Transcript Inapproximability of Combinatorial Public Projects Michael Schapira (Yale University and UC Berkeley) Joint work with Yaron Singer (UC Berkeley)

Inapproximability of
Combinatorial Public
Projects
Michael Schapira
(Yale University and UC Berkeley)
Joint work with Yaron Singer (UC Berkeley)
Overview of the Talk
• The Combinatorial Public Project Problem.
• The Submodular Case – Background.
• A Trivial Truthful Approximation-Algorithm.
• Our Main Result.
• Conclusions and Open Questions.
Algorithmic Mechanism Design
• Algorithmic Mechanism Design deals with
designing efficient mechanisms for
decentralized computerized settings [Nisan-Ronen].
• Takes into account both the strategic behavior
of the different participants and the usual
computational efficiency considerations.
• Target applications: protocols for Internet
environments.
Combinatorial Public Project
• Set of n users; Set of m resources;
• Each user i has a valuation function:
vi : 2[m] → R≥0
• Objective: Given a parameter k, choose a set of
resources S* of size k which maximizes the
social welfare:
S* = argmax Σi vi(S)
S [m], |S|=k
Assumptions Regarding Each Valuation
Function
• Normalized:
v(∅ ) = 0
• Non-decreasing:
v(S) ≤ v(T)
S T
• Subadditive:
v( S) + v(T) ≥ v( S υ T)
[ Submodular: v( S υ { j })− v(S) ≥ v( T υ { j })− v(T)
S T ]
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/]
Access Models
How can we access the input ?
• One possibility: succinct valuations
computational complexity approach.
• The “black box” approach: each bidder is
represented by an oracle which can
answer certain queries.
Communication complexity approach.
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
The Submodular Case [Papadimitriou-S-Singer]
• Computational Perspective:
A 1-1/e approximation ratio is achievable due to the
submodularity of the valuations (but not truthful)
 A tight lower bound exists [Feige].
• Strategic Perspective:
A truthful solution is achievable via VCG payments
(but NP-hard to obtain)
• What about achieving both simultaneously?
The Submodular Case: Truth and
Computation Don’t Mix
• Theorem [Papadimitriou-S-Singer]:
Any truthful algorithm for the combinatorial public project
problem which approximates better than √m requires
exponential communication in m.
Even for n=2.
• Implications for AMD: A huge gap between
truthful&polynomial algorithms, and truthful/polynomial
algorithms.
A trivial √m-approximation
Algorithm for Subadditiver Agents
• The algorithm:
If k≤√m, simply choose the single
resource j for which the social-welfare is
maximized.
If k>√m, divide the m resources to √m
disjoint sets of equal size and choose the
one that maximizes the social welfare.
The Algorithm is Truthful
• Fact: Maximal-in-range algorithms
are truthful (VCG).
• -> The trivial approximation algorithm
is (essentially) the best truthful
algorithm for the submodular case.
Also for the subadditive case.
Upper and Lower Bounds
constant
non- truthful
upper bounds
exist
?
Submodular
Subadditive
√m truthful
upper bound
√m truthful
upper bound
High Hopes
• Twin problem: combinatorial auctions.
• Theorem (Informal): There is a 2approximation algorithm for
combinatorial auctions with subadditive
bidders. [Feige]
Our Main Result
• Theorem: Any approximation algorithm for the
combinatorial public project problem with subadditive
agents which approximates better than O(m1/4)
requires exponential communication in m.
• Implications: The trivial truthful approximation
algorithm is nearly tight even from a purely
computational perspective.
Other Results and an Open Question
• A communication complexity lower bound for
general valuations.
• A computational complexity lower bound for
general valuations.
• Open question: prove a computational-complexity
analogue of our result.
Thanks!