We consider the online problem of scheduling jobs on unrelated machines so as to minimize the total weighted flow time. This problem has an unbounded competitive ratio even for very restricted settings. In this paper we show that if we allow the machines of the online algorithm to have more speed than those of the offline algorithm then we can get an O((1 + −1 ) 2 )-competitive algorithm.Our algorithm schedules jobs preemptively but without migration. However, we compare our solution to an offline algorithm which allows migration. Our analysis uses a potential function argument which can also be extended to give a simpler and better proof of the randomized immediate dispatch algorithm of Chekuri-Goel-Khanna-Kumar for minimizing average flow time on parallel machines.

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We address a version of the set-cover problem where we do not know the sets initially (and hence referred to as covert) but we can query an element to find out which sets contain this element as well as query a set to know the elements. We want to find a small set-cover using a minimal number of such queries. We present a Monte Carlo randomized algorithm that approximates an optimal set-cover of size OP T within O(log N ) factor with high probability using O(OP T ·log 2 N ) queries where N is the input size.We apply this technique to the network discovery problem that involves certifying all the edges and non-edges of an unknown n-vertices graph based on layered-graph queries from a minimal number of vertices. By reducing it to the covert set-cover problem we present an O(log 2 n)-competitive Monte Carlo randomized algorithm for the covert version of network discovery problem. The previously best known algorithm [4] has a competitive ratio of Ω( √ n log n) and therefore our result achieves an exponential improvement. * A Preliminary version of the results have appeared earlier in the 4th Workshop on Algorithms and Computation 2010 1 We have chosen n ′ , m ′ as notations to keep them distinct from graphs with n vertices and m edges.

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For Reed-Solomon Codes with block length n and dimension k, the Johnson theorem states that for a Hamming ball of radius smaller than n − √ nk, there can be at most O(n 2 ) codewords. It was not known whether for larger radius, the number of code words is polynomial. The best known list decoding algorithm for Reed-Solomon Codes due to Guruswami and Sudan [13] is also known to work in polynomial time only within this radius. In this paper we prove that when k < αn for any constant 0 < α < 1, we can overcome the barrier of the Johnson bound for list-decoding of Reed-Solomon Codes (even if the field size is exponential). More specifically in such a case, we prove that for Hamming ball of radius n − √ nk + c, (for any c > 0) there can be at most O(n) number of codewords. For any constant c, we describe a polynomial time algorithm to enumerate all of them, thereby also improving on the Guruswami-Sudan's algorithm. Although the improvement is modest this provides evidence for the first time that the n − √ nk bound is not sacrosanct for such a high rate. We apply our method to obtain sharper bounds on a list recovery problem introduced by Guruswami and Rudra [11] where they establish super polynomial lower bounds on the output size when the list size exceeds n k . We show that even for larger list sizes the problem can be solved in polynomial time for certain values of k.

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