Approximation Algorithms for Combinatorial Auctions with ComplementFree Bidders

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Approximation Algorithms forCombinatorial
Auctions with Complement-Free Bidders

• Speaker Shahar Dobzinski
• Joint work with Noam Nisan Michael Schapira

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Talk Structure

• ?Combinatorial Auctions
• Log(m)-approximation for CF auctions
• An incentive compatible O(m1/2)-approximation CF
  auctions
• A lower bound of 2-e for CF auctions
• 2-approximation for XOS auctions
• A lower bound of e/(e-1)-e for XOS auctions

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Combinatorial Auctions

• m items for sale.
• n bidders, each bidder i has a valuation function
  vi2M-gtR.
• Common assumptions
• Normalization vi(?)0
• Free disposal S?T ? vi(T) vi(S)
• Goal find a partition S1,,Sn such that social
  welfare Svi(Si) is maximized.
• Problem 1 finding an optimal allocation is
  NP-hard.
• Problem 2 valuation length is exponential in n
  and m.

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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.
• Demand queries are more powerful than value
  queries (Blumrosen-Nisan)

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Known Results

• Finding an exact solution requires exponential
  communication. Nisan-Segal
• Finding an O(m1/2-e)-approximation requires
  exponential communication. Nisan-Segal.
• This result holds for every possible type of
  oracle.
• Using demand oracles, a matching upper bound of
  O(m1/2) exists (Blumrosen-Nisan).
• Better results might be obtained by using
  restricted classes of valuations.

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The Hierarchy of CF Valuations
Lehmann, Lehmann, Nisan
OXS ? GS ? SM ? XOS ? CF

• Complement-Free v(S?T) v(S) v(T).
• XOS XOR of ORs of singletons
• Example (A2 OR B2) XOR (A3)
• Submodular v(S?T) v(S??T) v(S) v(T).
• 2-approximation by LLN.
• GS (Gross) Substitutes
• Solvable in polynomial time
• OXS OR of XORs of singletons

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Talk Structure

• Combinatorial Auctions
• ?Log(m)-approximation for CF auctions
• An incentive compatible O(m1/2)-approximation CF
  auctions
• A lower bound of 2-e for CF auctions
• 2-approximation for XOS auctions
• A lower bound of e/(e-1)-e for XOS auctions

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Intuition

• We will allow the auctioneer to allocate k
  duplicates from each item.
• Each bidder is still interested in at most one
  copy of each item (so valuations are kept the
  same).
• Using the assumption that all valuations are CF,
  we will find an approximate solution to the
  original auction, based on the k-duplicates
  allocation.

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The Algorithm Step 1

• Solve the linear relaxation of the problem
• Maximize Si,Sxi,Svi(S)
• Subject To
• For each item j Si,Sj?Sxi,S 1
• For each bidder i SSxi,S 1
• For each i,S xi,S 0
• Despite the exponential number of variables, the
  LP relaxation may still be solved in polynomial
  time using demand oracles.(Nisan-Segal).
• OPTSi,Sxi,Svi(S) is an upper bound for the
  value of the optimal integral allocation.

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The Algorithm Step 2

• Use randomized rounding to build a
  pre-allocation S1,..,Sn
• Each item j appears at most kO(log(m)) times in
  Sii.
• Sivi(Si) OPT/2.
• 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)
• We use the Chernoff bound to show that such
  pre-allocation is built with high probability.

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The Algorithm Step 3

• For each bidder i, partition Si into a disjoint
  union Si Si1?.. ?Sik such that for each1ilti
  n, 1tt k, Sit?Sit?.

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The Algorithm Step 3

• For each bidder i, partition Si into a disjoint
  union Si Si1?.. ?Sik such that for each1ilti
  n, 1tt k, Sit?Sit?.

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The Algorithm Step 3

• For each bidder i, partition Si into a disjoint
  union Si Si1?.. ?Sik such that for each1ilti
  n, 1tt k, Sit?Sit?.

S11 A,B,D
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The Algorithm Step 3

• For each bidder i, partition Si into a disjoint
  union Si Si1?.. ?Sik such that for each1ilti
  n, 1tt k, Sit?Sit?.

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The Algorithm Step 3

• For each bidder i, partition Si into a disjoint
  union Si Si1?.. ?Sik such that for each1ilti
  n, 1tt k, Sit?Sit?.

S22 A,D
S21 C,E
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The Algorithm Step 3

• For each bidder i, partition Si into a disjoint
  union Si Si1?.. ?Sik such that for each1ilti
  n, 1tt k, Sit?Sit?.

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The Algorithm Step 3

• For each bidder i, partition Si into a disjoint
  union Si Si1?.. ?Sik such that for each1ilti
  n, 1tt k, Sit?Sit?.

S32 C,E
S33 A
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The Algorithm Step 3

• For each bidder i, partition Si into a disjoint
  union Si Si1?.. ?Sik such that for each1ilti
  n, 1tt k, Sit?Sit?.

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The Algorithm Step 3

• For each bidder i, partition Si into a disjoint
  union Si Si1?.. ?Sik such that for each1ilti
  n, 1tt k, Sit?Sit?.

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The Algorithm Step 4

• Find the t maximizes Sivi(Sit)
• Return the allocation (S1t,...,Snt).
• All valuations are CF so
• StSivi(Sit) SiStvi(Sit) Sivi(Si) OPT/2
• ? For the t that maximizes Sivi(Sit), it holds
  that Sivi(Sit) OPT/2k OPT/O(log(m)).

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Talk Structure

• Combinatorial Auctions
• Log(m)-approximation for CF auctions
• ?An incentive compatible O(m1/2)-approximation CF
  auctions
• A lower bound of 2-e for CF auctions
• 2-approximation for XOS auctions
• A lower bound of e/(e-1)-e for XOS auctions

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Incentive Compatibility VCG Prices

• VCG is the only general technique known for
  making auctions incentive compatible (if bidders
  are not single-minded)
• Each bidder i pays Sk?ivi(Oi) - Sk?ivi(O-i)
• Oi is the optimal allocation, O-i the optimal
  allocation of the auction without the ith
  bidder.
• VCG requires an optimal allocation.
• Finding an optimal allocation requires
  exponential communication and is computationally
  intractable.
• Approximations do not suffice (Nisan-Ronen,
  Lehmann-OCallaghan-Shoham).

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VCG on a Subset of the Range

• Our solution limit the set of possible
  allocations.
• We will let each bidder to get at most one item,
  or well allocate all items to a single bidder.
• Optimal solution in the set can be found in
  polynomial time ? VCG prices can be computed ?
  incentive compatibility.
• We still need to prove that we achieve an
  approximation.

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The Algorithm

• Ask each bidder i for vi(M), and for vi(j), for
  each item j.
• (We have used only value queries)
• Construct a bipartite graph and find the maximum
  weighted matching P.
• can be done in polynomial time (Tarjan).

Bidders
Items
1
v1(A)
A
2
B
3
v3(B)
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The Algorithm (Cont.)

• Let i be the bidder that maximizes vi(M).
• If vi(M)gtP
• Allocate all items to i.
• else
• Allocate according to P.
• Let each bidder pay his VCG price (in respect to
  the restricted set).

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Proof of the Approximation Ratio

• Theorem If all valuations are CF, the algorithm
  provides an O(m1/2)-approximation.
• Proof Let OPT(T1,..,Tk,Q1,...,Ql), where for
  each Ti, Tigtm1/2, and for each Qi, Qim1/2.
  OPT Sivi(Ti) Sivi(Qi)

Case 1 Sivi(Ti) gt Sivi(Qi) (large bundles
contribute most of the social welfare) ? Sivi(Qi)
gt OPT/2 At most m1/2 bidders get at least m1/2
items in OPT. ? For the bidder i the bidder i
that maximizes vi(M), vi(M) gt OPT/2m1/2.
Case 2 Sivi(Qi) Sivi(Ti) (small bundles
contribute most of the social welfare) ? Sivi(Qi)
OPT/2 For each bidder i, there is an item
ci, such that vi(ci) gt vi(Qi) / m1/2. (The CF
property ensures that the sum of the values is
larger than the value of the whole bundle) cii
is an allocation with at most one item to each
bidder P Sici OPT/2m1/2.
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Talk Structure

• Combinatorial Auctions
• Log(m)-approximation for CF auctions
• An incentive compatible O(m1/2)-approximation CF
  auctions
• ?A lower bound of 2-e for CF auctions
• 2-approximation for XOS auctions
• A lower bound of e/(e-1)-e for XOS auctions

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A Lower Bound of 2-e

• Nisan Segal exhibit a set of 0/1 valuations,
  such that distinguishing between the following
  cases requires exponential communication
• There is an allocation in which vi(Si)1, for all
  i.
• In all allocations, at most one bidder gets a
  value of 1.

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A Lower Bound of 2-e (Cont.)

• Add 1 to all valuations in Nisan-Segal Auction
• vi(S)vi(S)1.
• The valuations are now CF.
• It requires exponential communication to
  distinguish between the following cases
• There is an allocation in which vi(Si)2, for
  all i.
• In all allocations, at most one bidder gets a
  value of 2. The rest of the bidders get a value
  of at most 1.
• ? Any approximation better than 2n/(n1) requires
  exponential communication.

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Talk Structure

• Combinatorial Auctions
• Log(m)-approximation for CF auctions
• An incentive compatible O(m1/2)-approximation CF
  auction
• A lower bound of 2-e for CF auctions
• ?2-approximation for XOS auctions
• A lower bound of e/(e-1)-e for XOS auctions

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Definition of XOS

• XOS XOR of ORs of Singletons.
• Singleton valuation (xp)
• v(S) p x?S
• 0 otherwise
• Example (A2 OR B2) XOR (A3)
• This is a XOR of additive valuations.

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XOS Properties

• The strongest bidding language syntactically
  restricted to represent only complement-free
  valuations.
• Can describe all submodular valuations (and also
  some non-submodular valuations)

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Supporting Prices

• p1,,pm supports the bundle S if
• v(S) Sj?Spj
• v(T) Sj?Tpj for all T ? S
• Claim a valuation is XOS iff every bundle S has
  supporting prices.
• Proof
• ? There is a clause that maximizes the value of a
  bundle S. The prices in this clause are the
  supporting prices.
• ? Take the prices of each bundle, and build a
  clause.

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The Algorithm

• Input n bidders, each represented a demand
  oracle and a supporting price oracle.
• Init p1pm0.
• For each bidder i1..n
• Let Si be the demand of the ith bidder at prices
  p1,,pm.
• For all i lt i take away from Si any items from
  Si.
• Let q1,,qm be the supporting prices for Si in
  vi.
• For all j ? Si update pj qj.

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Example

• Items A, B, C, D, E. 3 bidders.
• Price vector p0(0,0,0,0,0) v1 (A1 OR B1 OR
  C1) XOR (C2)Bidder 1 gets his demand A,B,C.
• Price vector p1(1,1,1,0,0) v2 (A1 OR B1 OR
  C9) XOR (D2 OR E2)Bidder 2 gets his demand
  C
• Price vector p2(1,1,9,0,0) v3 (C10 OR D1 OR
  E2)Bidder 3 gets his demand C,D,E
• Final allocation A,B to bidder 1, C,D,E to
  bidder 3.

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Proof

• For proving the approximation ratio, we will need
  these two simple lemmas
• Lemma The total social welfare generated by the
  algorithm is at least Spi.
• Lemma The optimal social welfare is at most 2Spi.

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Proof Lemma 1

• Lemma The total social welfare generated by the
  algorithm is at least Spi.
• Proof
• Each bidder i got a bundle Ti at stage i.
• At the end of the algorithm, he holds Ai ? Ti.
• The supporting prices guarantee that vi(Ai)
  Sj?Aipj

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Proof Lemma 2

• Lemma The optimal social welfare is at most
  2Spi.
• Proof
• Let O1,...,On be the optimal allocation. Let pi,j
  be the price of the jth item at the ith stage.
• Each bidder i ask for the bundle that maximizes
  his demand at the ith stage
• vi(Oi)-Sj?Oi pi,j Sj pi,j Sj p(i-1),j
• Since the prices are non-decreasing
• vi (Oi )-Sj?Oi pn,j Sj pi,j Sj p(i-1),j
• Summing up on both sides
• Si vi(Oi )-SiSj?Oi pn,j Si (Sj pi,j Sj
  p(i-1),j)
• Si vi(Oi )-Sj pn,j Sj pn,j
• Si vi(Oi ) 2Sj pn,j

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Talk Structure

• Combinatorial Auctions
• Log(m)-approximation for CF auctions
• An incentive compatible O(m1/2)-approximation CF
  auctions
• A lower bound of 2-e for CF auctions
• 2-approximation for XOS auctions
• ? A lower bound of e/(e-1)-e for XOS auctions

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k-Set Cover

• We will show a polynomial time reduction from
  k-Set-Cover.
• k-Set-Cover definition
• Input a set of Mm items, t subsets Si ? M, an
  integer k.
• Goal Find k subsets such that the number of item
  in their union, ?1ikSi, is maximized.
• Theorem approximating k-Set-Cover within a
  better factor than e/(e-1) is NP-hard (feige).

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The Reduction
k-Set-Cover Instance
XOS Auction with k bidders
v1 (A1 OR D1) XOR (A1 OR D1) XOR (D1 OR
E1 OR F1)
A
B
C
D
E
F
vk (A1 OR D1) XOR (A1 OR D1) XOR (D1 OR
E1 OR F1)

• Every solution to k-Set-Cover implies an
  allocation with the same value.
• Every allocation implies a solution to
  k-Set-Cover with at least that value.
• ? Same approximation lower bound.
• A communication lower bound can also be proven by
  reducing from the communication of this version
  (Nisan).