arXiv subjects

Frederick V. Qiu

Publications and source records attributed to Frederick V. Qiu.

3 recordsLinked to original sources

arXiv2025

The $k$-Fold Matroid Secretary Problem

In the matroid secretary problem, elements $N := [n]$ of a matroid $\mathcal{M} \subseteq 2^N$ arrive in random order. When an element arrives, its weight is revealed and a choice must be made to accept or reject the element, subject to the constraint that the accepted set $S \in \mathcal{M}$. Kleinberg'05 gives a $(1-O(1/\sqrt{k}))$-competitive algorithm when $\mathcal{M}$ is a $k$-uniform matroid. We generalize their result, giving a $(1-O(\sqrt{\log(n)/k}))$-competitive algorithm when $\mathcal{M}$ is a $k$-fold matroid union.

cs.DS↗

arXiv2025

The Communication Complexity of Combinatorial Auctions with Additional Succinct Bidders

We study the communication complexity of welfare maximization in combinatorial auctions with bidders from either a standard valuation class (which require exponential communication to explicitly state, such as subadditive or XOS), or arbitrary succinct valuations (which can be fully described in polynomial communication, such as single-minded). Although succinct valuations can be efficiently communicated, we show that additional succinct bidders have a nontrivial impact on communication complexity of classical combinatorial auctions. Specifically, let $n$ be the number of subadditive/XOS bidders. We show that for SA $\cup$ SC (the union of subadditive and succinct valuations): (1) There is a polynomial communication $3$-approximation algorithm; (2) As $n \to \infty$, there is a matching $3$-hardness of approximation, which (a) is larger than the optimal approximation ratio of $2$ for SA, and (b) holds even for SA $\cup$ SM (the union of subadditive and single-minded valuations); and (3) For all $n \geq 3$, there is a constant separation between the optimal approximation ratios for SA $\cup$ SM and SA (and therefore between SA $\cup$ SC and SA as well). Similarly, we show that for XOS $\cup$ SC: (1) There is a polynomial communication $2$-approximation algorithm; (2) As $n \to \infty$, there is a matching $2$-hardness of approximation, which (a) is larger than the optimal approximation ratio of $e/(e-1)$ for XOS, and (b) holds even for XOS $\cup$ SM; and (3) For all $n \geq 2$, there is a constant separation between the optimal approximation ratios for XOS $\cup$ SM and XOS (and therefore between XOS $\cup$ SC and XOS as well).

cs.GT↗

arXiv2024

Settling the Communication Complexity of VCG-based Mechanisms for all Approximation Guarantees

We consider truthful combinatorial auctions with items $M = [m]$ for sale to $n$ bidders, where each bidder $i$ has a private monotone valuation $v_i : 2^M \to R_+$. Among truthful mechanisms, maximal-in-range (MIR) mechanisms achieve the best-known approximation guarantees among all poly-communication deterministic truthful mechanisms in all previously-studied settings. Our work settles the communication necessary to achieve any approximation guarantee via an MIR mechanism. Specifically: Let MIRsubmod$(m,k)$ denote the best approximation guarantee achievable by an MIR mechanism using $2^k$ communication between bidders with submodular valuations over $m$ items. Then for all $k = Ω(\log(m))$, MIRsubmod$(m,k) = Ω(\sqrt{m/(k\log(m/k))})$. When $k = Θ(\log(m))$, this improves the previous best lower bound for poly-comm. MIR mechanisms from $Ω(m^{1/3}/\log^{2/3}(m))$ to $Ω(\sqrt{m}/\log(m))$. We also have MIRsubmod$(m,k) = O(\sqrt{m/k})$. Moreover, our mechanism is optimal w.r.t. the value query and succinct representation models. When $k = Θ(\log(m))$, this improves the previous best approximation guarantee for poly-comm. MIR mechanisms from $O(\sqrt{m})$ to $O(\sqrt{m/\log(m)})$. Let also MIRgen$(m,k)$ denote the best approximation guarantee achievable by an MIR mechanism using $2^k$ communication between bidders with general valuations over $m$ items. Then for all $k = Ω(\log(m))$, MIRgen$(m,k) = Ω(m/k)$. When $k = Θ(\log(m))$, this improves the previous best lower bound for poly-comm. MIR mechanisms from $Ω(m/\log^2(m))$ to $Ω(m/\log(m))$. We also have MIRgen$(m,k) = O(m/k)$. Moreover, our mechanism is optimal w.r.t. the value query and succinct representation models. When $k = Θ(\log(m))$, this improves the previous best approximation guarantee for poly-comm. MIR mechanisms from $O(m/\sqrt{\log(m)})$ to $O(m/\log(m))$.

cs.GT↗