From Convex Optimization to Randomized Mechanisms: Toward Optimal Combinatorial Auctions

We design an expected polynomial-time, truthful-in-expectation, (1-1/e)-approximation mechanism for welfare maximization in a fundamental class of combinatorial auctions. Our results apply to bidders with valuations that are m matroid rank sums (MRS), which encompass most concrete examples of submodular functions studied in this context, including coverage functions, matroid weighted-rank functions, and convex combinations thereof. Our approximation factor is the best possible, even for known and explicitly given coverage valuations, assuming P != NP. Ours is the first truthful-in-expectation and polynomial-time mechanism to achieve a constant-factor approximation for an NP-hard welfare maximization problem in combinatorial auctions with heterogeneous goods and restricted valuations. Our mechanism is an instantiation of a new framework for designing approximation mechanisms based on randomized rounding algorithms. A typical such algorithm first optimizes over a fractional relaxation of the original problem, and then randomly rounds the fractional solution to an integral one. With rare exceptions, such algorithms cannot be converted into truthful mechanisms. The high-level idea of our mechanism design framework is to optimize directly over the (random) output of the rounding algorithm, rather than over the input to the rounding algorithm. This approach leads to truthful-in-expectation mechanisms, and these mechanisms can be implemented efficiently when the corresponding objective function is concave. For bidders with MRS valuations, we give a novel randomized rounding algorithm that leads to both a concave objective function and a (1-1/e)-approximation of the optimal welfare.

💡 Research Summary

The paper tackles the classic problem of welfare maximization in combinatorial auctions, where a set of m heterogeneous items must be allocated among n bidders whose private valuation functions are unknown to the seller. The authors focus on a broad class of submodular valuations known as matroid rank sum (MRS) functions—non‑negative linear combinations of matroid rank functions. This class includes coverage functions, weighted‑rank matroid functions, and any convex combination thereof, thereby covering most concrete submodular examples studied in the literature.

The main contribution is a truthful‑in‑expectation mechanism that runs in expected polynomial time and achieves a (1 − 1/e) approximation to the optimal social welfare. This approximation ratio is provably optimal under the standard assumption P ≠ NP, even when the valuations are explicitly given coverage functions. Prior to this work, no polynomial‑time, truthful‑in‑expectation mechanism achieved a constant‑factor approximation for any NP‑hard welfare maximization problem in combinatorial auctions with heterogeneous items and non‑trivial valuation restrictions.

The mechanism is built on a novel design framework that departs from the traditional two‑step approach (solve a fractional relaxation, then apply a randomized rounding algorithm). The key insight is to optimize directly over the distribution of outcomes produced by the rounding algorithm, rather than over the fractional solution that serves as its input. Formally, let r(x) denote a randomized rounding procedure that maps a fractional allocation x∈