Ashwinkumar Badanidiyuru — Page 2

arXiv2011

Approximating Low-Dimensional Coverage Problems

We study the complexity of the maximum coverage problem, restricted to set systems of bounded VC-dimension. Our main result is a fixed-parameter tractable approximation scheme: an algorithm that outputs a $(1-\eps)$-approximation to the maximum-cardinality union of $k$ sets, in running time $O(f(\eps,k,d)\cdot poly(n))$ where $n$ is the problem size, $d$ is the VC-dimension of the set system, and $f(\eps,k,d)$ is exponential in $(kd/\eps)^c$ for some constant $c$. We complement this positive result by showing that the function $f(\eps,k,d)$ in the running-time bound cannot be replaced by a function depending only on $(\eps,d)$ or on $(k,d)$, under standard complexity assumptions. We also present an improved upper bound on the approximation ratio of the greedy algorithm in special cases of the problem, including when the sets have bounded cardinality and when they are two-dimensional halfspaces. Complementing these positive results, we show that when the sets are four-dimensional halfspaces neither the greedy algorithm nor local search is capable of improving the worst-case approximation ratio of $1-1/e$ that the greedy algorithm achieves on arbitrary instances of maximum coverage.

cs.CG↗

arXiv2011

Optimization with Demand Oracles

We study \emph{combinatorial procurement auctions}, where a buyer with a valuation function $v$ and budget $B$ wishes to buy a set of items. Each item $i$ has a cost $c_i$ and the buyer is interested in a set $S$ that maximizes $v(S)$ subject to $Σ_{i\in S}c_i\leq B$. Special cases of combinatorial procurement auctions are classical problems from submodular optimization. In particular, when the costs are all equal (\emph{cardinality constraint}), a classic result by Nemhauser et al shows that the greedy algorithm provides an $\frac e {e-1}$ approximation. Motivated by many papers that utilize demand queries to elicit the preferences of agents in economic settings, we develop algorithms that guarantee improved approximation ratios in the presence of demand oracles. We are able to break the $\frac e {e-1}$ barrier: we present algorithms that use only polynomially many demand queries and have approximation ratios of $\frac 9 8+ε$ for the general problem and $\frac 9 8$ for maximization subject to a cardinality constraint. We also consider the more general class of subadditive valuations. We present algorithms that obtain an approximation ratio of $2+ε$ for the general problem and 2 for maximization subject to a cardinality constraint. We guarantee these approximation ratios even when the valuations are non-monotone. We show that these ratios are essentially optimal, in the sense that for any constant $ε>0$, obtaining an approximation ratio of $2-ε$ requires exponentially many demand queries.

cs.GT↗