We study online mechanisms for preemptive scheduling with deadlines, with the goal of maximizing the total value of completed jobs. This problem is fundamental to deadline-aware cloud scheduling, but there are strong lower bounds even for the algorithmic problem without incentive constraints. However, these lower bounds can be circumvented under the natural assumption of deadline slackness, i.e., that there is a guaranteed lower bound s > 1 on the ratio between a job's size and the time window in which it can be executed.In this paper, we construct a truthful scheduling mechanism with a constant competitive ratio, given slackness s > 1. Furthermore, we show that if s is large enough then we can construct a mechanism that also satisfies a commitment property: it can be determined whether or not a job will finish, and the requisite payment if so, well in advance of each job's deadline. This is notable because, in practice, users with strict deadlines may find it unacceptable to discover only very close to their deadline that their job has been rejected.

We consider the k-server problem on trees and HSTs. We give an algorithm based on Bregman projections. This algorithm has a competitive ratios that match some of the recent results given by Bubeck et al. (STOC 2018), whose algorithm was based on mirror-descent-based continuous dynamics prescribed via a differential inclusion. *

Motivated by Internet targeted advertising, we address several ad allocation problems. Prior work has established these problems admit no randomized online algorithm better than $(1-\frac{1}{e})$-competitive (\cite{karp1990optimal,mehta2007adwords}), yet simple heuristics have been observed to perform much better in practice. We explain this phenomenon by studying a generalization of the bounded-degree inputs considered by Buchbinder et al.~\cite{buchbinder2007online}, graphs which we call $(k,d)-bounded$. In such graphs the maximal degree on the online side is at most $d$ and the minimal degree on the offline side is at least $k$. We prove that for such graphs, these problems' natural greedy algorithms attain competitive ratio $1-\frac{d-1}{k+d-1}$, tending to \emph{one} as $d/k$ tends to zero. We prove this bound is tight for these algorithms. Next, we develop deterministic primal-dual algorithms for the above problems achieving competitive ratio $1-(1-\frac{1}{d})^k>1-\frac{1}{e^{k/d}}$, or \emph{exponentially} better loss as a function of $k/d$, and strictly better than $1-\frac{1}{e}$ whenever $k\geq d$. We complement our lower bounds with matching upper bounds for the vertex-weighted problem. Finally, we use our deterministic algorithms to prove by dual-fitting that simple randomized algorithms achieve the same bounds in expectation. Our algorithms and analysis differ from previous ad allocation algorithms, which largely scale bids based on the spent fraction of their bidder's budget, whereas we scale bids according to the number of times the bidder could have spent as much as her current bid. Our algorithms differ from previous online primal-dual algorithms, as they do not maintain dual feasibility, but only primal-to-dual ratio, and only attain dual feasibility upon termination. We believe our techniques could find applications to other well-behaved online packing problems

We consider online fractional covering problems with a convex objective, where the covering constraints arrive over time. Formally, we want to solve

We design a randomized polynomial time algorithm which, given a 3-tensor of real numbers A = {a i jk } n i, j,k=1 such that for all i, j, k ∈ {1, . . . , n} we have a i jk = a ik j = a k ji = a jik = a ki j = a jki and a iik = a i j j = a i ji = 0, computes a number Alg(A) which satisfies with probability at leastOn the other hand, we show via a simple reduction from a result of Håstad and Venkatesh [22] that under the assumption NP DT I ME n (log n) O(1) , for every ε > 0 there is no algorithm that approximates max x∈{−1,1} n n i, j,k=1 a i jk x i x j x k within a factor of 2O(1) . Our algorithm is based on a reduction to the problem of computing the diameter of a convex body in R n with respect to the L 1 norm. We show that it is possible to do so up to a multiplicative error of O n log n , while no randomized polynomial time algorithm can achieve accuracy o n log n . This resolves a question posed by Brieden, Gritzmann, Kannan, Klee, Lovász and Simonovits in [10].We apply our new algorithm to improve the algorithm of Håstad and Venkatesh [22] for the Max-E3-Lin-2 problem. Given an over-determined system E of N linear equations modulo 2 in n ≤ N Boolean variables, such that in each equation appear only three distinct variables, the goal is to approximate in polynomial time the maximum number of satisfiable equations in E minus N 2 (i.e. we subtract the expected number of satisfied equations in a random assignment). Håstad and Venkatesh [22] obtained an algorithm which approximates this value up to a factor of O √ N . We obtain a O n log n approximation algorithm. By relating this problem to the refutation problem for random 3 − CNF formulas we give evidence that obtaining a significant improvement over this approximation factor is likely to be difficult.