An Improved Approximation Algorithm for Combinatorial Auctions with Submodular Bidders

1
An Improved Approximation Algorithm for
Combinatorial Auctions with Submodular Bidders
2
Combinatorial Auctions

• A set M1,,m of 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

3
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?

4
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)-Sj?Spj.
  (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
Lehmann, Lehmann, Nisan
OXS ? GS ? SM ? XOS ? CF

• Complement-Free v(S?T) v(S) v(T).
• XOS
• Submodular v(S?T) v(S??T) 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.

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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

(a1?? b2 ? c3)?? (a2)
v(a) 2
Examples
v(a,b) 3
v(a,b,c) 6
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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 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

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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-SSxi,S.
• The expected value of vi(Si) is SSxi,Svi(S)
• Bidder i got the bundle Si (x1p1i ??
  xmpmi)
• Give item j to bidder i such that pjj pji 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 (x11 ? x21) ? (x11)
• We now only need to prove that the number of
  items which are allocated (1-(1-1/n)n)(Si,sxi,s
  S).
• We will prove that each item is allocated with
  probability (1- (1-1/n)n)Si,Sj ?Sxi,s.

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The XOS Algorithm Proof

• Pr item j is not allocated Pni1(1-Sj?Sxi,S)
  ((Pni1(1-Sj?Sxi,S))1\n)n
• Due to the arithmetic/geometric mean
  inequality ((Sni1(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,Sj?Sxi,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.

MCG Instance
Auction with n XOS bidders
A
B
C
v1 (A1 ? D1) ? (D1 ? E1 ? F1)
v2 (B1 ? C1) ? (C1 ? F1)
D
E
F
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Approximate Disjointness

• n players, each holds a string of length t.
• The string of player i specifies a subsetAi ?
  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)

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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 CsiAi1.
• If the intersection wasnt 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-e) lower bound for XOS
• An m1/2-approximation algorithm for 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)
• 11/2m 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-e) lower bound for XOS

• Setting m items, m½ XOS bidders.
• Choose, uniformly at random, a partition T1,,Tn,
  where Tim½.
• Valuations
• vi (?j?T jm-½) ?S2m(¼e) (?j?S jm-¼)
  ?Sm(¾) (?j?S jm-¼)
• The optimal Allocation has value of m½ (according
  to the Tis).
• Lemma Exponential number of value queries is
  required to find a bundle R, Rltm¾, for which
  the maximizing clause is (?j?T jm-½).
• Corollary the best allocation has value of
  2m¼e.
• Proof (of lemma)
• The average intersection between a random bundle
  and Ti is m¼.
• By the chernoff bounds, the chance of finding a
  bundle whose intersection with Ti is greater
  than the average by e is exponentially small in
  e.
• 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(jS)v(S?j) -
  v(S)
• T?S?M v(jS) v(jT)
• 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 j1..m
• For each bidder i, let ti vi(jSi)n-1
• Assign to exactly one bidder the item j, where
  bidder i is chosen with probability ti / Sktk.
• 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 n2.

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Proof Sketch
v1(a)1, v1(b)1, v1(c)1 v1(S)min(2, Sj?Sv1(j))
v2(a)0, v2(b)1, v2(c)0 v2(S)min(1, Sj?Sv2(j))

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

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Proof Sketch
a
v1(a)1, v1(b)1, v1(c)1 v1(S)min(2, Sj?Sv1(j))
v2(a)0, v2(b)1, v2(c)0 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)
  .

v1(ba)1, v1(ca)1 v1(Sa)min(1, Sj?Sv1(ja))
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Proof Sketch
a
v1(ba)1, v1(ca)1 v1(Sa)min(1, Sj?Sv1(ja))
v2(a)0, v2(b)1, v2(c)0 v2(S)min(1, Sj?Sv2(j))

• 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 EALG SjEPj.
• 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 OPTj1
• i.e. how much did we lose by assigning item j to
  bidder i?
• We will prove that ELj / EPj 1.5, and the
  theorem will follow.

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Proof Sketch
a
v1(ba)1, v1(ca)1 v1(Sa)min(1, Sj?Sv1(ja))
v2(a)0, v2(b)1, v2(c)0 v2(S)min(1, Sj?Sv2(j))

• Lemma ELj / EPj 1.5
• Proof Notation vi v(jSi).
• EPj (v1(v1 / (v1v2)) v2(v1 / (v1v2)))
  (v12 v22) / (v1v2)

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Proof Sketch
b
a
v1(ba)1, v1(ca)1 v1(Sa)min(1, Sj?Sv1(ja))
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 LOPTj-OPT1jv2
• This happens with probability v2 / (v1v2)

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Proof Sketch
b
a
v1(ba)1, v1(ca)1 v1(Sa)min(1, Sj?Sv1(ja))
v2(a)0, v2(b)1, v2(c)0 v2(S)min(1, Sj?Sv2(j))

• Suppose we assign item j to bidder 1
• Bidder 1 loses at most v1 in OPT1j
• 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 v1v2
• This happens with probability v1 / (v1v2)
• ELj (v2(v2 / (v1v2)) (v1v2) (v1 /
  (v1v2))) (v12v22v1v2) / (v1v2)

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Proof Sketch

• We have
• ELj (v12v22v1v2) / (v1v2)
• EPj (v12 v22) / (v1v2)
• ELj / EPj (v12v22v1v2) / (v12v22)
  1v1v2 / (v12v22) 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 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.

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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-e) 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-e lower bound.
• Incentive compatible auctions with better
  approximation ratios.