The Geometry of Online Packing Linear Programs

Abstract

Abstract. We consider packing LP's with m rows where all constraint coefficients are normalized to be in the unit interval. The n columns arrive in random order and the goal is to set the corresponding decision variables irrevocably when they arrive to obtain a feasible solution maximizing the expected reward. Previous (1 − )-competitive algorithms require the right-hand side of the LP to be Ω( m 2 log n ), a bound that worsens with the number of columns and rows. However, the dependence on the number of columns is not required in the single-row case and known lower bounds for the general case are also independent of n. Our goal is to understand whether the dependence on n is required in the multirow case, making it fundamentally harder than the single-row version. We refute this by exhibiting an algorithm which is (1 − )-competitive as long as the righthand sides are Ω( m 2 2 log m ). Our techniques refine previous PAC-learning based approaches which interpret the online decisions as linear classifications of the columns based on sampled dual prices. The key ingredient of our improvement comes from a non-standard covering argument together with the realization that only when the columns of the LP belong to few 1-d subspaces we can obtain small such covers; bounding the size of the cover constructed also relies on the geometry of linear classifiers. General packing LP's are handled by perturbing the input columns, which can be seen as making the learning problem more robust.

Iterative Computation of Security Strategies of Matrix Games with Growing Action Set

“…. , x τ−1 , x * } when searching for the optimum of the old LP (10)(11). When the new constraint (26) is added, we can test the feasibility of the visited shadow vertices one by one starting from x 0 until we find one that fails the feasibility test.…”

“…Lemma 1 (Lemma 1.4 in [5]) Consider LP problem (10)(11). Suppose u is linearly independent with c. Let x 0 , x 1 , .…”

“…The corresponding dual problem looks similar to an online LP problem where new variables are introduced to the existing LP [1,6,8,10,11]. The key idea to approximate the optimum in online LP is to assign the new variable 1 if reward is greater than the shadow price based cost, 0 otherwise.…”

Adversarial Bandits with Knapsacks

“…Two players in the repeated game can be seen as the respective primal algorithm and dual algorithm. Compared to the rich literature on primal-dual algorithms (Williamson and Shmoys, 2011;Buchbinder and Naor, 2009b;Mehta, 2013) (including the more recent literature on stochastic online packing problems Devanur and Hayes, 2009;Devanur et al, 2011;Feldman et al, 2010;Molinaro and Ravi, 2012) LagrangeBwK has a very specific and modular structure dictated by the repeated game.…”

Competitive online algorithms for resource allocation over the positive semidefinite cone

“…A feature in these problems is that the variables denoting allocation amounts, the bid prices, and the coefficients in the resource constraints are all nonnegative, thus a resource can only be used up as time goes on (resource usage is nondecreasing) while the total available resource is fixed. In the context of linear programming (LP), these are sometimes called "packing" problems, e.g., [MR13,BN09b]. For an overview of online linear programming and more generally online convex problems with different models for online information arrival, we refer the reader to [LJ16,GM16,AD15,ESF14].…”