By Charles E. Leiserson (auth.), Torben Hagerup, Jyrki Katajainen (eds.)

ISBN-10: 3540223398

ISBN-13: 9783540223399

ISBN-10: 3540278109

ISBN-13: 9783540278108

This booklet constitutes the refereed court cases of the ninth Scandinavian Workshop on set of rules concept, SWAT 2004, held in Humlebaek, Denmark in July 2004.

The forty revised complete papers provided including an invited paper and the summary of an invited speak have been conscientiously reviewed and chosen from 121 submissions. The papers span the whole variety of theoretical algorithmics and functions in numerous fields together with graph algorithms, computational geometry, scheduling, approximation algorithms, community algorithms, information garage and manipulation, bioinformatics, combinatorics, sorting, looking out, on-line algorithms, optimization, etc.

**Read or Download Algorithm Theory - SWAT 2004: 9th Scandinavian Workshop on Algorithm Theory, Humlebæk, Denmark, July 8-10, 2004. Proceedings PDF**

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**Additional resources for Algorithm Theory - SWAT 2004: 9th Scandinavian Workshop on Algorithm Theory, Humlebæk, Denmark, July 8-10, 2004. Proceedings**

**Example text**

Hartline and A. R. Karlin, Competitive generalized auctions. In Proc. 34th Ann. ACM Symp. on Theory of Computing (STOC), 2002, 7-81. 6. Y. Fujishima, K. Leyton-Brown and Y. Shoham, Taming the computational complexity of combinatorial auctions: Optimal and approximate approaches. In Proc. 16th Int. Joint Conf. on Artiﬁcial Intelligence (IJCAI), 1999, 548–553. 7. A. V. Goldberg and J. D. Hartline, Competitiveness via consensus. In Proc. 14th Ann. ACM-SIAM Symp. on Discrete Algorithms (SODA) 2003, 215–222.

Theorem 1. There is an O(n2 |S|) time approximation algorithm for TSΓ with approximation ratio 1 + ln n for Γ = {{1}} and 1 + ln 2 + ln n for Γ = {{1}, ]{0, 2}}. There is an O(n2 |P | + L|P |) time approximation algorithm for MCPΣ (r) with approximation ratio 1+ln n+ln log2 (r +1), where r = min{r, n} and L is the total length of the sequences in S. 1 Proof of Theorem 1 for TS{1} In this section we provide a greedy heuristic for TS{1} running in time O(n2 |S|) time with an improved approximation ratio of 1 + ln n.

Auctions with Budget Constraints 31 Obviously, Algorithm RR is randomized, outputs a feasible allocation, and terminates in polynomial time complexity. The following theorem proves the approximation ratio of the algorithm. Theorem 4. 582. Proof. Let Zi be a random variable indicating the revenue from bidder i after the allocation. The expected total revenue is i E(Zi ). Therefore, it is suﬃcient to prove that for any bidder, the expected ratio between the beneﬁt of the fractional e . allocation and the ﬁnal integer allocation is at most e−1 We analyze separately the expected revenue from each bidder i.

### Algorithm Theory - SWAT 2004: 9th Scandinavian Workshop on Algorithm Theory, Humlebæk, Denmark, July 8-10, 2004. Proceedings by Charles E. Leiserson (auth.), Torben Hagerup, Jyrki Katajainen (eds.)

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