An Improved Approximation Algorithm for Combinatorial Auctions with Submodular Bidders

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Combinatorial Auctions

A set M={1,…,m} of items for sale. n bidders, each bidder i has a valuation function vi:2M->R+.

Common assumptions: Normalization: vi()=0 Free disposal: ST vi(T) ≥ vi(S)

Goal: find a partition S1,…,Sn such that social welfare vi(Si) is maximized

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Combinatorial Auctions Problem 1: finding an optimal allocation is

NP-hard. Therefore, we are interested in the possible approximation ratios.

Problem 2: the valuations’ length is exponential in m, while we wish our algorithms to be polynomial in m and n.

Problem 3: how can we be certain that the bidders do not lie?

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Access Models Common types of queries:

Value: given a bundle S, return v(S).

Demand: given a vector of prices (p1,…, pm) return the bundle S that maximizes v(S)-jSpj. (demand queries are strictly more powerful than value queries Blumrosen-Nisan, Dobzinski-Schapira ).

General: any possible type of query (the communication model).

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The Hierarchy of CF Valuations

Complement-Free: v(ST) ≤ v(S) + v(T). XOS Submodular: v(ST) + v(ST) ≤ v(S) + v(T).

Semantic Characterization: Decreasing Marginal Utilities. 2-approximation (Lehmann-Lehmann-Nisan). Recent result: an e/(e-1)-approximation (Dobzinski-

Schapira).

GS: (Gross) Substitutes: Solvable in polynomial time.

OXS GS SM XOS CF

Lehmann, Lehmann, Nisan

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Part I: Approximations Using Demand Queries An e/(e-1)-approximation for XOS

Also holds for submodular valuations.The previously known upper bound is 2

(Lehmann-Lehmann-Nisan, Dobzinski-Nisan-Schapira)

An e/(e-1) communication lower bound for XOS

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XOS

The maximum over additive valuations:

(a:1 b:2 c:3) (a:2)

v({a}) = 2

v({a,b}) = 3

v({a,b,c}) = 6

Examples:

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Intuition for the XOS algorithm

We exploit the syntax of the XOS class. We can regard the value each bidder

assigns a bundle as a sum of the values he assigns the items in that bundle.

We will analyze the expected contribution of each item separately.

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The XOS Algorithm – Step 1

Solve the linear relaxation of the problem:

Maximize: i,Sxi,Svi(S)

Subject To: For each item j: i,S|jSxi,S ≤ 1

For each bidder i: Sxi,S ≤ 1

For each i,S: xi,S ≥ 0

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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-Sxi,S. The expected value of vi(Si) is Sxi,Svi(S)

Bidder i got the bundle Si = (x1:p1

i … xm:pmi)

Give item j to bidder i such that pjj ≥ pj

i’ for all i’.

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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)(i,sxi,s|S|).

We will prove that each item is allocated with probability ≥ (1- (1-1/n)n)i,S:j Sxi,s.

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The XOS Algorithm Proof

Pr [item j is not allocated] ≤ n

i=1(1-jSxi,S) = ((ni=1(1-jSxi,S))1\n)n

Due to the arithmetic/geometric mean inequality:≤ ((n

i=1(1-jSxi,S))\n)n = (1-(i,jSxi,s)/n)n

Pr [item j is allocated] ≥ 1-(1-(i,jSxi,s)/n)n

≥ (1-(1-1/n)n)i,S:jSxi,s

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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.

A B C

D E F

v1: (A:1 D:1) (D:1 E:1 F:1)

v2: (B:1 C:1) (C:1 F:1)

MCG Instance Auction with n XOS bidders

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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 AiAj =

Theorem: Requires t/n4 bits of communication (Alon-Matias-Szegedy)

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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 = {Cs

i|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).

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Part II: Approximations Using Value Queries An O(m1/4- lower bound for XOS

An m1/2approximation algorithmfor CF is known (Dobzinski-Nisan-Schapira).

(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

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An O(m1/4- lower bound for XOS Setting: m items, m½ XOS bidders. Choose, uniformly at random, a partition T1,…,Tn, where |Ti|=m½. Valuations:

vi = (jT j:m-½) |S|=2m^(¼+ (jS j:m-¼) |S|=m^(¾ (jS 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 (jT j:m-½). Corollary: the best allocation has value of 2m¼+. Proof (of lemma):

The average intersection between a random bundle and T i is m¼. By the chernoff bounds, the chance of finding a bundle whose intersection with

Ti is greater than the average by is exponentially small in . By the union bound it requires an exponential number of value queries to find

such a bundle.

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A (2-1/n)-Approximation

An equivalent definition for submodular valuations (“decreasing marginal utilities”): Marginal utility of j given S: v(j|S):=v(S{j}) - v(S) TSM: v(j|S) ≤ v(j|T)

Fact: the marginal valuation of a submodular valuation is also submodular.

The greedy algorithm provides a 2-approximation (Lehmann-Lehmann-Nisan)

We use randomization to improve the approximation ratio.

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The Algorithm

For each item j=1..m For 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 / ktk.

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.

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Proof Sketch

v1(a)=1, v1(b)=1, v1(c)=1

v1(S)=min(2, jSv1(j))

v2(a)=0, v2(b)=1, v2(c)=0

v2(S)=min(1, jSv2(j))

Let OPTj denote the value of the optimal solution without the first (j-1) items.

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Proof Sketch

Let OPTj denote the value of the optimal solution without the first (j-1) items.With the submodular valuations v1(·|S1),

…,vn(·|Sn).

v1(a)=1, v1(b)=1, v1(c)=1

v1(S)=min(2, jSv1(j))

v2(a)=0, v2(b)=1, v2(c)=0

v2(S)=min(1, jSv2(j))

a

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Proof Sketch

Let Pj denote the random variable which indicates the “price” we got for item j. i.e. the contribution of item j to the total social welfare.

Observe the E[ALG] = jE[Pj]. Let OPTi

j 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

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.

v1(b|a)=1, v1(c|a)=1v1(S|a)=min(1, SjSv1(j|a))

v2(a)=0, v2(b)=1, v2(c)=0

v2(S)=min(1, jSv2(j))

a

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Proof Sketch

Lemma: E[Lj] / E[Pj] ≤ 1.5 Proof:

Notation: vi := v(j|Si). E[Pj] = (v1*(v1 / v1+v2)) + v2*(v1 / v1+v2)))

= (v12 + v2

2) / (v1+v2)

v1(b|a)=1, v1(c|a)=1v1(S|a)=min(1, SjSv1(j|a))

v2(a)=0, v2(b)=1, v2(c)=0

v2(S)=min(1, jSv2(j))

a

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Proof Sketch

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)

v1(b|a)=1, v1(c|a)=1v1(S|a)=min(1, SjSv1(j|a))

v2(a)=0, v2(b)=1, v2(c)=0

v2(S)=min(1, jSv2(j))

a

b

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Proof Sketch

Suppose we assign item j to bidder 1: Bidder 1 loses at most v1 in OPT1

j

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))) = (v1

2+v22+v1*v2) / v1+v2)

v1(b|a)=1, v1(c|a)=1v1(S|a)=min(1, SjSv1(j|a))

v2(a)=0, v2(b)=1, v2(c)=0

v2(S)=min(1, jSv2(j))

a

b

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Proof Sketch

We have:E[Lj] ≤ (v1

2+v22+v1*v2) / v1+v2)

E[Pj] = (v12 + v2

2) / (v1+v2)

E[Lj] / E[Pj] ≤ (v12+v2

2+v1*v2) / (v12+v2

2)

≤ 1+v1*v2 / (v12+v2

2) ≤ 1.5

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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.

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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 vS 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.

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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.)

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Summary

Demand Queries: e/(e-1) upper bound for XOS valuations

Also holds for submodular valuations e/(e-1) lower bound for XOS valuations

Holds for any type of queries

Value Queries: An O(m1/4-) lower bound for approximating CF

valuations using value queries only. 2-1/n approximation for submodular valuations.

e/(e-1) lower bound is known (Khot-Lipton-Markakis-Mehta).

Reminder: OXS GS SM XOS CF

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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- lower bound. Incentive compatible auctions with better approximation

ratios.