Approximation Algorithms for CAs with Complement-Free Bidders

1. Let us now choose the row (say row 2) that represents the allocation with the highest value. Let T1,…,T5, be that allocation. The CF property ensures that the sum of values of the 3 rows is at least Sivi(Si). Therefore Sivi(Ti) ≥ (1/3)(Sivi(Si)). 1 2 3 4 1 2 3 4 Observe that each row is an allocation of the 4 items to the 5 bidders. 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 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) 1 2 3 4 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). 1 2 3 4 Red bidder: 1 3 4 In the figure, we see an allocation, S1,…,S5 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. 1 2 3 4 Blue bidder: 1 2 Yellow bidder: 1 3 4 Orange bidder: 2 Green bidder: 2 3 4 • 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 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*=Si,Sxi,Svi(S) is an upper bound for the value of the optimal allocation. The Hierarchy of Complement Free Valuations: (Lehmann, Lehmann, Nisan, 2001) Shahar Dobzinski, Noam Nisan, Michael Schapira{shahard,noam,mikesch}@cs.huji.ac.ilThe Hebrew University of Jerusalem Approximation Bounds for Valuation Classes XOR of ORs of Singletons. Example: (x1:2 OR x2:2) XOR (x1:3) OR of XORs of Singletons. Approximation Algorithms for CAs with Complement-Free Bidders (LOS) (NS) (BGN) (DNS) OXS  GS  SM XOSCF (LLN) (NS) (new) (Gross) Substitutes Complement-Free: v(ST) ≤ v(S) + v(T) (AM) (LLN) Submodular: v(ST) + v(ST) ≤ v(S) + v(T) (NS) (NS) (NS) New A Combinatorial Auction In a combinatorial auction m items are sold to n bidders. Bidder i is defined by the valuation function vi. The goal is to find a partition of the items S1…Sn such that the total social welfare, Sivi(Si), is maximized. New Algorithms: Unless stated otherwise the lower bounds indicate that exponential communication is required. Given items’ prices, p1,…,pm, a demand oracle for v returns the bundle S that maximizes v(S)-SjSpj. An O(log(n + m))-Approximation Algorithm For CF Lemma: Let {Si}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 {Ti}1≤i≤n for a CA with a single copy of each item and the same valuations, such that Sivi(Ti)≥(1/k)(Sivi(Si))(the total social welfare of {Ti} is at least 1/k the total social welfare of {Si}). Proof sketch: We will illustrate the proof by giving an example of how this is done. The example can easily be generalized. A 2-Approximation Algorithm For XOS 2 1 Input: n valuations vi, given in the form of an XOS formula. Output: An allocation S1,...,Sn. 3 4 • Initialize S1 = ... = Sn = , and p1 ...pm = 0. • For each bidder i = 1...n : • Let Si be the demand of bidder i at prices p1...pm. • For all i’ < i take away from Si’ any items from Si: • Si’ ← Si’ – Si • Let (x1:q1 OR ... OR xm:qm) be the OR clause • maximizing the value of Si. • For all jSi, update pj = qj. The Algorithm: Input: n demand oracles for the valuations vi. Output: An allocation T1 ,...,Tn which is an O(log(n + m)) approximation to the optimal allocation. • 2.Use randomized rounding to build a multi­set C of pairs <i,S> such that: • For each bidder i there exist at most O(log(n+m)) pairs <i,S> in C. • For each item j there exist at most O(log(n+m)) pairs <i,S>, where jS, in C. • S<i,S>C vi(S) ≥ O(log(n+m))OPT*. Construction: Repeat O(log(n+m)) times for each i,S: insert <i,S> into C with probability xi,S. If any of the constraints are violated repeat this from scratch. 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 Spi. Lemma: The optimal social welfare is at most 2Spi. 3.For each bidder i, let Si be the set S (<i,S>C) that maximizes vi(S). We can view the sets {Si} as an allocation in an O(log(n+m))-duplicates CA. After this stage Sivi(Si) ≥ O(OPT*). 4.Use the lemma to obtain an allocation, T1, … ,Tn, such that Sivi(Ti)≥ (Sivi(Si))/(log(n+m)).