Pattern Matching and Consensus Problems on Weighted Sequences and Profiles

We study pattern matching problems on two major representations of uncertain sequences used in molecular biology: weighted sequences (also known as position weight matrices, PWM) and profiles (i.e., scoring matrices). In the simple version, in which only the pattern or only the text is uncertain, we obtain efficient algorithms with theoretically-provable running times using a variation of the lookahead scoring technique. We also consider a general variant of the pattern matching problems in which both the pattern and the text are uncertain. Central to our solution is a special case where the sequences have equal length, called the consensus problem. We propose algorithms for the consensus problem parameterized by the number of strings that match one of the sequences. As our basic approach, a careful adaptation of the classic meet-in-the-middle algorithm for the knapsack problem is used. On the lower bound side, we prove that our dependence on the parameter is optimal up to lower-order terms conditioned on the optimality of the original algorithm for the knapsack problem.Comment: 22 page

Selfish Knapsack

We consider a selfish variant of the knapsack problem. In our version, the items are owned by agents, and each agent can misrepresent the set of items she owns---either by avoiding reporting some of them (understating), or by reporting additional ones that do not exist (overstating). Each agent's objective is to maximize, within the items chosen for inclusion in the knapsack, the total valuation of her own chosen items. The knapsack problem, in this context, seeks to minimize the worst-case approximation ratio for social welfare at equilibrium. We show that a randomized greedy mechanism has attractive strategic properties: in general, it has a correlated price of anarchy of $2$ (subject to a mild assumption). For overstating-only agents, it becomes strategyproof; we also provide a matching lower bound of $2$ on the (worst-case) approximation ratio attainable by randomized strategyproof mechanisms, and show that no deterministic strategyproof mechanism can provide any constant approximation ratio. We also deal with more specialized environments. For the case of $2$ understating-only agents, we provide a randomized strategyproof $\frac{5+4\sqrt{2}}{7} \approx 1.522$-approximate mechanism, and a lower bound of $\frac{5\sqrt{5}-9}{2} \approx 1.09$. When all agents but one are honest, we provide a deterministic strategyproof $\frac{1+\sqrt{5}}{2} \approx 1.618$-approximate mechanism with a matching lower bound. Finally, we consider a model where agents can misreport their items' properties rather than existence. Specifically, each agent owns a single item, whose value-to-size ratio is publicly known, but whose actual value and size are not. We show that an adaptation of the greedy mechanism is strategyproof and $2$-approximate, and provide a matching lower bound; we also show that no deterministic strategyproof mechanism can provide a constant approximation ratio