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Online Algorithms for Covering and Packing Problems with Convex Objectives

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Online Algorithms for Covering and Packing Problems with Convex Objectives Online Algorithms for Covering and Packing Problems with Convex Objectives Yossi Azar∗, Niv Buchbindery, T-H. Hubert Chanz, Shahar Chenx, Ilan Reuven Coheny, Anupam Gupta{, Zhiyi Huangz, Ning Kangz, Viswanath Nagarajank, Joseph (Seffi) Naorx, Debmalya Panigrahi∗∗ ∗Blavatnik School of Computer Science, Tel Aviv University, Tel Aviv, Israel. Email: [email protected], [email protected] yDepartment of Statistics and Operations Research, Tel Aviv University, Tel Aviv, Israel. Email: [email protected] zDepartment of Computer Science, The University of Hong Kong, Hong Kong, China. Email: fhubert,zhiyi,[email protected] xDepartment of Computer Science, Technion - Israel Institute of Technology, Haifa, Israel. Email:[email protected], [email protected] {Department of Computer Science, Carnegie Mellon University, Pittsburgh, PA 15213, USA. Email: [email protected] kDepartment of Industrial and Operations Engineering, University of Michigan, Ann Arbor, MI 48109, USA. Email: [email protected] ∗∗Department of Computer Science, Duke University, Durham, NC 27708, USA. Email: [email protected] Abstract—We present online algorithms for covering and a primal-dual algorithm, and then rounding it, has proved to be packing problems with (non-linear) convex objectives. The convex a very powerful general technique (see [4] for a survey). This covering problem is defined as: minx2 n f(x) s.t. Ax ≥ 1, where R+ framework is applicable to many central online problems, like f : n ! A m × n R+ R+ is a monotone convex function, and is an the classic ski rental problem, online set-cover [5], generalized matrix with non-negative entries. In the online version, a new row of the constraint matrix, representing a new covering constraint, paging [6], [7], k-server and metrical task systems [8], [9], is revealed in each step and the algorithm is required to maintain graph connectivity [10], [11], routing [12], load balancing and a feasible and monotonically non-decreasing assignment x over machine scheduling [13], [14], matching [15], and budgeted time. We also consider a convex packing problem defined as: Pm T n allocation [16]. Via this approach, not only can we unify maxy2 m yj − g(A y), where g : ! + is a monotone R+ j=1 R+ R previously known results, but we can also resolve important convex function. In the online version, each variable y arrives j open questions in competitive analysis. The use of the online online and the algorithm must decide the value of yj on its arrival. This represents the Fenchel dual of the convex covering program, primal-dual method mirrors (and is inspired by) the use of when g is the convex conjugate of f. We use a primal-dual linear programming relaxations for approximation algorithms, approach to give online algorithms for these generic problems, with one major difference. In the online setting, generic and use them to simplify, unify, and improve upon previous linear programming solvers cannot be used for even obtaining results for several applications. Index Terms—online algorithm; convex optimization; primal- a fractional solution, since constraints are presented online. dual algorithm Hence, new algorithms have been developed for solving linear programs online [17], [18]. I. INTRODUCTION The basic setting of the online primal-dual framework is In the area of online algorithms, the method of modeling a grounded on the well-studied covering/packing framework for problem as a linear program, obtaining a fractional solution via combinatorial optimization problems. Covering/packing linear formulations are a special class of linear formulations in which This paper presents results independently obtained by Y. Azar, I. R. Cohen, all coefficients are non-negative. In a covering problem, the and D. Panigrahi [1], N. Buchbinder, S. Chen, A. Gupta, V. Nagarajan, and goal is to minimize a non-negative cost function subject to J. Naor [2], and T.-H. H. Chan, Z. Huang, and N. Kang [3]. Y. Azar and I. R. Cohen are supported in part by the Israel Science non-negative covering constraints. In a packing problem the Foundation (grant No. 1506/16), by the I-CORE program (Center No. 4/11), goal is to maximize a non-negative profit function subject to and by the Blavatnik Fund. N. Buchbinder is supported in part by ISF grant non-negative packing constraints. Packing and covering form 1585/15 and by BSF grant 2014414. S. Chen and J. Naor are supported in part by ISF grant 1585/15 and BSF grant 2014414. A. Gupta is supported in a primal-dual pair – for consistency, we refer to the covering part by NSF awards CCF-1319811, CCF-1536002, CCF-1540541 and CCF- problem as the primal problem and to the packing problem as 1617790. D. Panigrahi is supported in part by NSF Awards CCF-1527084 and the dual problem, although in some cases, it is the dual packing CCF-1535972. T-H. H. Chan is supported in part by the Hong Kong RGC grant 17202715. Z. Huang is supported in part by the Hong Kong RGC grant problem in which we are interested in. The covering/packing 17202115. framework captures a large class of (relaxations) of well- studied combinatorial problems. produces a quantity zj of each item j 2 [n], at total cost ? Although extremely successful, the online primal-dual ap- f (z). Now, if yj denotes the value obtained from the bundle proach has been mainly studied for linear objective functions. of goods assigned to buyer j, and Ay ≤ z captures the Yet, what about convex optimization problems? In the offline constraints that no item is sold in greater quantities than its setting, the Ellipsoid and interior-point methods can solve production, then we recover precisely the dual problem. This these problems with linear convergence rates. But, online, it social-welfare maximization problem was studied by Blum was not known how to solve convex optimization problems et al. [24] and Huang et al. [14] for the “separable” special in general prior to this work. In contrast, specific online case in which the production cost of each item is a univariate problems with non-linear convex/concave objectives have been convex function, and the total cost is the sum over the items: ? P ? studied in recent years, in areas such as energy-efficient f (z) = i fi (zi). This case is somewhat restrictive in that scheduling [19], [20], [21], paging [22], network routing [23], it does not capture production costs such as energy that are ? P α combinatorial auctions [24], [14], matching [25], and budgeted typically modeled as f (z) = ( zi) for some α > 1. This allocation [26]. In this paper, we develop a general framework is a non-separable function, and so not captured by previous for a broad class of covering/packing programs with convex work, but it falls neatly into our framework. objective functions. The competitive ratios we obtain are optimal. This already improves and substantially generalizes B. Our Results previous results for problem classes such as mixed packing covering LPs considered in [13]. We then show how to round Our main theorem for the general online covering/packing these fractional solutions online for specific optimization prob- framework is for monotone convex functions with monotone y lems, obtaining improved results for non-linear optimization gradients (we denote the gradient rf). Our primal-dual problems considered recently in machine scheduling, network approach simultaneously gives us solutions to both the primal routing, and combinatorial auctions. and dual convex programs. A. The Convex Optimization Framework Theorem 1 [Online Convex Covering/Packing Theorem] Let n f : R+ ! R+ be a function that is monotone, differentiable, The next convex covering problem is our primal problem. hrf(x);xi convex, and rf is monotone. Let p = supx≥0 f(x) . Then, min C(x) := f(x) subject to Ax ≥ 1: (I.1) (a) There exists an O(p log d)p-competitive algorithm that x2 n R+ produces a monotone primal solution. n p Here, f : R+ ! R is a convex function and Am×n is a (b) There exists an O(p log(ρd)) -competitive algorithm that non-negative matrix. The rows of A, the covering constraints, produces a monotone dual solution. arrive online over time. At any point of time we must maintain Here d is the maximal row sparsity of the matrix and ρ is ai0j a feasible fractional solution x, non-decreasing over time. maxj max 0 f g. i;i jai;j 6=0 aij This convex program captures, e.g., the mixed covering- packing problem [13], the machine scheduling and online The conditions of the theorem are satisfied, for exam- routing problems from [23], [20], and the nested set cover ple, by convex polynomials of degree p (with non-negative f(x) = 1 kxkp problem [27]. coefficients). One such example is p p, whose ? 1 q 1 1 As in solving LPs online, we use a primal-dual technique Fenchel conjugate function is f = q kxkq where p + q = 1. where we simultaneously obtain a solution to both primal and Intuitively, p bounds the growth of the objective around a given dual programs. In the convex setting, the dual of (I.1) is the point and shows the (necessary) effects of discretization of following packing problem: the feasible region performed by the algorithm. In fact, our competitive ratios are the best possible for all p: see xIII-A. P ? T max P (y) := yj−f (z) subject to A y ≤ z: m n j2[m] Observe that the dual result of Theorem1(b) makes as- y2R+ ;z2R (I.2) sumptions on the function f and then solves the dual problem ? Here f ? is the Fenchel dual or conjugate function of f, using f . However, if our goal is to solve the dual problem, which is always a convex function∗, and hence another convex e.g., for social-welfare maximization with production costs, optimization problem.
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