The complexity of Boolean formula minimization

The Minimum Equivalent Expression problem is a natural optimization problem in the second level of the Polynomial-Time Hierarchy. It has long been conjectured to be Σ^P_2-complete and indeed appears as an open problem in Garey and Johnson (1979) [5]. The depth-2 variant was only shown to be Σ^P_2-complete in 1998 (Umans (1998) [13], Umans (2001) [15]) and even resolving the complexity of the depth-3 version has been mentioned as a challenging open problem. We prove that the depth-k version is Σ^P_2-complete under Turing reductions for all k ≥ 3. We also settle the complexity of the original, unbounded depth Minimum Equivalent Expression problem, by showing that it too is Σ^P_2-complete under Turing reductions

Inapproximability of Truthful Mechanisms via Generalizations of the VC Dimension

Algorithmic mechanism design (AMD) studies the delicate interplay between computational efficiency, truthfulness, and optimality. We focus on AMD's paradigmatic problem: combinatorial auctions. We present a new generalization of the VC dimension to multivalued collections of functions, which encompasses the classical VC dimension, Natarajan dimension, and Steele dimension. We present a corresponding generalization of the Sauer-Shelah Lemma and harness this VC machinery to establish inapproximability results for deterministic truthful mechanisms. Our results essentially unify all inapproximability results for deterministic truthful mechanisms for combinatorial auctions to date and establish new separation gaps between truthful and non-truthful algorithms

Progress on pricing with peering

This paper examines a simple model of how a provider ISP charges customer ISPs by assuming the provider ISP wants to maximize its revenue when customer ISPs have the possibility of setting up peering connections. It is shown that finding the optimal pricing is NP-complete, and APX-complete. Customers can respond to price in many ways, including throttling traffic as well as peering. An algorithm is studied which obtains a 1/4 approximation for a wide range of customer responses

The Complexity of Formula Minimization

The Minimum Equivalent Expression problem is a natural optimization problem in the second level of the Polynomial-Time Hierarchy. It has long been conjectured to be Σ2-complete and indeed appears as an open problem in Garey and Johnson. The depth-2 variant was only shown to be Σ2-complete in 1998, and even resolving the complexity of the depth-3 version has been mentioned as a challenging open problem. We prove that the depth-k version is Σ2-complete under Turing reductions for all k &gt;= 3. We also settle the complexity of the original, unbounded depth Minimum Equivalent Expression problem, by showing that it too is Σ2-complete under Turing reductions. We also consider a three-level model in which the third level is composed of parity gates, called SPPs. SPPs allow for small representations of Boolean functions and have efficient heuristics for minimization. However, little has been known about the complexity of SPP minimization. Here, we show that SPP minimization is Σ2-complete under Turing reductions.</p

Limits on Computationally Efficient VCG-Based Mechanisms for Combinatorial Auctions and Public Projects

A natural goal in designing mechanisms for auctions and public projects is to maximize the social welfare while incentivizing players to bid truthfully. If these are the only concerns, the problem is easily solved by use of the VCG mechanism. Unfortunately, this mechanism is not computationally efficient in general and there are currently no other general methods for designing truthful mechanisms. However, it is possible to design computationally efficient VCG-based mechanisms which approximately maximize the social welfare. We explore the design space of computationally efficient VCG-based mechanisms under submodular valuations and show that the achievable approximation guarantees are poor, even compared to efficient non-truthful algorithms. Some of these approximation hardness results stem from an asymmetry in the information available to the players versus that available to the mechanism. We develop an alternative Instance Oracle model which reduces this asymmetry by allowing the mechanism to access some computational capabilities of the players. By building assumptions about player computation into the model, a more realistic study of mechanism design can be undertaken. Finally, we give VCG-based mechanisms for some problems in the Instance Oracle model which achieve provably better approximations than the best VCG-based mechanism in the standard model. However, for other problems we give reductions in the Instance Oracle model which prove inapproximability results as strong as those shown in the standard model. These provide more robust hardness results that are not simply artifacts of the asymmetry in the standard model.</p

Limits on the Social Welfare of Maximal-In-Range Auction Mechanisms

Many commonly-used auction mechanisms are “maximal-in-range”. We show that any maximalin-range mechanism for n bidders and m items cannot both approximate the social welfare with a ratio better than min(n, m η) for any constant η &lt; 1/2 and run in polynomial time, unless NP ⊆ P/poly. This significantly improves upon a previous bound on the achievable social welfare of polynomial time maximal-in-range mechanisms of 2n/(n + 1) for constant n. Our bound is tight, as a min(n, 2m 1/2) approximation of the social welfare is achievable.

Abstract

The Minimum Equivalent Expression problem is a natural optimization problem in the second level of the Polynomial-Time Hierarchy. It has long been conjectured to be Σ P 2-complete and indeed appears as an open problem in Garey and Johnson [GJ79]. The depth-2 variant was only shown to be Σ P 2-complete in 1998 [Uma98], and even resolving the complexity of the depth-3 version has been mentioned as a challenging open problem. We prove that the depth-k version is Σ P 2-complete under Turing reductions for all k ≥ 3. We also settle the complexity of the original, unbounded depth Minimum Equivalent Expression problem, by showing that it too is Σ P 2-complete under Turing reductions. ∗ Supported by NSF CCF-0346991 and BSF 2004329