[Paper Review] On the Hardness of Welfare Maximization in Combinatorial Auctions with Submodular Valuations

Shahar Dobzinski, J. Vondrák|arXiv (Cornell University)|Feb 13, 2012

Complexity and Algorithms in Graphs8 references3 citations

TL;DR

This paper establishes the first constant-factor inapproximability threshold for welfare maximization in combinatorial auctions with submodular valuations, proving that no algorithm can achieve better than a (1 − 1/(2e) + ε)-approximation under both communication and computational models. The authors introduce multi-peak submodular functions to construct hardness instances that demonstrate this limit holds even for constant-size valuations and in the demand oracle model.

ABSTRACT

We present a new type of monotone submodular functions: \emph{multi-peak submodular functions}. Roughly speaking, given a family of sets $\cF$, we construct a monotone submodular function $f$ with a high value $f(S)$ for every set $S \in {\cF}$ (a "peak"), and a low value on every set that does not intersect significantly any set in $\cF$. We use this construction to show that a better than $(1-\frac{1}{2e})$-approximation ($\simeq 0.816$) for welfare maximization in combinatorial auctions with submodular valuations is (1) impossible in the communication model, (2) NP-hard in the computational model where valuations are given explicitly. Establishing a constant approximation hardness for this problem in the communication model was a long-standing open question. The valuations we construct for the hardness result in the computational model depend only on a constant number of items, and hence the result holds even if the players can answer arbitrary queries about their valuation, including demand queries. We also study two other related problems that received some attention recently: max-min allocation (for which we also get hardness of $(1-\frac 1 {2e}+ε)$-approximation, in both models), and combinatorial public projects (for which we prove hardness of $(3/4+ε)$-approximation in the communication model, and hardness of $(1 -\frac 1 e+ε)$-approximation in the computational model, using constant size valuations).

Motivation & Objective

• To close a long-standing open question on the communication complexity hardness of submodular welfare maximization.
• To establish NP-hardness of (1−1/(2e)+ε)-approximation for constant-size submodular valuations.
• To extend the hardness result to related problems such as max-min allocation and combinatorial public projects.
• To show that the hardness threshold holds across all reasonable oracle models, including demand oracles.
• To provide a new construction of multi-peak submodular functions as a key technical tool for proving inapproximability.

Proposed method

• Introduces multi-peak submodular functions that achieve high values on a family of sets F and low values elsewhere.
• Uses a reduction from Feige's Max k-cover inapproximability result to construct submodular valuations for combinatorial auctions.
• Constructs constant-size submodular valuations based on the structure of Feige's Max k-cover instance with uniform set sizes and bounded intersections.
• Applies the multi-peak function construction to create valuation functions where each player's valuation peaks on a specific family of bundles.
• Demonstrates that any (1−1/(2e)+ε)-approximation would imply P=NP in the computational model and requires exponential communication in the communication model.
• Extends the hardness to max-min allocation and combinatorial public projects using similar constructions and reductions.

Experimental results

Research questions

• RQ1 Is a (1−1/(2e)+ε)-approximation for submodular welfare maximization possible in the communication model?
• RQ2 Can (1−1/(2e)+ε)-approximation be achieved in polynomial time for constant-size submodular valuations?
• RQ3 Does the hardness result extend to other problems such as max-min allocation and combinatorial public projects?
• RQ4 Can the inapproximability threshold be proven for all reasonable oracle models, including demand oracles?
• RQ5 What new structural tools are needed to prove constant-factor inapproximability in submodular combinatorial auctions?

Key findings

• A (1−1/(2e)+ε)-approximation for submodular welfare maximization requires exponential communication, resolving a long-standing open question.
• For constant-size submodular valuations, achieving a (1−1/(2e)+ε)-approximation implies P=NP.
• The hardness result holds for all reasonable oracle models, including demand oracles, due to their simulatability in communication and implementability in constant time.
• The same construction yields (1−1/(2e)+ε)-hardness for max-min allocation in both models.
• A (3/4+ε)-approximation is hard in the communication model and a (1−1/e+ε)-approximation is hard in the computational model for combinatorial public projects.
• The paper introduces multi-peak submodular functions as a novel technical tool that enables the first constant-factor inapproximability results in this setting.