1. VC v. VCG: Inapproximability of Combinatorial Auctions via Generalizations of the VC Dimension Michael Schapira Yale University and UC Berkeley Joint work with Christos Papadimitriou and Yaron Singer (2008) and Elchanan Mossel, Christos Papadimitriou and Yaron Singer (2009)

2. Auctions: Different Concerns Computational concerns:  bounded computational resources  optimization  … Economic concerns:  truthful behaviour  fairness  … computational efficiency incentive- compatibility

3. Algorithmic Mechanism Design Can these different desiderata coexist? The central problem in Algorithmic Mechanism Design [Nisan-Ronen]

4. Illustration: Restricted Combinatorial Auctions A set of m items for sale {1,…m}. n bidders {1,…,n}. Each bidder i has an additive valuation with a spending constraint v i. –per-item values a i1,…,a im –“maximum spending” value b i –For every bundle S, v i (S)=min {  j in S a ij, b i }, Goal: find a partition of the items between the bidders S 1,…,S n such that social welfare  i v i (S i ) is maximized

5. What Do We Want? Quality of the solution: As close to the optimum as possible. Computationally tractable: Polynomial running time (in n and m). Truthful: Motivate (via payments) agents to report their true values. –The utility of each user is u i = v i (S) – p i –Solution concepts: dominant strategies, ex- post Nash.

6. State of the Art Easy from an economic perspective. –VCG! Easy to solve computationally. –NP-hard (even for n=2) [Lehmann-Lehmann-Nisan] but… –We can get arbitrarily close to the optimum for any constant n (PTAS)! [Andelman-Mansour] Can both be achieved simultaneously?

7. Huge Gap! ? non-truthful: get arbitrarily close to opt. truthful: 1/n-appx mechanism

8. More Generally… easy + easy = easy? NO! [Papadimitriou-S-Singer] Hard (Clique) Easy (in APX, e.g., matching) Easy (social- welfare max. in auctions) Hard (max-min fairness in auctions) Computation Incentives

9. Truthfulness and Computation Clash: Combinatorial Public Projects Problem (CPPP) Orthogonal to combinatorial auctions (elections, overlay networks). Easy from a purely economic perspective (VCG), and from a purely computational perspective (in APX). Theorem (Informal) [Papadimitriou-S-Singer] : No truthful and computationally-efficient mechanism for CPPP obtains a constant approximation ratio.

10. Combinatorial Public Projects: The Proof Complexity theory mechanism design combinatorics (the embedding of NP-hard problems) (Characterization of truthful mechanisms, based on Roberts’ Theorem) (VC dimension)

11. What About Combinatorial Auctions? Complexity theory mechanism design combinatorics (the embedding of NP-hard problems) (Characterization of truthful mechanisms, based on Roberts’ Theorem) (VC dimension) consider a specific class of mechanisms (VCG-based). generalize the VC dimension to handle partitions of a universe.

12. Back to Our Problem… A set of items for sale {1,…m}. n bidders {1,…,n}. Each bidder i has additive valuation with a spending constraint v i. –per-item values a i1,…,a im –“maximum spending” value b i –For every bundle S, v i (S)=min {  j in S a ij, b i }, Goal: find a partition S 1,…,S n such that social welfare  i v i (S i ) is maximized

13. Maximal-In-Range Mechanisms No good characterizations of truthful mechanisms for combinatorial auctions. We must focus on specific classes of mechanisms. In this talk: Maximal-In-Range (MIR) mechanisms (= VCG based mechanisms)

14. VCG-Based Mechanisms VCG-based = Maximal-In-Range (MIR). MIR mechanisms provide the best known (deterministic) approximations for a large variety of problems: –Combinatorial auctions (general, subadditive, submodular). –Multi-unit auctions. –Unrelated machine scheduling. In fact, sometimes MIR is all you can do. [Roberts, Lavi-Mu’alem-Nisan, Dobzinski-Sundararajan, Papadimitriou-S- Singer]

15. Maximal-In-Range Mechanisms A mechanism M is MIR (= VCG-based) if: –There’s a fixed subset R M of the possible outcomes (allocations of the m items between the n bidders) = “M’s range”. –For every valuation profile (v 1,…v n ) M outputs the optimal partition in R M. Example: The trivial (1/n-appx.) mechanism –Bundle all items together. –Allocated them to the highest bidder. Fact: MIR mechanisms are truthful (VCG…). RMRM all partitions

16. Can We Do Better Than the Trivial MIR Mechanism? Can we choose R M such that –the optimum in R M always provides a constant approximation to the global optimum. –optimizing over R M can be done in a computationally- efficient manner. Not for the more general class of submodular valuations! [Dobzinski-Nisan] But… the “input” there is assumed to be exponentially large! (exp. communication) –What about succinctly-described valuations? –No computational-complexity results are known!

17. The Case of 2 Bidders Not trivial even for n=2! –We shall focus on this case. Let us first consider the (more easy) allocate- all-items case. –all outcomes in R M do not leave any item unallocated. Theorem: No computationally-efficient MIR mechanism M obtains an appx-ratio of ½+  in the allocate-all-items case. – unless NP has polynomial size circuits.

18. Proof Let M be a MIR mechanism for the 2- bidder case. Assume, by contradiction, that M obtains an appx-ratio of (1/2+  ). We shall prove that optimizing over R M implicitly means solving an NP-hard problem.

19. Proof (intuition) 1 2 3 4 5 items Mechanism M 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 RMRM M is (implicitly) optimally solving a 2-item auction

20. Proof So, we wish to prove the existence of a subset of items E that is “shattered” by M’s range (R M ). –“Embed” a smaller auction in E. –Not too small! (|E| ≥ m  ) VC dimension!

21. Proof Lemma: If a MIR mechanism M obtains an appx-ratio of ½+  in the allocate-all-items case then |R M | ≥ 2  m (for some constant  >0). –Proof by probabilistic construction. Corollary: Bidder 1 can be assigned at least 2  m different subsets of items by M. –Denote this collection of subsets by R M,1

22. Proof The Sauer-Shelah Lemma: Let R be a collection of subsets of a universe U. Then, there exists a subset E of U such that: –R’s projection on E is 2 E. –|E| ≥  ( log(|R|)/log(|U|) ). Corollary (set R=R M,1 ): There is a subset of items E, |E| ≥ m , s.t. bidder 1 can be assigned all subsets of E in M. Corollary: All partitions of E are induced by R M. –Because all items are allocated.

23. Proof We can now conclude that if M optimizes over its range then it is optimally solving an identical auction with m  items. –An NP-hard task. A non-uniform reduction. –We do not know how to find E in polynomial time. So… No computationally-efficient MIR mechanism M obtains an appx-ratio of ½+  in the allocate-all-items case (unless NP has polynomial size circuits). QED

24. Getting Rid of the Allocate-All- Items Assumption Not trivial! –If we just allocate unallocated items arbitrarily we might lose the MIR property! Our approach: Generalizing the VC dimension. –Of independent interest.

25. An Analogue of the Sauer-Shelah Lemma Definition: A partition of a universe is a pair of disjoint subsets of the universe. –Does not necessarily exhaust the universe! Definition: Two partitions, (T 1,T 2 ) and (T’ 1,T’ 2 ), are said to be b-far (or at distance b) if |T 1 U T’ 2 | + |T’ 1 U T 2 | ≥ b.

26. An Analogue of the Sauer-Shelah Lemma Lemma: Let   > 0 be some constant. Let R be a collection of partitions of a universe U, such that every two partitions in R are  |U|-far. Then, there exists a subset E of U such that: –R’s projection on E is all partitions of E. –|E| ≥  ( log(|R|)/log(|U|) ).

27. A Lower Bound Theorem: For any MIR mechanism M that obtains an appx-ratio of ¾ + , there exists some R  R M such that –R is exponential in m. –Every two partitions in R are  m-far (for some constant  >0) Theorem: No computationally-efficient MIR mechanism M obtains an appx-ratio of ¾+  –unless NP has polynomial size circuits.

28. Directions for Future Research A recent result [Buchfuhrer-Umans] : For any constant n, no MIR mechanism M obtains an appx-ratio of 1/n+  (unless NP has polynomial size circuits). –Tight for all constant n’s. –Non-constant n’s? Other classes of valuation functions. Characterizing truthful mechanisms for combinatorial auctions. Relaxing the computational assumption. Many intriguing questions regarding the VC dimension of partitions.

29. Thank You