By Yossi Azar (auth.), Giuseppe Persiano, Roberto Solis-Oba (eds.)

ISBN-10: 354024574X

ISBN-13: 9783540245742

The second Workshop on Approximation and on-line Algorithms (WAOA 2004) serious about the layout and research of algorithms for on-line and computationally challenging difficulties. either types of difficulties have lots of functions bobbing up from a number of ?elds. WAOA 2004 happened in Bergen, Norway, from September 14 to September sixteen, 2004. The workshop used to be a part of the ALGO 2004 occasion which additionally hosted ESA, WABI, IWPEC, and ATMOS. TopicsofinterestsforWAOA2004were:applicationstogametheory,appr- imation sessions, coloring and partitioning, aggressive research, computational ?nance, cuts and connectivity, geometric difficulties, inapproximability effects, mechanism layout, community layout, routing, packing and masking, paradigms, randomization thoughts, and scheduling difficulties. in line with our name we acquired forty seven submissions. each one submission used to be reviewed by way of no less than three referees, who judged the paper on originality, caliber, and consistency with the themes of the convention. in line with the experiences, this system Committee chosen 21 papers. This quantity includes the 21 chosen papers and the 2 invited talks given via Yossi Azar and Klaus Jansen. We thank the entire authors who submitted papers to the workshop and we additionally kindly thank the neighborhood organizers of ALGO 2004.

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**Extra info for Approximation and Online Algorithms: Second International Workshop, WAOA 2004, Bergen, Norway, September 14-16, 2004, Revised Selected Papers**

**Example text**

We ﬁrst assume that there is ∈ U such that β (1 + (m + 1)ε)(1 + ε) ≥ 1 1. In this case α ≥ (1+(m+1)ε)(1+ε) 2 , and therefore COST (OP T ) ≥ OP T1 + n n (n − OP T1 )α ≥ nα ≥ (1+(m+1)ε)(1+ε) 2 ≥ (1+(m+1)ε)(1+ε)(1+2ε) . Note that the returned solution costs at most n, and therefore in this case the returned solution pays at most (1+(m+1)ε)(1+ε)(1+2ε)COST (OP T ). Therefore, we can assume that for all , β (1 + (m + 1)ε)(1 + ε) < 1. 42 L. Epstein and A. Levin We denote by τ = (1 + ε)(1 + (m + 1)ε). The cost of the candidate solution is at most: OP T1 + (n − OP T1 ) · 1− (1 − β (1 + (m + 1)ε)) (1) ∈U ≤ OP T1 + (n − OP T1 ) · 1− (1 − β (1 + (m + 1)ε)) + 2ε (2) ∈L ≤ OP T1 + (n − OP T1 ) · 1− (1 − τ α ) + 2ε (3) + 2ε (4) (1 − α ) (5) ∈L ≤ OP T1 + (n − OP T1 )τ · 1− (1 − α ) ∈L ≤ (τ + 2ε) OP T1 + (n − OP T1 ) · 1− ∈L ≤ τ (1 + 2ε) OP T1 + (n − OP T1 ) · 1− (1 − α ) (6) ∈L ≤ τ (1 + 2ε) OP T1 + (n − OP T1 ) · 1− (1 − α ) (7) ∈U = τ (1 + 2ε)COST (OP T ), (8) where (1) follows from Lemma 6 and the monotonicity of the goal function (increasing the probability of not ﬁnding a user in the ﬁrst round only increases the solution cost).

Tm ). First, assume that ti ≥ 1. By Lemma 5, the contribution of type T cells to the probability that the candidate solution ﬁnds i during the second round, is at most 1 + ε times the contribution of type T cells to the probability that OP T ﬁnds i only during the second round. Next, consider a type T such that ti = 0. For such a type we deﬁne the leader of T to be the ﬁrst entry of the type vector that relates to the largest interval (the interval which contains the point 1). There exists at least one such entry as A PTAS for Delay Minimization in Establishing Wireless Conference Calls 41 ˜ j there is at least one unit entry.

In this case, TMH only needs “to pay” for the blue bins. It packs a volume of k(b − α − ε) using only (1 − α)k blue bins. The total weight 1−α according to WA is 1 + b−α−ε (1 − b/2). Balancing the weights gives that the best choice is α = 1/(b − α − ε) = (2b − 8)/(3b2 − 10b + 4). 5 2b−2 4−b and a ratio of Results As mentioned in Section 4, we can determine valid upper and lower bounds on the asymptotic performance ratio for this problem on any interval by sampling a ﬁnite number of points. In fact, since we have given an analytical solution for the algorithm TMH, it is not necessary to do any sampling for the upper bound on the interval (12/7, 2].

### Approximation and Online Algorithms: Second International Workshop, WAOA 2004, Bergen, Norway, September 14-16, 2004, Revised Selected Papers by Yossi Azar (auth.), Giuseppe Persiano, Roberto Solis-Oba (eds.)

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