EECS 495: Combinatorial Optimization
Mechanism Design with Rounding
Lecture Manolis, Nima
Rn−1 : E[vi (f (vi , v−i )) − pi (vi , v−i )] ≥
E[vi (f (bi , v−i )) − pi (bi , v−i )]
Motivation
• Make a social choice that (approximately) maximizes the social welfare
subject to the economic constraints of
truthfulness
Problem Formulation
Multi Unit Combinatorial Auctions
• When optimizing the social welfare is
NP-hard and the standard solution, the
VCG mechanism, cannot be computed
efficiently find truthful approximation
algorithms
• A set [m] of items
• A number B representing how many
copies we have of each item
• A set [n] of players
• Player i has valuation function vi :
2[m] → ℜ+ , vi (∅) = 0, vi (A) ≤ vi (B)
for A ⊆ B (monotone)
Mechanism Design Basics
• A set of possible outcomes A and a set
of bidders N .
• Feasible solution: allocation (S1 , . . . , Sn )
such that each item is allocated to at
most B players (these sets must be disjoint for B = 1)
• For each i ∈ N a private function vi :
A → R+
0
• A Mechanism elicits bid bi from each i ∈
N and outputs the set f (b1 , . . . , bn ) ∈
A and and charges player I ∈ N :
pi (b1 , . . . , bn )
• i’s value for (S1 , . . . , Sn ) is vi (Si )
• Goal:
Maximize
P
v
(S
)
i
i i
Social
Welfare:
• We relax f and p to be random variables
over the possible outcomes and the Rn Lotteries and Value Oracles
payment vectors
• Value oracle takes set and returns value
• The expected utility of a player i ∈
• Analogous oracle for lotteries
N is given by E[vi (f (bi , . . . , bn )) −
• For x ∈ [0, 1]m , Dx define
pi (b1 , . . . , bn )]
X
Y Y
Fv (x) = ES∼Dx [v(S)] =
v(S)
xj
(1 − xj )
• A mechanism is truthful in expectaS
j∈S
j
∈S
/
tion if for every bi 6= vi and v−i ∈
1
• That is the value of the lottery can be
computed as the expected value of the
outcome w.r.t. the distribution of the
lottery
following
max
X
vi (S)xi,S
i,S6=∅
subject to
A general technique for constructing truthful approximation
algorithms
X
xi,S ≤ 1 for each player i
S6=∅
X X
i
xi,S ≤ 1 for each item j
S:j∈S
xi,S ≥ 0 for each i, S
Deterministic Support Mechanism
• Any randomized mechanism can be
viewed as a deterministic mechanism
with a different outcome space: the set
of all lotteries over the original outcomes,
the deterministic support mechanism
Theorem 0.1 Given any α-approximation
algorithm for a problem with the ”packing
property” that proves also an integrality gap
of at most α there is a randomized truthful in
expectation α-approximation algorithm
• Given any truthful in expectation mechThe mechanism is defined as follows:
anism the corresponding deterministic
support mechanism that charges the ex1. Compute the fractional optimal solution
pected payment vector is truthful
x∗
• The reverse is also true. A similar rule
Clark-Groves could be used to ensure individual rationality in expectation.
2. Scale the solution by α: x′ = x∗ /α
3. Express x′ as a convex combination of
integer solutions where there are polynomially many non-zero weights
• These transformations preserve the social welfare at any input
4. Convert the deterministic support allocation into its corresponding randomized
version.
The new framework
• The packing property of a problem is Claim: This algorithm returns a truthful in
that its corresponding natural LP sat- expectation randomized mechanism
isfies the following fact: if vector x is a
solution to the problem then any 0 ≤ Proof:
x′ ≤ x is also a solution
• Consider the mechanism that did not
scale by α and could probably express
• The mutli-unit combinatorial auction
the best fractional solution as a convex
problem has the packing property:
combination of exponentially many integral ones
• The LP formulation we will use is the
2
• This coincides with the VCG mechanism
on lotteries therefore its randomization is
also truthful
• If the optimal solution to the primal is
1 then the variables λ correspond to the
weights of the convex combination
• The randomization obtained in the case
where we scale by α has simply scaled
the utility of each agent at every input,
hence, the truthfulness equation is satisfied
• Primal P has exponentially many variables
• It has polynomially many constrains;
Since x∗ is an extreme point of the original problem, it has at most m + n non
zero variables, i.e., |E| ≤ m + n.
2 Claim: We can express the
scaled allocation as a convex combination of
polynomially many integral solutions
• Equivalently, dual D has exponentially many constraints but polynomially
many variables
Proof:
• We will show how we can solve the Dual
restricting the feasible space only to the
optimal solutions since then we have a
separation oracle
• Let I be the set of integral solutions
• Let x∗ be the fractional optimal solution
• Let E = {(i, S) | x∗i,S > 0}
+
= max(wi,S , 0). Given any
Claim: Let wi,S
integral x to the original problem one can find
• we define the following primal P and its integral solution xl such that
dual D
X
X
+
xli,S wi,S =
x̂i,S wi,S
(i,S)∈E
(i,S)∈E
Primal
min
s.t.
P
P
l
P
l∈I
l∈I
λl
Proof:
λl xli,S = x∗i,S /α for all (i, S) ∈ E
1. if wi,S > 0 set xli,S = xi,S
λl ≥ 1
2. Otherwise set xli,S = 0.
λl ≥ 0 for all l ∈ I
Dual
max α1
s.t.
P
P
(i,S)∈E
(i,S)∈E
2
Claim: For any weight vector w we can compute in polynomial time an integral solution
such that
X
X
1
max
xi,S wi,S
xli,S wi,S ≥
α feasible x
x∗i,S wi,S + z
xli,S wi,S + z ≤ 1 for all l ∈ I
(i,S)∈E
z≥0
Proof:
λl ≥ 0 for all l ∈ I
3
(i,S)∈E
• If some components of w are negative
take w+
• First prove that the optimum of the dual
is 1: Set z = 1 and wi,S = 0 for all
(i, S) ∈ E.
• We can monotonize
′
wi,S
=
max
T ⊆S:(i,T )∈E
• Assume we can find something better using solution x∗ : Then by the previous
claim we can find an integer solution that
is α approximate ofx∗ which implies the
that corresponding constrain is violated
+
wi,T
• Apply approximation algorithm with
valuations w′ and get solution y
• Using this argument we add the inequality
1 X ∗
·
xi,S wi,S + z ≥ 1
α
′
• By construction wi,S
≥ wi,S for all
(i, S) ∈ E.
• Note that we don’t care about as singing
weigths for (i, S) ∈
/ E such the respective
wi,S are zero.
(i,S)∈E
• Run Ellipsoid method on D to identify a
dual program with polynomial size set of
inequalities that is equivalent to D (the
violated inequalities that are returned by
the separation oracle that are used to cut
the ellipsoid)
• We know that
X
′
yi,S wi,S
(i,S)∈E
is greater than α fraction of the objective
• The primal of this program has polynomial number of constraints and variables.
• We need to find another y such that the
same holds with weights w+
• If we put some weight to some (i, S) such
+
′
that wi,S
6= wi,S
, i.e., we increased this
value due to monotonicity constraints we
put all of this weight then we simply
put weight 1 to (i, T ) where T ⊆ S and
+
′
wi,T
= wi,S
• Therefore non zero variables to the integer solution are at most the number
of variables plus the number of the constrains.
• The separation oracle used is the following: If
• Use the previous claim to conclude that
there is an integral solution xl such that
is greater than α fraction of the objective
when we use values w
1 X ∗
xi,S wi,S + z > 1
α
(i,S)∈E
then we identify the violated constrain
according to the previous Claim or otherwise we use the half space that is defined
by this constraint to cut the ellipsoid
2
Claim: We can find a convex combination of
x∗
with polynomially many non zero weights
α
in polynomial time
Proof:
2
4
2
Applications of this technique
Convex Rounding and MIDR
• MIDR: Fix some algorithm. Let distribution Dw be outcome for objective
function w. Let D be the set of all possible Dw . The algorithm is MIDR if
∀w, D ∈ D, Ex∼Dw [w(x)] ≥ Ex∼D [w(x)]
These technique can be used on existing approximation algorithms for the MUCA problem and derive the following results:
• For short valuation (each player is inter1
ested in one set and its subsets) O(m B+!
for any B ≥ 1 and (1 + ǫ) for B =
Ω(log m).
• We can convert any MIDR algorithm to
a truthful mechanism, with the same approximation guarantee, using VCG payments
• For general valuation functions the same
bound but with with ex post Nash equilibrium as a solution concept
• Observation: Instead of solving the relaxation and then rounding, why not optimize over the outcome of the rounding
scheme?
• For the special case of multi unit auctions this technique can be proved to be
2 approximate
maximize E[w(r(x))]
subject to x ∈ R
Convex Rounding Framework
(2)
• We don’t know if it is tractable
Relaxations
Lemma 0.2 Program 2 is MIDR
• Π an optimization problem, ∀(S, w) ∈ Π
maximize w(x)
subject to x ∈ S
(1)
• For most typical roundings, 2 is hard to
solve
• e.g., if r(x) = x for x ∈ S, then it is more
general than 1
• Π′ relaxation: ∀(S, w) ∈ Π defines convex and compact relaxed feasible set R ∈
ℜm and an extension wR : R → ℜ of the
objective
• So we probably should have the unusual
property that r(x) 6= x for x ∈ S
• So we have the following
• Rounding scheme r is α-approximate if
w(x) ≥ E[w(r(x))] ≥ αw(x), ∀x ∈ S
maximize wR (x)
subject to x ∈ R
Lemma 0.3 If r is α-approximate, then
2 is an α-approximation to the original
problem 1
• A rounding scheme r : R → S (possibly
randomized)
• Call r convex if E[w(r(x))] concave
• If ∀x ∈ R, E[w(r(x))] ≥ αwR (x), then
this is an α-approximation
• For r convex 2 can be solve efficiently
5
• So the problem is to find α-approximate
convex rounding
Poisson Rounding
Lemma 0.6 Consider f : 2V → ℜ
monotone, submodular, and normalizes
(f (∅) = 0). Consider set S ⊆ V and
random set S ′ by choosing each element
of S independently with prob p. Then
E[f (S ′ )] ≥ p · f (S).
Theorem 0.4 If ∃ polynomial, αapproximate, convex r for Π′ , then
∃ truthful-in-expectation, polynomial,
α-approximate mechanism for Π
Combinatorial Auctions
Proof:
Matroid Rank Sum
– Fix an ordering on elements of S
• Set function v : 2[m] → ℜ is an MRS
function if ∃u1 . . . , uk (all matroid rank
+
functions), and weights
Pk w 1 , . . . , w k ∈ ℜ
such that v(S) = ℓ=1 wℓ uℓ (S)
– Let Si be the first i elements of S
(similarly for Si′ )
P
– f (S) =
1≤i≤|S| f (Si ) − f (Si−1 ),
where f (S0 ) = 0
• Includes weighted coverage functions,
matroid weighted rank functions, and all
convex combinations of them
E[f (S ′ )] = E[
X
′
f (Si′ ) − f (Si−1
)]
1≤i≤|S ′ |
• Negative result: no universally truthful, polynomial, VCG-based mechanism
achieves constant factor assuming N P *
P |P oly
≥
X
p · (f (Si ) − f (Si−1 ))
1≤i≤|S|
= p · f (S)
2
Results
• Now we define the Poisson Rounding
Theorem 0.5 ∃(1−1/e)-approximate mechanism for combinatorial auctions with MRS
valuations
• Given fractional solution x, independently for each item assign j to i with
prob 1 − e−xij
• Formulation
P
• Note 1 − e−xij ≤ xij
maximize w(x) = i vi ({j : xij = 1})
P
subject to
∀j
i xij ≤ 1
Lemma 0.7 Poisson rounding is (1 −
xij ∈ {0, 1}
∀i, j 1/e)-approximate for submodular valuations
• R is relaxation to 0 ≤ xij ≤ 1
Proof:
• For x ∈ [0, 1]m , Dx define the extension
of the value function
– Rounding applied to integer soluX
Y Y
Fv (x) = ES∼Dx [v(S)] =
v(S)
xj
(1 − xj ) tion cancels each allocated item
S
j∈S
j ∈S
/
with prob 1/e
6
– Consider
(S1 , . . . , Sn )
integer
and
corresponding
(random)
′
′
(S1 , . . . , Sn )
– Si′ includes any j ∈ Si independently with prob 1 − 1/e
– Then E[vi (Si′ )] ≥ (1 − 1/e)vi (Si )
2
Lemma 0.8 Poisson rounding is concave for
MRS valuations
Proof:
• Let (S1 , . . . , Sn ) = rpoiss (x)
• Want
P to prove E[w(rposs (x))]
E[ vi (Si )] concave
=
• Show E[vi (Si )] concave
• We prove concavity for a subclass: Coverage functions
Definition 0.1 A function f : 2V → ℜ is a
coverage function if ∃ a set A (different from
V ) of “activities”, a value vi for each activity
i ∈ A, and
P a set Xj ⊆ A for eachSj, such that
v(S) = i∈c(S) vi , where c(S) = j∈S vj .
Proof:
X
E[v(S)] = E[
vi ]
i∈c(S)
=
X
vi P r[i ∈ c(S)]
i∈A
For each i ∈ A, define Yi = {j ∈ S|i ∈ Xj }.
We have
P r[i ∈ c(S)] = P r[S ∩ Xi 6= ∅]
= 1 − e−
P
j∈Yi
xj
2
2
7