On the Hardness of Being Truthful

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On the Hardness of Being Truthful

• Michael Schapira
• Joint work with Christos Papadimitriou and Yaron
  Singer (UC Berkeley)

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The Hebrew University of Jerusalem
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Overview of the Talk

• The Combinatorial Public Project Problem.
• The Communication Hardness of Truthfulness.
• The Computational Hardness of Truthfulness.
• Conclusion and Open Questions.

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Combinatorial Public Project

• Set of n users Set of m resources
• Each user i has a valuation function vi 2m
  ? R0
• Objective Given a parameter k, choose a set of
  resources S of size k which maximizes the social
  welfare

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Assumptions Regarding Each Valuation Function

• Normalized
• v(Ø ) 0
• Non-decreasing
• v(S) v(T) S T
• Submodular
• v( S ? j )- v(S) v( T ? j )- v(T) S
  T

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Motivating Examples

• Elections for a committee The agents are voters,
  resources are potential candidates.
• Overlay networks We wish to select a subset of
  nodes in a graph that will function as an overlay
  network. http//nms.csail.mit.edu/ron/

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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 regardless of other
  agents reports.
• The utility of each user is ui vi(S) - pi

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Are Combinatorial Public Projects Easy?

• Computational PerspectiveA 1-1/e approximation
  ratio is achievable due to the submodularity of
  the valuations (but not truthful)
• A tight lower bound exists Feige.
• Strategic PerspectiveA truthful solution is
  achievable via VCG payments (but NP-hard to
  obtain)
• What about achieving both simultaneously?

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Central Open Question in Algorithmic Mechanism
Design Nisan-Ronen
easy easy easy? canonically hard problems
- Feigenbaum-Shenker
Conjecture NO!
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Overview of the Talk

• The Combinatorial Public Project Problem.
• The Communication Hardness of Truthfulness.
• The Computational Hardness of Truthfulness.
• Conclusion and Open Questions.

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Truth and Computation Dont Mix

• Theorem Any truthful algorithm for the
  combinatorial public project problem which
  approximates better than vm requires exponential
  communication in m.
• Even for n2.
• Implications for AMD A huge gap between
  truthfulpolynomial algorithms, and
  truthful/polynomial algorithms.
• Remark This lower bound is tight.

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Proving the Lower Bound

• Lemma 1 Any affine maximizer for the
  combinatorial public project problem which
  approximates better than vm requires exponential
  communication in m.
• Lemma 2 (!) An algorithm for the combinatorial
  public project problem is truthful iff its an
  affine maximizer

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Affine Maximizers

• Def (informal) A is an affine maximizer if
  there is some RA S k S m
  we shall refer to RA as As range.

s.t. A(v1,vn) argmaxS in RASi vi(S)(for
all v1,,vn)
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Lower Bound For Affine Maximizers

• Lemma 1 Any affine maximizer for the
  combinatorial public project problem which
  approximates better than vm requires exponential
  communication in m.
• Proof in two steps Dobzinski-Nisan
• Proposition 1 In order to get an approximation
  better than vm, the range must be exponentially
  large (in m)
• Even for n1.
• Proposition 2 Maximizing over a range RA
  requires communicating RA bits.
• Even for n2.

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Lower Bound For Affine Maximizers

• Proposition 1 In order to get an approximation
  better than vm, the range must be exponentially
  large (in m)
• Probablistic construction.
• Let kvm. Choose uniformly at random a set of
  resources T s.t. Tvm.
• v1(Q)QnT for every Q.
• For every set S in RA (Svm) Pr SnT cme
  is exp. small.
• Proposition 2 Maximizing over a range RA
  requires communicating RA bits.
• Reduction from the SET-DISJOINTNESS problem.
• Two parties, Alice and Bob, each holding a subset
  of 1,,t.
• It requires O(t) bits to find out if Alice and
  Bob share a common element.
• Identify RA with 1,,t.

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Characterization Lemma

• Characterization Lemma An algorithm for the
  combinatorial public project problem is truthful
  iff its an affine maximizer!
• Theorem (Roberts 79) For unrestricted valuation
  functions any truthful mechanism is an affine
  maximizer
• We use machinery from simplified proofs of
  Roberts Theorem Lavi-Mualem-Nisan.
• But our domain is severely restricted!
• But our domain isnt open!

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Characterizating Truthfulness (cntd)
single-parameterdomains
unrestricted valuations
Only affine maximizers!(Roberts 1979)
Manynon-affine-maximizers(truthfulnessis
well-understood)
Not always affine-maximizersauction settings
Lavi-Mualem-Nisan, Bartal-Gonen-Nisan
Always affine-maximizersfor the case of
combinatorial public projects!
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Overview of the Talk

• The Combinatorial Public Project Problem.
• The Communication Hardness of Truthfulness.
• The Computational Hardness of Truthfulness.
• Conclusion and Open Questions.

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Computational Hardness of Truthfulness

• To prove our results we had to assume that the
  input can be exponential in m.
• Realistic?
• If users have succinctly described valuations
  then computational-complexity techniques are
  required.
• No such impossibility results are known.

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Computational Hardness of Truthfulness

• Theorem There is a class of succinctly-described
  valuations C s.t.
• There exists a polynomial-time algorithm for
  combinatorial public project with valuations in C
  that obtains an approximation ratio of 1-1/e.
• Any truthful polynomial-time approximation
  algorithm cannot obtain an approximation ratio
  better than vm unless NP BPP.

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Our Proof Revisited

• Characterization Lemma an algorithm is truthful
  iff it is an affine-maximizer.
• Observation The proof only requires
  succinctly-described valuations.
• Inapproximability Lemma Any affine maximizer
  which approximates better than vm requires
  exponential communication.
• Proposition 1 In order to get an approximation
  better than vm, the range must be exponential.
• Proposition 2 Maximizing over a range RA
  requires communicating RA bits.

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New Proof

• Characterization Lemma an algorithm is truthful
  iff it is an affine-maximizer.
• Inapproximability Lemma No affine maximizer can
  approximate better than vm unless computational
  assumption is false.
• Proposition 1 In order to get an approximation
  better than vm, the range must be exponential.
• New Challenge Maximizing over an
  exponential-size range in polynomial time implies
  that computational assumption is false.
• New Technique.

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Computational-Complexity Hardness

• For many families of succinctly described
  valuations combinatorial public projects are
  NP-hard.
• Special case MAX-K-COVER Feige
• So, optimizing over the set of all possible
  solutions is hard.
• What about optimizing over a set of solutions of
  exponential size?
• Intuition - also hard!

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SATL

• You are given a language L 0,1n s.t. L is
  exponentially dense, i.e., L 2na (for some
  constant 0lta1)
• SATL Given a CNF determine whether there is a
  satisfying assignment in L.
• Conjecture SATL is NP-hard for every
  exponentially dense L.

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Intuition

• Let L s s is of the form 000xxx
• For this L, SATL is obviously NP-hard.
• General approach Find a smaller SAT hiding in
  SATL.
• Not too small!

n/2
n/2
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Sauer-Shelah Lemma (for SATL)

• Let L be some exponentially dense language.
• Then, there exists a set N of nb variables (for
  some constant 0ltb1) s.t. all assignments for
  these variables are in L.
• N is shattered by L.
• Are we done? Did we prove that SATL is NP-hard?

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No!

• We do not know how to find (approximate) N in
  polynomial time.
• Hard! Papadimitriou-Yannakakis, Schaefer,
  Mossel-Umans
• Theorem If SATL is in P then SAT has polynomial
  circuits.
• What about a probabilistic reduction from SAT?
• A naïve approach fails.
• Ajtais probabilistic version of the Sauer-Shelah
  Lemma helps in our case!
• What about SATL?

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CIRCUIT SATL

• You are given a language L 0,1n s.t. L is
  exponentially dense, i.e., L 2na (for some
  constant 0lta1)
• CIRCUIT SATL Given a boolean circuit determine
  whether it has a satisfying input-assignment in
  L.
• Theorem If CIRCUIT SATL is in P then SAT is in
  BPP.
• Hashing

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Overview of the Talk

• The Combinatorial Public Project Problem.
• The Communication Hardness of Truthfulness.
• The Computational Hardness of Truthfulness.
• Conclusion and Open Questions.

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Conclusion

• There is a huge gap between unrestricted and
  truthful (polynomial-time) algorithms.
• This is true in both the communication model and
  the computational model.
• Surprising connections between mechanism design
  and complexity theory.

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Open Questions What are the Limitations of Our
Techniques?

• Characterizing truthfulness (combinatorial
  auctions?).
• Proving computational-complexity lower bounds for
  AMD.
• SATL?
• Other problems (combinatorial auctions?)
  Mossel-Papadimitriou-S-Singer, work in progress)

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Open Questions

• Other notions of the hardness of truthfulness
  Babaioff-Blumrosen-Naor-S
• The approximability of combinatorial public
  projects S-Singer, work in progress.

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Thanks!