VC v. VCG: Inapproximability of Combinatorial

Auctions via Generalizations of the VC Dimension

Michael SchapiraYale University and UC Berkeley

Joint work with Christos Papadimitriou and Yaron Singer

(2008) and Elchanan Mossel, Christos Papadimitriou and Yaron Singer

(2009)

Auctions: Different Concerns• Computational concerns:

bounded computational resourcesoptimization…

• Economic concerns: truthful behaviourfairness…

computational

efficiency

incentive-compatibilit

y

Algorithmic Mechanism Design• Can these different desiderata coexist?

• The central problem in Algorithmic Mechanism Design [Nisan-Ronen]

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 vi.– per-item values ai1,…,aim – “maximum spending” value bi– For every bundle S, vi(S)=min {j in S aij , bi},

• Goal: find a partition of the items between the bidders S1,…,Sn such that social welfare i vi(Si) is maximized

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 ui = vi(S) – pi– Solution concepts: dominant strategies, ex-post Nash.

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?

Huge Gap!

?non-truthful:get arbitrarilyclose to opt.

truthful: 1/n-appx mechanism

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.

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)

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 specificclass of

mechanisms(VCG-based).

generalize the VC dimension to

handle partitions of a universe.

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]

Maximal-In-Range Mechanisms

• A mechanism M is MIR (= VCG-based) if:– There’s a fixed subset RM of the possible

outcomes (allocations of the m items between the n bidders) = “M’s range”.

– For every valuation profile (v1,…vn) M outputs the optimal partition in RM.

• Example: The trivial (1/n-appx.) mechanism – Bundle all items together.– Allocated them to the highest bidder.

• Fact: MIR mechanisms are truthful (VCG…).

RM

allpartitions

Can We Do Better Than the Trivial MIR Mechanism?

• Can we choose RM such that

– the optimum in RM always provides a constant approximation to the global optimum.

– optimizing over RM 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!

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 RM 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.

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 RM implicitly means solving an NP-hard problem.

Proof (intuition)

1 2 3 4 5items

Mechanism M

1 2 3 4 5

1 2 3 4 5

1 2 3 4 51 2 3 4 5

RM

M is (implicitly)optimally solvinga 2-item auction

Proof• So, we wish to prove the existence of a

subset of items E that is “shattered” by M’s range (RM).

– “Embed” a smaller auction in E.

– Not too small! (|E| ≥ m)

• VC dimension!

Proof

• Lemma: If a MIR mechanism M obtains an appx-ratio of ½+ in the allocate-all-items case then |RM| ≥ 2m (for some constant >0).– Proof by probabilistic construction.

• Corollary: Bidder 1 can be assigned at least 2m different subsets of items by M.– Denote this collection of subsets by RM,1

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 2E.– |E| ≥ ( log(|R|)/log(|U|) ).

• Corollary (set R=RM,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 RM.– Because all items are allocated.

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

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.

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, (T1,T2) and (T’1,T’2) , are said to be b-far (or at distance b) if |T1 U T’2| + |T’1 U T2| ≥ b.

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|) ).

A Lower Bound• Theorem: For any MIR mechanism M that

obtains an appx-ratio of ¾ + , there exists some R RM 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.

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.

Thank YouThank You