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 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: 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 queriesBlumrosen-Nisan, Dobzinski-Schapira ). • General: any possible type of query (the communication model).

5. 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.

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. XOS • The maximum over additive valuations: (a:1 b:2  c:3) (a:2) Examples: v({a}) = 2 v({a,b}) = 3 v({a,b,c}) = 6

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. 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

10. 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’.

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

12. 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,S:jSxi,s

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. MCG Instance Auction with n XOS bidders A B C v1: (A:1  D:1)  (D:1  E:1  F:1) v2: (B:1 C:1)  (C:1 F:1) D E F

14. 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)

15. 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).

16. Part II: Approximations Using Value Queries • An O(m1/4-e) lower bound for XOS • An m1/2-approximation 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

17. An O(m1/4-e) lower bound for XOS • 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 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.

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. 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 / 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 n=2.

20. 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.

21. 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). 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)) v1(b|a)=1, v1(c|a)=1 v1(S|a)=min(1, SjSv1(j|a))

22. 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] = 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 • 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. 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))

23. 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 + v22) / (v1+v2) 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))

24. 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) 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))

25. Proof Sketch • 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 ≤ 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) 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))

26. Proof Sketch • We have: • E[Lj] ≤ (v12+v22+v1*v2) / (v1+v2) • E[Pj] = (v12 + v22) / (v1+v2) • E[Lj] / E[Pj] ≤ (v12+v22+v1*v2) / (v12+v22) ≤ 1+v1*v2 / (v12+v22) ≤ 1.5

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. 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.

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. 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.)

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(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

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