General Bounds for Incremental Maximization

We propose a theoretical framework to capture incremental solutions to cardinality constrained maximization problems. The defining characteristic of our framework is that the cardinality/support of the solution is bounded by a value $k\in\mathbb{N}$ that grows over time, and we allow the solution to be extended one element at a time. We investigate the best-possible competitive ratio of such an incremental solution, i.e., the worst ratio over all $k$ between the incremental solution after $k$ steps and an optimum solution of cardinality $k$. We define a large class of problems that contains many important cardinality constrained maximization problems like maximum matching, knapsack, and packing/covering problems. We provide a general $2.618$-competitive incremental algorithm for this class of problems, and show that no algorithm can have competitive ratio below $2.18$ in general. In the second part of the paper, we focus on the inherently incremental greedy algorithm that increases the objective value as much as possible in each step. This algorithm is known to be $1.58$-competitive for submodular objective functions, but it has unbounded competitive ratio for the class of incremental problems mentioned above. We define a relaxed submodularity condition for the objective function, capturing problems like maximum (weighted) ($b$-)matching and a variant of the maximum flow problem. We show that the greedy algorithm has competitive ratio (exactly) $2.313$ for the class of problems that satisfy this relaxed submodularity condition. Note that our upper bounds on the competitive ratios translate to approximation ratios for the underlying cardinality constrained problems.Comment: fixed typo

Truthful Assignment without Money

We study the design of truthful mechanisms that do not use payments for the generalized assignment problem (GAP) and its variants. An instance of the GAP consists of a bipartite graph with jobs on one side and machines on the other. Machines have capacities and edges have values and sizes; the goal is to construct a welfare maximizing feasible assignment. In our model of private valuations, motivated by impossibility results, the value and sizes on all job-machine pairs are public information; however, whether an edge exists or not in the bipartite graph is a job's private information. We study several variants of the GAP starting with matching. For the unweighted version, we give an optimal strategyproof mechanism; for maximum weight bipartite matching, however, we show give a 2-approximate strategyproof mechanism and show by a matching lowerbound that this is optimal. Next we study knapsack-like problems, which are APX-hard. For these problems, we develop a general LP-based technique that extends the ideas of Lavi and Swamy to reduce designing a truthful mechanism without money to designing such a mechanism for the fractional version of the problem, at a loss of a factor equal to the integrality gap in the approximation ratio. We use this technique to obtain strategyproof mechanisms with constant approximation ratios for these problems. We then design an O(log n)-approximate strategyproof mechanism for the GAP by reducing, with logarithmic loss in the approximation, to our solution for the value-invariant GAP. Our technique may be of independent interest for designing truthful mechanisms without money for other LP-based problems.Comment: Extended abstract appears in the 11th ACM Conference on Electronic Commerce (EC), 201