1. OR of XORs of Singletons. Approximation Algorithms for CAs with Complement-Free Bidders ClassUpper BoundLower Bound Generalmin {n, O(m 1/2 )} min {n, O(m 1/2-   } K-Duplicatesmin{n/k,O(km 1/(k-1) )}min {n/k, O(m 1/(k+1) )} Submodular2 1+1/(4m), 1.02-  (NP) Budget Additivee/(e-1)NP-Complete (Gross) Substitutes1 Procurementln(m)log(m)/2 Complement-FreeO(log(n+m))2 XOS2 e/(e-1)-  (NP), 4/3-  OXS  GS  SM  XOS  CF 1. Initialize S 1 =... = S n = , and p 1...p m = 0. 2. For each bidder i = 1...n : a) Let S i be the demand of bidder i at prices p 1...p m. b) For all i’ < i take away from S i’ any items from S i : S i’ ← S i’ – S i c) Let (x 1 :q 1 OR... OR x m :q m ) be the OR clause maximizing the value of S i. d) For all j  S i, update p j = q j. XOR of ORs of Singletons. Example: (x 1 :2 OR x 2 :2) XOR (x 1 :3) (Gross) Substitutes Submodular: v(S  T) + v(S  T) ≤ v(S) + v(T) Complement-Free: v(S  T) ≤ v(S) + v(T) Input: n valuations v i, given in the form of an XOS formula. Output: An allocation S 1,...,S n. Theorem: The algorithm provides a 2­approximation to the optimal social welfare. Proof sketch: 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. Shahar Dobzinski, Noam Nisan, Michael Schapira {shahard,noam,mikesch}@cs.huji.ac.il The Hebrew University of Jerusalem The Hierarchy of Complement Free Valuations : (Lehmann, Lehmann, Nisan, 2001) New Algorithms: A 2-Approximation Algorithm For XOS 3. For each bidder i, let S i be the set S (  C) that maximizes v i (S). An O(log(n + m))-Approximation Algorithm For CF Approximation Bounds for Valuation Classes Unless stated otherwise the lower bounds indicate that exponential communication is required. A Combinatorial Auction In a combinatorial auction m items are sold to n bidders. Bidder i is defined by the valuation function v i. The goal is to find a partition of the items S 1 …S n such that the total social welfare,  i v i (S i ), is maximized. New (LOS)(NS) (new) (DNS) (LLN) (BGN) (LLN)(AM) (NS) References AM – Andelman, Mansour (2004) BGN – Bartal, Gonen, Nisan (2003) DNS – Dobzinski, Nisan, Schapira (2004) LLN – Lehmann, Lehmann, Nisan (2001) LOS - Lehmann, O’Callaghan, Shoam (1999) NS – Nisan, Segal (2002) Input: n demand oracles for the valuations v i. Output: An allocation T 1,...,T n which is an O(log(n + m)) approximation to the optimal allocation. 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. This can be done using the ellipsoid method on the dual problem (with the bidders’ demand oracles used as “separation oracles”). OPT*=  i,S x i,S v i (S) is an upper bound for the value of the optimal allocation. 2. Use randomized rounding to build a multi­set C of pairs such that: For each bidder i there exist at most O(log(n+m)) pairs in C. For each item j there exist at most O(log(n+m)) pairs, where j  S, in C.   C v i (S) ≥ O(log(n+m))OPT*. Construction: Repeat O(log(n+m)) times for each i,S: insert into C with probability x i,S. If any of the constraints are violated repeat this from scratch. Lemma: Let {S i } 1≤i≤n be an allocation in a CA with k duplicates of each item and CF valuations. Then, it is possible to find an allocation {T i } 1≤i≤n for a CA with a single copy of each item and the same valuations, such that  i v i (T i )≥(1/k)(  i v i (S i )) (the total social welfare of {T i } is at least 1/k the total social welfare of {S i }). Proof sketch: We will illustrate the proof by giving an example of how this is done. The example can easily be generalized. 134 12 2 In the figure, we see an allocation, S 1, …,S 5 in a CA with 3 duplicates of each item, and five bidders. The red bidder is assigned items 1,3,4. The blue bidder is assigned items 1,2, and so on. 134 234 Red bidder: Blue bidder: Yellow bidder: Orange bidder: Green bidder: 2134 Since we have 3 duplicates of each item, a simple greedy algorithm can arrange them in 3 “rows” (each row contains 1 copy of each item). 2134 2134 2134 2134 2134 Observe that each row is an allocation of the 4 items to the 5 bidders. 2134 2134 2134 Let us now choose the row (say row 2) that represents the allocation with the highest value. Let T 1,…,T 5, be that allocation. The CF property ensures that the sum of values of the 3 rows is at least  i v i (S i ). Therefore  i v i (T i ) ≥ (1/3)(  i v i (S i )). 4. Use the lemma to obtain an allocation, T 1, …,T n, such that  i v i (T i )≥ (  i v i (S i ))/(log(n+m)). The Algorithm: 1 34 2 We can view the sets {S i } as an allocation in an O(log(n+m))-duplicates CA. After this stage  i v i (S i ) ≥ O(OPT*). Given items’ prices, p 1,…,p m, a demand oracle for v returns the bundle S that maximizes v(S)-  j  S p j.