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!