1. Approximation Algorithms for Combinatorial Auctions with Complement-Free Bidders Speaker: Shahar Dobzinski Joint work with Noam Nisan & Michael Schapira

2. 2 Talk Structure  Combinatorial Auctions Log(m)-approximation for CF auctions An incentive compatible O(m 1/2 )- approximation CF auctions A lower bound of 2-  for CF auctions 2-approximation for XOS auctions A lower bound of e/(e-1)-  for XOS auctions

3. 3 Combinatorial Auctions m 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. Problem 1: finding an optimal allocation is NP-hard. Problem 2: valuation length is exponential in n and m.

4. 4 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)-  p i.  General: any type of possible query. Demand queries are more powerful than value queries (Blumrosen-Nisan)

5. 5 Known Results Finding an exact solution requires exponential communication. Nisan-Segal Finding an O(m 1/2-  )-approximation requires exponential communication. Nisan-Segal. This result holds for every possible type of oracle. Using demand oracles, a matching upper bound of O(m 1/2 ) exists (Blumrosen-Nisan). Better results might be obtained by using restricted classes of valuations.

6. 6 The Hierarchy of CF Valuations Complement-Free: v(S  T) ≤ v(S) + v(T). XOS: XOR of ORs of singletons  Example: (A:2 OR B:2) XOR (A:3) 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 OXS  GS  SM  XOS  CF Lehmann, Lehmann, Nisan

7. 7 Talk Structure Combinatorial Auctions  Log(m)-approximation for CF auctions An incentive compatible O(m 1/2 )- approximation CF auctions A lower bound of 2-  for CF auctions 2-approximation for XOS auctions A lower bound of e/(e-1)-  for XOS auctions

8. 8 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.

9. 9 The 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 Despite the exponential number of variables, the LP relaxation may still be solved in polynomial time using demand oracles. (Nisan-Segal). OPT*=  i,S x i,S v i (S) is an upper bound for the value of the optimal integral allocation.

10. 10 The Algorithm – Step 2 Use randomized rounding to build a “pre- allocation” S 1,..,S n :  Each item j appears at most k=O(log(m)) times in {S i } i.   i v i (S i ) ≥ OPT*/2. 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) We use the Chernoff bound to show that such “pre-allocation” is built with high probability.

11. 11 The Algorithm – Step 3 For each bidder i, partition S i into a disjoint union S i = S i 1 ..  S i k such that for each 1≤i<i’≤ n, 1≤t≤t’≤ k, S i t  S i’ t’ = .

12. 12 The Algorithm – Step 3 For each bidder i, partition S i into a disjoint union S i = S i 1 ..  S i k such that for each 1≤i<i’≤ n, 1≤t≤t’≤ k, S i t  S i’ t’ = . ABD

13. 13 The Algorithm – Step 3 For each bidder i, partition S i into a disjoint union S i = S i 1 ..  S i k such that for each 1≤i<i’≤ n, 1≤t≤t’≤ k, S i t  S i’ t’ = . ABD S 1 1 = {A,B,D}

14. 14 The Algorithm – Step 3 For each bidder i, partition S i into a disjoint union S i = S i 1 ..  S i k such that for each 1≤i<i’≤ n, 1≤t≤t’≤ k, S i t  S i’ t’ = . ABD CE AD

15. 15 The Algorithm – Step 3 For each bidder i, partition S i into a disjoint union S i = S i 1 ..  S i k such that for each 1≤i<i’≤ n, 1≤t≤t’≤ k, S i t  S i’ t’ = . CE AD S 2 1 = {C,E} S 2 2 = {A,D}

16. 16 The Algorithm – Step 3 For each bidder i, partition S i into a disjoint union S i = S i 1 ..  S i k such that for each 1≤i<i’≤ n, 1≤t≤t’≤ k, S i t  S i’ t’ = . ABD CE AD CE A

17. 17 The Algorithm – Step 3 For each bidder i, partition S i into a disjoint union S i = S i 1 ..  S i k such that for each 1≤i<i’≤ n, 1≤t≤t’≤ k, S i t  S i’ t’ = . CE A S 3 2 = {C,E} S 3 3 = {A}

18. 18 The Algorithm – Step 3 For each bidder i, partition S i into a disjoint union S i = S i 1 ..  S i k such that for each 1≤i<i’≤ n, 1≤t≤t’≤ k, S i t  S i’ t’ = . ABD CE AD CE A B D

19. 19 The Algorithm – Step 3 For each bidder i, partition S i into a disjoint union S i = S i 1 ..  S i k such that for each 1≤i<i’≤ n, 1≤t≤t’≤ k, S i t  S i’ t’ = . ABD CE AD CE A B DBCE

20. 20 The Algorithm – Step 4 Find the t maximizes  i v i (S i t ) Return the allocation (S 1 t,...,S n t ). All valuations are CF so:  t  i v i (S i t ) =  i  t v i (S i t ) ≥  i v i (S i ) ≥ OPT*/2  For the t that maximizes  i v i (S i t ), it holds that:  i v i (S i t ) ≥ OPT*/2k = OPT*/O(log(m)).

21. 21 Talk Structure Combinatorial Auctions Log(m)-approximation for CF auctions  An incentive compatible O(m 1/2 )- approximation CF auctions A lower bound of 2-  for CF auctions 2-approximation for XOS auctions A lower bound of e/(e-1)-  for XOS auctions

22. 22 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:  k≠i v i (O i ) -  k≠i v i (O -i ) O i is the optimal allocation, O -i the optimal allocation of the auction without the i’th bidder. VCG requires an optimal allocation. Finding an optimal allocation requires exponential communication and is computationally intractable.  Approximations do not suffice (Nisan-Ronen, Lehmann-O’Callaghan- Shoham).

23. 23 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 we’ll 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.

24. 24 The Algorithm Ask each bidder i for v i (M), and for v i (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). 1 2 3 A B Items Bidders v 1 (A) v 3 (B)

25. 25 The Algorithm (Cont.) Let i be the bidder that maximizes v i (M). If v i (M)>|P|  Allocate all items to i. else  Allocate according to P. Let each bidder pay his VCG price (in respect to the restricted set).

26. 26 Proof of the Approximation Ratio Theorem: If all valuations are CF, the algorithm provides an O(m 1/2 )-approximation. Proof: Let OPT=(T 1,..,T k,Q 1,...,Q l ), where for each T i, |T i |>m 1/2, and for each Q i, |Q i |≤m 1/2. |OPT|=  i v i (T i ) +  i v i (Q i ) Case 1:  i v i (T i ) >  i v i (Q i ) (“large” bundles contribute most of the social welfare)   i v i (Q i ) > |OPT|/2 At most m 1/2 bidders get at least m 1/2 items in OPT.  For the bidder i the bidder i that maximizes v i (M), v i (M) > |OPT|/2m 1/2. Case 2:  i v i (Q i ) ≥  i v i (T i ) (“small” bundles contribute most of the social welfare)   i v i (Q i ) ≥ |OPT|/2 For each bidder i, there is an item c i, such that: v i (c i ) > v i (Q i ) / m 1/2. (The CF property ensures that the sum of the values is larger than the value of the whole bundle) {c i } i is an allocation with at most one item to each bidder: |P| ≥  i c i ≥ |OPT|/2m 1/2.

27. 27 Talk Structure Combinatorial Auctions Log(m)-approximation for CF auctions An incentive compatible O(m 1/2) - approximation CF auctions  A lower bound of 2-  for CF auctions 2-approximation for XOS auctions A lower bound of e/(e-1)-  for XOS auctions

28. 28 A Lower Bound of 2-  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 v i (S i )=1, for all i.  In all allocations, at most one bidder gets a value of 1.

29. 29 A Lower Bound of 2-  (Cont.) Add 1 to all valuations in Nisan-Segal Auction:  v’ i (S)=v i (S)+1. The valuations are now CF. It requires exponential communication to distinguish between the following cases:  There is an allocation in which v’ i (S i )=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/(n+1) requires exponential communication.

30. 30 Talk Structure Combinatorial Auctions Log(m)-approximation for CF auctions An incentive compatible O(m 1/2 )- approximation CF auction A lower bound of 2-  for CF auctions  2-approximation for XOS auctions A lower bound of e/(e-1)-  for XOS auctions

31. 31 Definition of XOS XOS: XOR of ORs of Singletons. Singleton valuation (x:p)  v(S) =p x  S 0 otherwise Example: (A:2 OR B:2) XOR (A:3) This is a XOR of additive valuations.

32. 32 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)

33. 33 Supporting Prices p 1,…,p m supports the bundle S if:  v(S) =  j  S p j  v(T) ≥  j  T p j 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.

34. 34 The Algorithm Input: n bidders, each represented a demand oracle and a supporting price oracle. Init: p 1 =…=p m =0. For each bidder i=1..n  Let S i be the demand of the i’th bidder at prices p 1,…,p m.  For all i’ < i take away from S i’ any items from S i.  Let q 1,…,q m be the supporting prices for S i in v i.  For all j  S i update p j = q j.

35. 35 Example Items: {A, B, C, D, E}. 3 bidders. Price vector: p 0 =(0,0,0,0,0) v 1 : (A:1 OR B:1 OR C:1) XOR (C:2) Bidder 1 gets his demand: {A,B,C}. Price vector: p 1 =(1,1,1,0,0) v 2 : (A:1 OR B:1 OR C:9) XOR (D:2 OR E:2) Bidder 2 gets his demand: {C} Price vector: p 2 =(1,1,9,0,0) v 3 : (C:10 OR D:1 OR E:2) Bidder 3 gets his demand: {C,D,E} Final allocation: {A,B} to bidder 1, {C,D,E} to bidder 3.

36. 36 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  p i. Lemma: The optimal social welfare is at most 2  p i.

37. 37 Proof – Lemma 1 Lemma: The total social welfare generated by the algorithm is at least  p i. Proof: Each bidder i got a bundle T i at stage i. At the end of the algorithm, he holds A i  T i. The supporting prices guarantee that: v i (A i ) ≥  j  Ai p j

38. 38 Proof – Lemma 2 Lemma: The optimal social welfare is at most 2  p i. Proof: Let O 1,...,O n be the optimal allocation. Let p i,j be the price of the j’th item at the i’th stage. Each bidder i ask for the bundle that maximizes his demand at the i’th stage: v i (O i )-  j  Oi p i,j ≤  j p i,j –  j p (i-1),j Since the prices are non-decreasing: v i (O i )-  j  Oi p n,j ≤  j p i,j –  j p (i-1),j Summing up on both sides:  i v i (O i )-  i  j  Oi p n,j ≤  i (S j p i,j –  j p (i-1),j )  i v i (O i )-  j p n,j ≤  j p n,j  i v i (O i ) ≤ 2  j p n,j

39. 39 Talk Structure Combinatorial Auctions Log(m)-approximation for CF auctions An incentive compatible O(m 1/2 )- approximation CF auctions A lower bound of 2-  for CF auctions 2-approximation for XOS auctions  A lower bound of e/(e-1)-  for XOS auctions

40. 40 k-Set Cover We will show a polynomial time reduction from k-Set-Cover. k-Set-Cover definition:  Input: a set of |M|=m items, t subsets S i  M, an integer k.  Goal: Find k subsets such that the number of item in their union, |  1≤i≤k S i |, is maximized. Theorem: approximating k-Set-Cover within a better factor than e/(e-1) is NP-hard (feige).

41. 41 The Reduction ABC DEF v 1 : (A:1 OR D:1) XOR (A:1 OR D:1) XOR (D:1 OR E:1 OR F:1) v k : (A:1 OR D:1) XOR (A:1 OR D:1) XOR (D:1 OR E:1 OR F:1) 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). k-Set-Cover InstanceXOS Auction with k bidders