An Improved
Approximation Algorithm
for Combinatorial Auctions
with Submodular Bidders
Speaker: Shahar Dobzinski
Joint work with Michael Schapira
Combinatorial Auctions


m items for sale.
n bidders, each bidder i has a valuation function
vi:2M->R+.
Common assumptions:
 Normalization: vi()=0
 Free disposal: ST  vi(T) ≥ vi(S)

Goal: find a partition S1,…,Sn such that the total
social welfare Svi(Si) is maximized.

Problem 1: NP-hard.
Problem 2: valuation length is exponential in n
and m.

2
Access Models



One possibility: bidding languages.
In this talk: each bidder is represented by an
oracle which can answer only a specific type of
queries.
Common types of queries:
 Value:
given a bundle S, return v(S).
 Demand: given a vector of prices p, return the bundle
S that maximizes v(S) - Spi.
 General: any type of possible query.

Bidders are computationally unlimited
3
Known Results

Finding an exact solution requires
exponential communication. (Nisan-Segal)
 Holds

for every possible type of oracle.
Finding an O(m1/2-e)-approximation
requires exponential communication. (NisanSegal).

Using demand oracles, a matching upper
bound of O(m1/2) exists (Blumrosen-Nisan, DobzinskiSchapira).
4
The Hierarchy of CF Valuations
OXS  GS  SM  XOS  CF
(Lehmann, Lehmann, Nisan)



Complement-Free: v(S) + v(T) ≥ v(ST).
XOS
Submodular: v(S) + v(T) ≥ v(ST) + v(ST).
 Semantic
Characterization: Decreasing Marginal
Utilities

GS: (Gross) Substitutes
 Solvable
in polynomial time
5
Part I: Approximations Using
Demand Queries

An e/(e-1)-approximation for XOS
 Also
holds for submodular valuations.
 Previously known upper bounds are 2 (LehmannLehmann-Nisan, Dobzinski-Nisan-Schapira)

An e/(e-1) communication lower bound for
XOS
6
XOS

The maximum of additive valuations:
(a:1 b:2  c:3) (a:2)
Examples: v({a}) = 2
v({a,b}) = 3
v({a,b,c}) = 6
7
Intuition for the XOS algorithm
We exploit the syntax of the XOS class.
 We will “reduce” the XOS valuations to
additive valuations (using the randomized
rounding).
 We will analyze the expected contribution
of each item separately.

8
The XOS Algorithm – Step 1

Solve the linear relaxation of the problem:
Maximize: Si,Sxi,Svi(S)
Subject To:
 For each item j: Si,S|jSxi,S ≤ 1
 For each bidder i: SSxi,S ≤ 1
 For each i,S: xi,S ≥ 0
9
The XOS Algorithm – Steps 2-3

Randomized Rounding: For each bidder i,
let Si be the bundle S with probability xi,S,
and the empty set with probability 1-SSxi,S.
 The
expected value of vi(Si) is SSxi,Svi(S)
Bidder i got the bundle
Si = (x1:p1i … xm:pmi)
 Give item j to bidder i such that pjj ≥ pji’ for
all i’.

10
The XOS Algorithm


Theorem: The algorithm is an e/(e-1)approximation.
Proof: only for the special case where all prices
are equal.
 Example:


(x1:1  x2:1)  (x1:1)
We now only need to prove that the number of
items which are allocated ≥ (1-(1-1/n)n)Si,sxi,s.
We will prove that each item is allocated with
probability ≥ (1- (1-1/n)n)Si,Sxi,s.
11
The XOS Algorithm Proof
Pr [item j is not allocated] ≤
Pni=1(1-SjSxi,S) = ((Pni=1(1-SjSxi,S))1\n)n
 Due to the arithmetic/geometric mean
inequality:
≤ ((Sni=1(1-SjSxi,S))\n)n = (1-(Si,jSxi,s)/n)n
 Pr [item j is allocated] ≥ 1-(1-(Si,jSxi,s)/n)n
≥ 1-(1-1/n)n Si,jSxi,s

12
An e/(e-1) Lower Bound for XOS

Theorem: Any approximation better than e/(e-1) of a combinatorial
auctions with XOS bidders requires exponential communication.


Unconditional Lower bound
We will prove the lower bound for the MCG problem (Chekuri-Kumar):


We are given a set of M items, and n groups of subsets of the M items
The goal is to choose one subset from each group, such that their union
is maximized.
MCG Instance
A
D
B
E
Auction with n XOS bidders
C
F
v1: (A:1  D:1)  (D:1  E:1  F:1)
v2: (B:1  C:1)  (C:1  F:1)
13
Approximate Disjointness




n players, each holds a string of length t.
The string of player i specifies a subset
Ai  {1,…,t}.
The goal is to distinguish between the following two
extreme cases:

NO: iAi ≠ 

YES: for every i≠j AiAj = 
Theorem: Requires t/n4 bits of communication
(Alon-Matias-Szegedy)
14
The Reduction


Denote a partition C of M to n parts as {C1,…,Cn).
We build a set of partitions F=(C1,…,Cexp(m/n)), such that every n sets
from different parts cover at most
(1-(1-1/n)n)m elements.

Existence is proved using probabilistic construction.








Randomly build each partition: place each item in exactly one of the n sets.
Given n sets the probability that an item is covered is (1-(1-1/n)n)
The expectation is (1-(1-1/n)n)m
By the chernoff bounds the probability that we are far from the optimum is
exponentially small  we have an exponential number of sets.
Each player i who got Ai as input, constructs the collection Bi =
{Csi|Ai=1}.
If the intersection wasn’t empty, all the elements can be covered.
If the intersection was empty, the construction guarantees that no
more than (1-(1-1/n)n)m elements can be covered.
Corollary: exponential communication is required for any
approximation better than (1-(1-1/n)n).
15
Part II: Approximations Using Value
Queries

A O(m1/4-e) lower bound for XOS
O(m1/2-e)-approximation algorithm for CF is known
(Dobzinski-Nisan-Schapira).
 An

(2-1/n)- approximation for submodular
valuations.
 The
Previously known upper bound for submodular
valuations is 2 (Lehmann-Lehmann-Nisan)
 1+1/2m communication lower bound for submodular
valuations is known (Nisan-Segal)
 e/(e-1) lower bound – conditional in P≠NP
(Khot-Lipton-Markakis-Mehta)
Reminder: OXS  GS  SM  XOS  CF
16
1/4-e
O(m )
An
XOS







lower bound for
Setting: m items, m½ XOS bidders.
Choose, uniformly at random, a partition T1,…,Tn, where |Ti|=m½.
Valuations:
vi = (jT j:m-½) |S|=2m^(¼+e) (jS j:m-¼) |S|=m^(¾) (jS j:m-¼)
The optimal Allocation has value of m½ (according to the Ti’s).
Lemma: Exponential number of value queries is required to find a bundle R,
|R|<m¾, for which the maximizing clause is (jT j:m-½).
Corollary: the best allocation has value of 2m¼+e.
Proof (of lemma):
The average intersection between a Random bundle and Ti is m¼.
By the chernoff bounds, the chance for exceeding the average by e is
exponentially small in e.
 By the union bound it requires an exponential number of value queries to find
that bundle R.


17
A (2-1/n)-Approximation

An equivalent definition for submodular
valuations (“decreasing marginal utilities”):
 Marginal
 TSM:



utility of j given S: v(j|S):=v(S{j}) - v(S)
v(j|S) ≤ v(j|T)
Fact: the marginal valuation of a submodular
valuation is also submodular.
The greedy algorithm provides a 2approximation (Lehmann-Lehmann-Nisan)
We use randomization to improve the
approximation ratio.
18
The Algorithm

For each item j=1..m
each bidder i, let ti = vi(j|Si)n-1
 Assign to exactly one bidder the item j, where bidder i
is chosen with probability ti / Sktk.
 For

Theorem: the algorithm produces an allocation
which is in expectation a (2-1/n)-approximation
to the optimal total social welfare.
 We
will prove the theorem for n=2.
19
Proof Sketch

v1(a)=1, v1(b)=1, v1(c)=1
v2(a)=0, v2(b)=1, v2(c)=0
v1(S)=min(2, SjSv1(j))
v2(S)=min(1, SjSv2(j))
Let OPTj denote the value of the optimal
solution without the first (j-1) items.
20
Proof Sketch
a

v1(a)=1, v1(b)=1, v1(c)=1
v2(a)=0, v2(b)=1, v2(c)=0
v1(S)=min(2, SjSv1(j))
v2(S)=min(1, SjSv2(j))
Let OPTj denote the value of the optimal
solution without the first (j-1) items.
 With
the submodular valuations
v1(·|S1),…,vn(·|Sn).
21
Proof Sketch
a
v1(b|a)=1, v1(c|a)=1
v1(S|a)=min(1, SjSv1(j|a))



i.e. the contribution of item j to the total social welfare.
Observe the E[ALG] = SjE[Pj].
Let OPTij denote the optimal solution given that item j was assigned to bidder i.
Lj denotes the random variable that gets the value of OPTj – OPTj+1


v2(S)=min(1, SjSv2(j))
Let Pj denote the random variable which indicates the “price” we got for item j.


v2(a)=0, v2(b)=1, v2(c)=0
i.e. how much did we lose by assigning item j to bidder i?
We will prove that E[Lj] / E[Pj] ≤ 1.5, and the theorem will follow.
22
Proof Sketch
a
v1(b|a)=1, v1(c|a)=1
v1(S|a)=min(1, SjSv1(j|a))



v2(a)=0, v2(b)=1, v2(c)=0
v2(S)=min(1, SjSv2(j))
Lemma: E[Lj] / E[Pj] ≤ 1.5
Proof:
Notation: vi := v(j|Si).
E[Pj] = (v1*(v1 / (v1+v2)) + v2*(v1 / (v1+v2)))
= (v12 + v22) / (v1+v2)
23
Proof Sketch
b
a
v1(b|a)=1, v1(c|a)=1
v1(S|a)=min(1, SjSv1(j|a))



v2(a)=0, v2(b)=1, v2(c)=0
v2(S)=min(1, SjSv2(j))
WLOG bidder 2 gets item j in OPTj.
If we assign item j to bidder 2:
L=OPTj-OPT1j=v2
This happens with probability v2 / (v1+v2)
24
Proof Sketch
b
a
v1(b|a)=1, v1(c|a)=1
v1(S|a)=min(1, SjSv1(j|a))

Bidder 1 loses at most v1 in OPT1j




v2(S)=min(1, SjSv2(j))
Suppose we assign item j to bidder 1:


v2(a)=0, v2(b)=1, v2(c)=0
the marginal value of j given the bundle he gets in OPT1j is smaller than v1.
Bidder 2 loses at most v2 in OPT1j
 L ≤ v1+v2
This happens with probability v1 / (v1+v2)
E[Lj] ≤ (v2*(v2 / (v1+v2)) +(v1+v2) *(v1 / (v1+v2))) = (v12+v22+v1*v2) / (v1+v2)
25
Proof Sketch

We have:
≤ (v12+v22+v1*v2) / (v1+v2)
 E[Pj] = (v12 + v22) / (v1+v2)
 E[Lj]

E[Lj] / E[Pj] ≤ (v12+v22+v1*v2) / (v12+v22)
≤ 1+v1*v2 / (v12+v22) ≤ 1.5
26
Online Combinatorial Auctions
Items arrive one by one.
 Each item must be assigned as it arrives.
 The type of queries the algorithm is
allowed to ask is restricted.
 We suggest two natural restrictions.
 Our algorithm provides a 2-1/n upper
bound for both variants.

27
Variant I: Look Backwards

Before assigning item j the algorithm may only query the any bundle
S  {1,..j}.

Online Matching (Karp-Vazirani-Vazirani)

Bipartite graph. The goal is to find the maximum bipartite matching.
Vertices from side I arrive one by one, and the edges of a vertex are
revealed as the vertex arrive.
 Reduction: the set of vertices from side I is the set of items, and the set
of vertices from side II is the set of bidders. Vi(S)=1 if there exists some
vS such that the edge (v,i) exists. Otherwise Vi(S)=0.
 e/(e-1) randomized upper bound.


Other problems: Online b-Matching (Kalayanasundaram-Pruhs), Adwords
(Mehta-Saberi-Vazirani-Vazirani).
All have an e/(e-1) randomized upper bound.
28
Variant II: Look Ahead

Before assigning item j the algorithm may only query the
marginal value of item j given any bundle S  M.

Bounded-Delay buffer (Kesselman et al.)



Packets arrive one by one, each has a value and a deadline. We
can handle one packet at a time. The goal is to maximize the
sum of values of packets which have been transferred before
their deadline.
Reduction: let set of time slots be the set of items, each packet is
reduced to a bidder. Vi(S)=1 if S contains a time slot between the
arrival and the expiration of the corresponding packet.
Otherwise, Vi(S)=1.
e/(e-1) randomized upper bound (Bartal et al.)
29
Summary

Demand Queries:
 e/(e-1) upper bound for XOS valuations
 Also holds for submodular valuations
 e/(e-1) lower bound for XOS
 Holds for any type of queries

valuations
Value Queries:
O(m1/4-e) lower bound for approximating CF
valuations using value queries only.
 2-1/n approximation for submodular valuations.
 An

e/(e-1) lower bound is known (Khot-Lipton-Markakis-Mehta).
Reminder: OXS  GS  SM  XOS  CF
30
Open Questions




Is there an e/(e-1) upper bound for combinatorial auctions
with submodular valuations using value queries only?
 An upper bound of e/(e-1) is known for many special
cases.
 Online: online matching, bounded delay buffer, …
 Offline: budget additive valuations (Andelman-Mansour),
coverage valuations.
Is there a constant lower bound for approximation of
submodular valuations using demand oracles?
Close the gap between the O(log m)-approximation for CF
valuations and the 2-e lower bound.
Incentive compatible Auctions with better approximation
ratios.
31