[Paper Review] Online Packing and Covering Framework with Convex Objectives

Niv Buchbinder, Shahar Chen+1 more|arXiv (Cornell University)|Dec 29, 2014

Optimization and Search Problems15 references12 citations

TL;DR

This paper presents a novel online primal-dual framework for fractional covering and packing problems with convex objectives, where constraints and variables arrive online. It achieves competitive ratios dependent on smoothness and monotonicity parameters of the convex function f, generalizing prior linear-objective frameworks and enabling new or improved competitive algorithms for problems like online set cover with multiple costs, capacitated facility location, and multicast with non-separable production costs.

ABSTRACT

We consider online fractional covering problems with a convex objective, where the covering constraints arrive over time. Formally, we want to solve min{f(x)∣Ax≥1,x≥0}, where the objective function f:Rn→R is convex, and the constraint matrix Am×n is non-negative. The rows of A arrive online over time, and we wish to maintain a feasible solution x at all times while only increasing coordinates of x. We also consider "dual" packing problems of the form max{c⊺y−g(μ)∣A⊺y≤μ,y≥0}, where g is a convex function. In the online setting, variables y and columns of A⊺ arrive over time, and we wish to maintain a non-decreasing solution (y,μ). <br>We provide an online primal-dual framework for both classes of problems with competitive ratio depending on certain "monotonicity" and "smoothness" parameters of f; our results match or improve on guarantees for some special classes of functions f considered previously. <br>Using this fractional solver with problem-dependent randomized rounding procedures, we obtain competitive algorithms for the following problems: online covering LPs minimizing ℓp-norms of arbitrary packing constraints, set cover with multiple cost functions, capacity constrained facility location, capacitated multicast problem, set cover with set requests, and profit maximization with non-separable production costs. Some of these results are new and others provide a unified view of previous results, with matching or slightly worse competitive ratios.

Motivation & Objective

• To develop a general online primal-dual algorithm for fractional convex covering and packing problems with online constraint arrivals.
• To extend the classical online primal-dual framework—previously limited to linear objectives—to handle general convex objectives.
• To provide competitive ratio guarantees that depend on smoothness and monotonicity parameters of the convex function f.
• To apply the framework to diverse online problems via problem-specific randomized rounding, yielding new or improved competitive algorithms.
• To unify and generalize prior results on problems such as set cover with multiple costs, facility location, and multicast with capacity and production cost constraints.

Proposed method

• Proposes two online primal-dual algorithms: one allowing dual variables y to decrease, and another enforcing monotonicity on all variables including y.
• Derives competitive ratio bounds based on the ratio of gradients ∇ℓf(z)/∇ℓf(cz) and the term (z⊺∇f(z) − f(z))/f(cz), capturing smoothness and monotonicity.
• Applies the framework to convex relaxations of problems like mixed covering-packing LPs, set cover with multiple cost functions, and capacitated facility location.
• Uses problem-dependent randomized rounding to convert fractional solutions into integral ones, with controlled additive error.
• Employs a scaling and rounding scheme that ensures concentration bounds via Chernoff inequalities, maintaining feasibility with high probability.
• Introduces a dual variable transformation to map problems with non-separable production costs into a form compatible with the framework.

Experimental results

Research questions

• RQ1 Can the online primal-dual framework be extended from linear to convex objectives in covering and packing problems?
• RQ2 What competitive ratio can be achieved for convex objectives, and how do smoothness and monotonicity parameters of f affect it?
• RQ3 Can this framework unify or improve existing results for online problems like set cover with multiple costs or capacitated facility location?
• RQ4 What is the impact of randomized rounding with additive error on the competitive ratio in problems with non-separable production costs?
• RQ5 How do the derived competitive ratios compare to known lower bounds for special cases like linear objectives?

Key findings

• The framework achieves an O(log d)-competitive ratio for monotone primal variables in the linear objective case, matching known lower bounds.
• For general convex objectives, the competitive ratio depends on smoothness and monotonicity parameters of f, with explicit bounds provided in terms of gradient ratios.
• An O(p log d)-competitive algorithm is obtained for mixed covering-packing LPs minimizing ℓp-norms of load vectors.
• An O(log²m)-competitive fractional solution is achieved for capacitated facility location, which rounds to O(log²m log mn)-competitive randomized online algorithm.
• For the capacitated multicast problem, the framework yields an O(log²m log mn)-competitive algorithm with O((d + log²m) log mn) capacity violation.
• For profit maximization with non-separable production costs, the framework achieves (1−1/m)OPT/O(p log Rd)p −g(O(log m)·1) expected profit, with additive error unavoidable in convex relaxations.