence Spectrum (InfSpec), the maximum influence at each possible seed set size k within a given range [ k lower , k upper ], thus providing optimal decision making for any availability of budget or influence requirements. As none of the existing methods for InfMax are efficient enough for the task in large networks, we propose LISA (sub-Linear Influence Spectrum Approximation), an efficient approximation algorithm for InfSpec (and also InfMax) with the best-known worst-case guarantees for billion-scale networks. LISA returns an (1-1/e -ϵ)-approximate influence spectrum with high probability (1-δ), where ϵ, δ are precision parameters provided by users. Using statistical decision theory, LISA has an asymptotic optimal running time (in addition to optimal approximation guarantee). In practice, LISA surpasses the state-of-the-art InfMax methods, taking less than 15 minutes to process a network of 41.7 million nodes and 1.5 billions edges.

Surgery scheduling with recovery resources

ABSTRACTSurgical services are large revenue sources that account for a large portion of hospital expenses. Thus, efficient resource allocation is crucial in this system; however, this is a challenging problem, in part due to the interaction of the different stages of the surgery delivery system and the uncertainty of surgery and recovery durations. This article focuses on single-day in-patient elective surgery scheduling considering surgeons, operating rooms (ORs), and the post-anesthesia care unit (recovery). We propose a mixed-integer programming formulation of this problem and then present a fast two-phase heuristic: phase 1 is used for determining the number of ORs to open for the day and surgeon-to-OR assignments, and phase 2 is used for surgical case sequencing. Both phases have provable worst-case performance guarantees and excellent average case performance. We evaluate schedules under uncertainty using a discrete-event simulation model based on data provided by a mid-sized hospital. We show that the fast and easy-to-implement two-phase heuristic performs extremely well, in both deterministic and stochastic settings. The new methods developed reduce the computational barriers to implementation and demonstrate that hospitals can realize substantial benefits without resorting to sophisticated optimization software implementations.

The impact of processing order on performance: A taxonomy of semi-FIFO policies

Modern network processors increasingly deal with packets that require heterogeneous processing. We consider a bounded size input queue buffer where each packet requires several rounds of processing before transmission. Usually the transmission order of packets is induced by processing order, but processing order can have significant impact on the performance of buffer management policies even if the transmission order is fixed. We introduce the class of Semi-FIFO policies that decouple processing order from transmission order, restricting the latter to First-In-First-Out (FIFO). We build a taxonomy of Semi-FIFO policies and provide worst case guarantees for different processing orders. We consider various special cases and properties of Semi-FIFO policies: greedy, work-conserving, lazy, and push-out policies, and show how these properties affect performance. We generalize our results to additional constraints related to copying cost and conduct a comprehensive simulation study that validates our results.

Imperfect Public Monitoring with a Fear of Signal Distortion

This paper proposes a model of signal distortion in a two-player game with imperfect public monitoring. We construct a tractable theoretical framework where each player has the opportunity to distort the true public signal and each player is uncertain about the distortion technologies available to the other player. We show that when players evaluate strategies according to their worst-case guarantees — i.e., are ambiguity averse over certain distributions in the environment — perceived continuation payoffs endogenously lie on a positively sloped line. We then provide examples showing that, counterintuitively, identifying deviators can be harmful in enforcing a strategy profile; moreover, we illustrate how the presence of such signal distortion can sustain cooperation when it is impossible in standard settings. We finally extend our model to a repeated game where our concept is a natural generalization of strongly symmetric equilibria. In this setting, we prove an anti-folk theorem, showing that payoffs under our equilibrium concept are under general conditions bounded away from efficiency..

Optimally Approximating the Coverage Lifetime of Wireless Sensor Networks

We address a classical problem concerning energy efficiency in sensor networks. In particular, we consider the problem of maximizing the lifetime of coverage of targets in a wireless sensor network with battery-limited sensors. We first show that the problem cannot be approximated within a factor less than $\ln n$ by any polynomial time algorithm, where $n$ is the number of targets. This provides closure to the long-standing open problem of showing optimality of previously known $\ln n$ approximation algorithms. We also derive a new $\ln n$ approximation to the problem by showing the $\ln n$ approximation to the related maximum disjoint set cover problem. We show that this approach has many advantages over algorithms in the literature, including a simple and optimal extension that solves the problem with multiple coverage constraints. For the 1-D network topology, where sensors can monitor contiguous line segments of possibly different lengths, we show that the optimal coverage lifetime can be found in polynomial time. Finally, for the 2-D topology in which coverage regions are unit squares, we combine the existing results to derive a $1+\epsilon $ approximation algorithm for the problem. Extensive simulation experiments validate our theoretical results, showing that our algorithms not only have optimal worst case guarantees but also match the performance of the existing algorithms on special network topologies. In addition, our algorithms sometimes run orders of magnitude faster than the existing state of the art.

Exact Worst-Case Performance of First-Order Methods for Composite Convex Optimization

We provide a framework for computing the exact worst-case performance of any algorithm belonging to a broad class of oracle-based first-order methods for composite convex optimization, including those performing explicit, projected, proximal, conditional and inexact (sub)gradient steps. We simultaneously obtain tight worst-case guarantees and explicit instances of optimization problems on which the algorithm reaches this worst-case. We achieve this by reducing the computation of the worst-case to solving a convex semidefinite program, generalizing previous works on performance estimation by Drori and Teboulle [13] and the authors [43]. We use these developments to obtain a tighter analysis of the proximal point algorithm and of several variants of fast proximal gradient, conditional gradient, subgradient and alternating projection methods. In particular, we present a new analytical worst-case guarantee for the proximal point algorithm that is twice better than previously known, and improve the standard worst-case guarantee for the conditional gradient method by more than a factor of two. We also show how the optimized gradient method proposed by Kim and Fessler in [22] can be extended by incorporating a projection or a proximal operator, which leads to an algorithm that converges in the worst-case twice as fast as the standard accelerated proximal gradient method [2].

Algorithms for Group Isomorphism via Group Extensions and Cohomology

The isomorphism problem for finite groups of order $n$ (GpI) has long been known to be solvable in $n^{\log n+O(1)}$ time, but only recently were polynomial-time algorithms designed for several interesting group classes. Inspired by recent progress, we revisit the strategy for GpI via the extension theory of groups. The extension theory describes how a normal subgroup $N$ is related to $G/N$ via $G$, and this naturally leads to a divide-and-conquer strategy that “splits” GpI into two subproblems: one regarding group actions on other groups, and one regarding group cohomology. When the normal subgroup $N$ is abelian, this strategy is well known. Our first contribution is to extend this strategy to handle the case when $N$ is not necessarily abelian. This allows us to provide a unified explanation of all recent polynomial-time algorithms for special group classes. Guided by this strategy, to make further progress on GpI, we consider central-radical groups, proposed in Babai et al. [Code equivalence and group isomorphism, in Proceedings of the 22nd Annual ACM--SIAM Symposium on Discrete Algorithms (SODA'11), SIAM, Philadelphia, 2011, ACM, New York, pp. 1395--1408]: the class of groups such that $G$ modulo its center has no abelian normal subgroups. This class is a natural extension of the group class considered by Babai et al. [Polynomial-time isomorphism test for groups with no abelian normal subgroups (extended abstract), in International Colloquium on Automata, Languages, and Programming (ICALP), 2012, pp. 51--62], namely those groups with no abelian normal subgroups. Following the above strategy, we solve GpI in $n^{O(\log \log n)}$ time for central-radical groups, and in polynomial time for several prominent subclasses of central-radical groups. We also solve GpI in $n^{O(\log\log n)}$ time for groups whose solvable normal subgroups are elementary abelian but not necessarily central. As far as we are aware, this is the first time there have been worst-case guarantees on an $n^{o(\log n)}$-time algorithm that tackles both aspects of GpI---actions and cohomology---simultaneously. Prior to this work, the best proven upper bounds on algorithms for groups with central radicals were $n^{O(\log n)}$, even for groups with a central radical of constant size, such as ${Rad}(G) = Z(G)=\mathbb{Z}_2$. To develop our new algorithms we utilize several mathematical results on the detailed structure of cohomology classes, as well as algorithmic results for code equivalence, coset intersection, and cyclicity testing of modules over finite-dimensional associative algebras. We also suggest several promising directions for future work.

Provably Near-Optimal Balancing Policies for Multi-Echelon Stochastic Inventory Control Models

We develop the first algorithmic approach to compute provably good ordering policies for a multi-echelon, stochastic inventory system facing correlated, nonstationary and evolving demands over a finite horizon. Specifically, we study the serial system. Our approach is computationally efficient and provides worst-case guarantees. That is, the expected cost of the algorithms is guaranteed to be within a constant factor of the optimal expected cost; depending on the assumption the constant varies between two and three. Our algorithmic approach is based on an innovative scheme to account for costs in a multi-echelon, multi-period environment, as well as repeatedly balancing between opposing cost. The cost-accounting scheme, called a cause-effect cost-accounting scheme, is significantly different from traditional cost-accounting schemes in that it reallocates costs with the goal of assigning every unit of cost to the decision that caused the cost to be incurred. We believe it will have additional applications in other multi-echelon inventory models.

Cost Minimizing Online Algorithms for Energy Storage Management With Worst-Case Guarantee

The fluctuations of electricity prices in demand response schemes and intermittency of renewable energy supplies necessitate the adoption of energy storage in microgrids. However, it is challenging to design effective real-time energy storage management strategies that can deliver assured optimality, without being hampered by the uncertainty of volatile electricity prices and renewable energy supplies. This paper presents a simple effective online algorithm for the charging and discharging decisions of energy storage that minimizes the electricity cost in the presence of electricity price fluctuations and renewable energy supplies, without relying on the future information of prices, demands or renewable energy supplies. The proposed algorithm is supported by a near-best worst-case guarantee (i.e., competitive ratio), as compared to the offline optimal decisions based on full future information. Furthermore, the algorithm can be adapted to take advantage of limited future information, if available. By simulations on real-world data, it is observed that the proposed algorithms can achieve satisfactory outcome in practice.

Automated Credit Rating Prediction in a competitive framework

Automated credit rating prediction (ACRP) algorithms are used to predict the ratings of bonds without having to trust one rating agency, like Moody’s, Fitch or S&P. Nevertheless, for the moment, the accuracy of ACRP algorithms is investigated by empirical tests. In this paper, the framework for a competitive analysis is set and afterwards in this framework, the definition of competitive ACRP algorithms and its demonstration is given. In this way, for a competitive ACRP algorithm, a worst-case guarantee concerning the misclassification error is offered. Furthermore, several ACRP algorithms from the literature are compared according their competitiveness.

Short and Simple Cycle Separators in Planar Graphs

We provide an implementation of an algorithm that, given a triangulated planar graph with m edges, returns a simple cycle that is a 3/4-balanced separator consisting of at most √8 m edges. An efficient construction of a short and balanced separator that forms a simple cycle is essential in numerous planar graph algorithms, for example, for computing shortest paths, minimum cuts, or maximum flows. To the best of our knowledge, this is the first implementation of such a cycle separator algorithm with a worst-case guarantee on the cycle length. We evaluate the performance of our algorithm and compare it to the planar separator algorithms recently studied by Holzer et al. [2009]. Out of these algorithms, only the Fundamental Cycle Separator (FCS) produces a simple cycle separator. However, FCS does not provide a worst-case size guarantee. We demonstrate that (1) our algorithm is competitive across all test cases in terms of running time, balance, and cycle length; (2) it provides worst-case guarantees on the cycle length, significantly outperforming FCS on some instances; and (3) it scales to large graphs.

Feature selection for linear SVM with provable guarantees

We give two provably accurate feature-selection techniques for the linear SVM. The algorithms run in deterministic and randomized time respectively. Our algorithms can be used in an unsupervised or supervised setting. The supervised approach is based on sampling features from support vectors. We prove that the margin in the feature space is preserved to within ϵ-relative error of the margin in the full feature space in the worst-case. In the unsupervised setting, we also provide worst-case guarantees of the radius of the minimum enclosing ball, thereby ensuring comparable generalization as in the full feature space and resolving an open problem posed in Dasgupta et al. (2007) [7]. We present extensive experiments on real-world datasets to support our theory and to demonstrate that our method is competitive and often better than prior state-of-the-art, for which there are no known provable guarantees.

Almost Optimal Local Graph Clustering Using Evolving Sets

Spectral partitioning is a simple, nearly linear time algorithm to find sparse cuts, and the Cheeger inequalities provide a worst-case guarantee for the quality of the approximation found by the algorithm. A local graph partitioning algorithm finds a set of vertices with small conductance (i.e., a sparse cut) by adaptively exploring part of a large graph G , starting from a specified vertex. For the algorithm to be local, its complexity must be bounded in terms of the size of the set that it outputs, with at most a weak dependence on the number n of vertices in G . Previous local partitioning algorithms find sparse cuts using random walks and personalized PageRank [Spielman and Teng 2013; Andersen et al. 2006]. In this article, we introduce a simple randomized local partitioning algorithm that finds a sparse cut by simulating the volume-biased evolving set process , which is a Markov chain on sets of vertices. We prove that for any ϵ > 0, and any set of vertices A that has conductance at most φ, for at least half of the starting vertices in A our algorithm will output (with constant probability) a set of conductance O (√φ /ϵ). We prove that for a given run of the algorithm, the expected ratio between its computational complexity and the volume of the set that it outputs is vol( A ) ϵ φ -1/2 polylog( n ), where vol( A ) = Σ v ∈ A d ( v ) is the volume of the set A . This gives an algorithm with the same guarantee (up to a constant factor) as the Cheeger's inequality that runs in time slightly superlinear in the size of the output. This is the first sublinear (in the size of the input) time algorithm with almost the same guarantee as the Cheeger's inequality. In comparison, the best previous local partitioning algorithm, by Andersen et al. [2006], has a worse approximation guarantee of O (√φ log n ) and a larger ratio of φ -1 polylog( n ) between the complexity and output volume. As a by-product of our results, we prove a bicriteria approximation algorithm for the expansion profile of any graph. For 0 < k ≤ vol( V )/2, let φ( k ) : min S : vol( S ) ≤ k φ( S ). There is a polynomial time algorithm that, for any k , ϵ > 0, finds a set S of volume vol( S ) ≤ O ( k 1 + ϵ ) and expansion φ( S )≤ O (√φ ( k )/ϵ). As a new technical tool, we show that for any set S of vertices of a graph, a lazy t -step random walk started from a randomly chosen vertex of S will remain entirely inside S with probability at least (1 - φ( S )/2) t . This itself provides a new lower bound to the uniform mixing time of any finite state reversible Markov chain.

A Geometric Approach to Server Selection for Interactive Video Streaming

Many distributed interactive multimedia applications, such as live video conferencing and video sharing, require each participating client to transmit its captured video stream to other clients via relay servers. We consider connecting multiple clients through multiple relay servers and study the server selection problem from a dense pool of content delivery network edge locations and datacenters to reduce the end-to-end delays between clients. To achieve scalability in the presence of a large number of candidate servers, we formulate server selection as a geometric problem in a delay space instead of in a graph, which turns out to be an extension of the well-known Euclidean $k$ -median problem. We propose practical approximation schemes when using only one or two servers with theoretical worst-case guarantees as well as fast heuristics when using $k$ servers. We demonstrate the benefit of our optimized multiserver selection schemes through extensive evaluation based on real-world traces collected from the PlanetLab and Seattle platforms, containing personal mobile devices as well as real network experiments based on a prototype implementation.

ProMARTES: Accurate network and computation delay prediction for component-based distributed systems

This paper proposes a cycle-accurate performance analysis method for real-time component-based distributed systems (CB-RTDS). The method involves the following phases: (a) profiling SW components at cycle execution level and modeling the obtained performance measurements in MARTE-compatible component resource models, (b) guided composition of the system architecture from available SW and HW components, (c) automated generation of a system model, specifying both computation and network loads, and (d) performance analysis (scheduling, simulation and network analysis) of the composed system model. The method is demonstrated for a real-world case study of 3 autonomously navigating robots with advanced sensing capabilities. The case study is challenging because of the SW/HW mapping, real-time requirements and data synchronization among multiple nodes. This case-study proved that, thanks to the adopted low-level performance metrics, we are able to obtain accurate performance predictions of both computation and network delays. Moreover, the combination of analytical and simulation analysis methods enables the computation of both the guaranteed Worst Case Execution Time (WCET) and the detailed execution time-line data for real-time tasks. As a result, the analysis yields the identification of an optimal architecture, with respect to real-time deadlines, robustness and system costs. The paper main contributions are the cycle-accurate performance analysis workflow and supportive open-source ProMARTES tool-chain, both incorporating a network prediction model in all the performance analysis phases.