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

2. 2 Combinatorial Auctions A set M={1,…,m} of items for sale. n bidders, each bidder i has a valuation function v i :2 M ->R +. Common assumptions: Normalization: v i (  )=0 Free disposal: S  T  v i (T) ≥ v i (S) Goal: find a partition S 1,…,S n such that social welfare  v i (S i ) is maximized

3. 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. 4 Access Models Common types of queries:  Value: given a bundle S, return v(S).  Demand: given a vector of prices (p 1,…, p m ) return the bundle S that maximizes v(S)-  j  S p j. (demand queries are strictly more powerful than value queries Blumrosen-Nisan, Dobzinski-Schapira ).  General: any possible type of query (the communication model).

5. 5 The Hierarchy of CF Valuations 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. OXS  GS  SM  XOS  CF Lehmann, Lehmann, Nisan

6. 6 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

7. 7 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:

8. 8 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.

9. 9 The XOS Algorithm – Step 1 Solve the linear relaxation of the problem: Maximize:  i,S x i,S v i (S) Subject To:  For each item j:  i,S|j  S x i,S ≤ 1  For each bidder i:  S x i,S ≤ 1  For each i,S: x i,S ≥ 0

10. 10 The XOS Algorithm – Steps 2-3 Randomized Rounding: For each bidder i, let S i be the bundle S with probability x i,S, and the empty set with probability 1-  S x i,S.  The expected value of v i (S i ) is  S x i,S v i (S) Bidder i got the bundle S i = (x 1 :p 1 i  …  x m :p m i ) Give item j to bidder i such that p j j ≥ p j i’ for all i’.

11. 11 The XOS Algorithm Theorem: The algorithm is an e/(e-1)- approximation. Proof: only for the special case where all prices are equal.  Example: (x 1 :1  x 2 :1)  (x 1 :1) We now only need to prove that the number of items which are allocated ≥ (1-(1- 1/n) n )(  i,s x i,s |S|). We will prove that each item is allocated with probability ≥ (1- (1-1/n) n )  i,S:j  S x i,s.

12. 12 The XOS Algorithm Proof Pr [item j is not allocated] ≤  n i=1 (1-  j  S x i,S ) = ((  n i=1 (1-  j  S x i,S )) 1\n ) n Due to the arithmetic/geometric mean inequality: ≤ ((  n i=1 (1-  j  S x i,S ))\n) n = (1-(  i,j  S x i,s )/n) n Pr [item j is allocated] ≥ 1-(1-(  i,j  S x i,s )/n) n ≥ (1-(1-1/n) n )  i,S:j  S x i,s

13. 13 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. ABC DEF v 1 : (A:1  D:1)  (D:1  E:1  F:1 ) v 2 : (B:1  C:1)  (C:1  F:1) MCG InstanceAuction with n XOS bidders

14. 14 Approximate Disjointness n players, each holds a string of length t. The string of player i specifies a subset A i  {1,…,t}. The goal is to distinguish between the following two extreme cases:  NO:  i A i ≠   YES: for every i≠j A i  A j =  Theorem: Requires t/n 4 bits of communication (Alon-Matias-Szegedy)

15. 15 The Reduction Denote a partition C of M to n parts as {C 1,…,C n ). We build a set of partitions F=(C 1,…,C exp(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 A i as input, constructs the collection B i = {C s i |A i =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 ).

16. 16 Part II: Approximations Using Value Queries An O(m 1/4-   lower bound for XOS  An m 1/2  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)  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

17. 17 An O(m 1/4-   lower bound for XOS Setting: m items, m ½ XOS bidders. Choose, uniformly at random, a partition T 1,…,T n, where |T i |=m ½. Valuations: v i = (  j  T j:m -½ )  |S|=2m^(¼+  (  j  S j:m -¼ )  |S|=m^(¾  (  j  S j:m -¼ ) The optimal Allocation has value of m ½ (according to the T i ’s). Lemma: Exponential number of value queries is required to find a bundle R, |R|<m ¾, for which the maximizing clause is (  j  T j:m -½ ). Corollary: the best allocation has value of 2m ¼+ . Proof (of lemma):  The average intersection between a random bundle and T i is m ¼.  By the chernoff bounds, the chance of finding a bundle whose intersection with T i 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.

18. 18 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)  T  S  M: 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.

19. 19 The Algorithm For each item j=1..m  For each bidder i, let t i = v i (j|S i ) n-1  Assign to exactly one bidder the item j, where bidder i is chosen with probability t i /  k t k. 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.

20. 20 Proof Sketch v 1 (a)=1, v 1 (b)=1, v 1 (c)=1 v 1 (S)=min(2,  j  S v 1 (j)) v 2 (a)=0, v 2 (b)=1, v 2 (c)=0 v 2 (S)=min(1,  j  S v 2 (j)) Let OPT j denote the value of the optimal solution without the first (j-1) items.

21. 21 Proof Sketch Let OPT j denote the value of the optimal solution without the first (j-1) items.  With the submodular valuations v 1 (·|S 1 ),…,v n (·|S n ). v 1 (a)=1, v 1 (b)=1, v 1 (c)=1 v 1 (S)=min(2,  j  S v 1 (j)) v 2 (a)=0, v 2 (b)=1, v 2 (c)=0 v 2 (S)=min(1,  j  S v 2 (j)) a v 1 (b|a)=1, v 1 (c|a)=1 v 1 (S|a)=min(1,  j  S v 1 (j|a))

22. 22 Proof Sketch Let P j 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] =  j E[P j ]. Let OPT i j denote the optimal solution given that item j was assigned to bidder i. L j denotes the random variable that gets the value of OPT j – OPT j+1  i.e. how much did we lose by assigning item j to bidder i? We will prove that E[L j ] / E[P j ] ≤ 1.5, and the theorem will follow. v 1 (b|a)=1, v 1 (c|a)=1 v 1 (S|a)=min(1, S j  S v 1 (j|a)) v 2 (a)=0, v 2 (b)=1, v 2 (c)=0 v 2 (S)=min(1,  j  S v 2 (j)) a

23. 23 Proof Sketch Lemma: E[L j ] / E[P j ] ≤ 1.5 Proof: Notation: v i := v(j|S i ). E[P j ] = (v 1 *(v 1 /  v 1 +v 2 )) + v 2 *(v 1 /  v 1 +v 2 ))) = (v 1 2 + v 2 2 ) / (v 1 +v 2 ) v 1 (b|a)=1, v 1 (c|a)=1 v 1 (S|a)=min(1, S j  S v 1 (j|a)) v 2 (a)=0, v 2 (b)=1, v 2 (c)=0 v 2 (S)=min(1,  j  S v 2 (j)) a

24. 24 Proof Sketch WLOG bidder 2 gets item j in OPT j. If we assign item j to bidder 2: L=OPT j -OPT 1 j =v 2 This happens with probability v 2 / (v 1 +v 2 ) v 1 (b|a)=1, v 1 (c|a)=1 v 1 (S|a)=min(1, S j  S v 1 (j|a)) v 2 (a)=0, v 2 (b)=1, v 2 (c)=0 v 2 (S)=min(1,  j  S v 2 (j)) a b

25. 25 Proof Sketch Suppose we assign item j to bidder 1:  Bidder 1 loses at most v 1 in OPT 1 j the marginal value of j given the bundle he gets in OPT 1 j is smaller than v 1.  Bidder 2 loses at most v 2 in OPT 1 j   L ≤ v 1 +v 2 This happens with probability v 1 / (v 1 +v 2 ) E[L j ] ≤ (v 2 *(v 2 /  v 1 +v 2 )) +(v 1 +v 2 ) *(v 1 /  v 1 +v 2 ))) = (v 1 2 +v 2 2 +v 1 *v 2 ) /  v 1 +v 2 ) v 1 (b|a)=1, v 1 (c|a)=1 v 1 (S|a)=min(1, S j  S v 1 (j|a)) v 2 (a)=0, v 2 (b)=1, v 2 (c)=0 v 2 (S)=min(1,  j  S v 2 (j)) a b

26. 26 Proof Sketch We have:  E[L j ] ≤ (v 1 2 +v 2 2 +v 1 *v 2 ) /  v 1 +v 2 )  E[P j ] = (v 1 2 + v 2 2 ) / (v 1 +v 2 ) E[L j ] / E[P j ] ≤ (v 1 2 +v 2 2 +v 1 *v 2 ) / (v 1 2 +v 2 2 ) ≤ 1+v 1 *v 2 / (v 1 2 +v 2 2 ) ≤ 1.5

27. 27 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.

28. 28 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. V i (S)=1 if there exists some v  S such that the edge (v,i) exists. Otherwise V i (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.

29. 29 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. V i (S)=1 if S contains a time slot between the arrival and the expiration of the corresponding packet. Otherwise, V i (S)=1.  e/(e-1) randomized upper bound (Bartal et al.)

30. 30 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(m 1/4-  ) 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

31. 31 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.