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We present online algorithms for covering and packing problems with (non-linear) convex objectives. The convex covering problem is defined as: min x∈R n + f (x) s.t. Ax ≥ 1, where f : R n + → R+ is a monotone convex function, and A is an m × n matrix with non-negative entries. In the online version, a new row of the constraint matrix, representing a new covering constraint, is revealed in each step and the algorithm is required to maintain a feasible and monotonically non-decreasing assignment x over time. We also consider a convex packing problem defined as: max y∈R m + m j=1 yj − g(A T y), where g : R n + → R+ is a monotone convex function. In the online version, each variable yj arrives online and the algorithm must decide the value of yj on its arrival. This represents the Fenchel dual of the convex covering program, when g is the convex conjugate of f. We use a primal-dual approach to give online algorithms for these generic problems, and use them to simplify, unify, and improve upon previous results for several applications.

In this paper we study the classic online matching problem, introduced in the seminal work of Karp, Vazirani and Vazirani (STOC 1990), in regular graphs. For such graphs, an optimal deterministic algorithm as well as efficient algorithms under stochastic input assumptions were known. In this work, we present a novel randomized algorithm with competitive ratio tending to one on this family of graphs, under adversarial arrival order. Our main contribution is a novel algorithm which achieves competitive ratio 1In contrast, we show that all previously-studied online algorithms have competitive ratio strictly bounded away from one. Moreover, we show the convergence rate of our algorithm's competitive ratio to one is nearly tight, as no algorithm achieves competitive ratio better than 1 − O 1/ √ d . Finally, we show that our algorithm yields a similar competitive ratio with high probability, as well as guaranteeing each vertex a probability of being matched tending to one.

In this paper, we exploit linear programming duality in the online setting, where input arrives on the fly, from the unique perspective of designing lower bounds (i.e., hardness results) on the competitive ratio. In particular, we provide a systematic method (as opposed to ad hoc case analysis that is typically done) for obtaining online deterministic and randomized lower bounds on the competitive ratio for a wide variety of problems. We show the usefulness of our approach by providing new, tight hardness results for three diverse online problems: the Vector Bin Packing problem, Ad-auctions (and various online matching problems), and the Capital Investment problem. Our methods are sufficiently general that they can also be used to reconstruct existing lower bounds.Our approach is in stark contrast to previous works, which exploit linear programming duality to obtain positive results, often via the useful primaldual scheme. We design a general recipe with the opposite aim of obtaining negative results via duality. The general idea behind our approach is to construct a parameterized family of primal linear programs based on a candidate collection of input sequences for proving the lower bound, where the objective function corresponds to optimizing the competitive ratio. Solving the parameterized family of primal linear programs optimally would yield a valid lower bound, but is a challenging task and limits the tools that can be applied, since analysis must be done precisely and exact optimality needs to be proved. To this end, we consider the corresponding parameterized family of dual linear programs and provide feasible solutions, where the objective function yields