This paper takes a game theoretic approach to the design and analysis of online algorithms and illustrates the approach on the finite-horizon ski-rental problem. This approach allows beyond worst-case analysis of online algorithms. First, we define "subgame optimality" which is stronger than worst case optimality in that it requires the algorithm to take advantage of an adversary not playing a worst case input. Algorithms only focusing on the worst case can be far from subgame optimal. Second, we consider prior-independent design and analysis of online algorithms, where rather than choosing a worst case input, the adversary chooses a worst case independent and identical distribution over inputs. Prior-independent online algorithms are generally analytically intractable; instead we give a fully polynomial time approximation scheme to compute them. Highlighting the potential improvement from these paradigms for the finite-horizon ski-rental problem, we empirically compare worst-case, subgame optimal, and prior-independent algorithms in the prior-independent framework.

Patterned self-assembly tile set synthesis (PATS) aims at finding a minimum tile set to uniquely self-assemble a given rectangular color pattern. For k ≥ 1, k-PATS is a variant of PATS that restricts input patterns to those with at most k colors. We prove the NP-hardness of 29-PATS, where the best known is that of 60-PATS.

Patterned self-assembly tile set synthesis (Pats) aims at finding a minimum tile set to uniquely self-assemble a given rectangular (color) pattern. For k ≥ 1, k-Pats is a variant of Pats that restricts input patterns to those with at most k colors. A computer-assisted proof has been recently proposed for 2-Pats by Kari et al. [arXiv:1404.0967 (2014]. In contrast, the best known manually-checkable proof is for the NP-hardness of 29-Pats by Johnsen, Kao, and Seki [ISAAC 2013, LNCS 8283, pp. 699-710]. We propose a manually-checkable proof for the NP-hardness of 11-Pats.

This paper gives a theoretical model for design and analysis of mechanisms for online marketplaces where a bidding dashboard enables the bid-optimization of long-lived agents. We assume that a good allocation algorithm exists when given the true values of the agents and we develop online winner-pays-bid and all-pay mechanisms that implement the same outcome of the algorithm with the aid of a bidding dashboard. The bidding dashboards that we develop work in conjunction with the mechanism to guarantee that bidding according to the dashboard is strategically equivalent (with vanishing utility difference) to bidding truthfully in the sequential truthful implementation of the allocation algorithm. Our dashboard mechanism makes only a single call to the allocation algorithm in each stage.

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