This paper considers a generalized version of the coin weighing problem with a spring scale that lies at the intersection of group testing and compressed sensing problems. Given a collection of @math coins of total weight @math (for a known integer @math ), where the weight of each coin is an unknown integer in the range of @math (for a known integer @math ), the problem is to determine the weight of each coin by weighing subsets of coins in a spring scale. The goal is to minimize the average number of weighings over all possible weight configurations. For @math , an adaptive bisecting weighing strategy is known to be optimal. However, even the case of @math , which is the simplest non-trivial case of the problem, is still open. For this case, we propose and analyze a simple and effective adaptive weighing strategy. A numerical evaluation of the exact recursive formulas, derived for the analysis of the proposed strategy, shows that this strategy requires about @math weighings on average. To the best of our knowledge, this is the first non-trivial achievable upper bound on the minimum expected required number of weighings for the case of @math . As @math grows unbounded, the proposed strategy, when compared to an optimal strategy within the commonly-used class of nested strategies, requires about @math less number of weighings on average; and in comparison with the information-theoretic lower bound, it requires at most about @math extra number of weighings on average.
The worst-case setting of the CW problem was originally proposed in @cite_12 for @math and unknown @math , and was later studied for @math and known @math , e.g., in @cite_11 @cite_3 @cite_8 . Various order-optimal strategies were previously proposed for unknown @math , see, e.g., @cite_11 @cite_7 , and for known @math , see, e.g., @cite_10 @cite_2 @cite_4 @cite_0 @cite_5 . Recently, in @cite_4 , Bshouty proposed the first and only known order-optimal strategy for any @math and unknown @math , and no such result exists for any @math and known @math . Despite the rich literature on the worst-case setting, there was no result for the average-case setting of the CW problem prior to the present work, excluding the results that trivially carry over from worst case into average case.
@cite_4 @cite_7 @cite_8 @cite_3 @cite_0 @cite_2 @cite_5 @cite_10 @cite_12 @cite_11
2000931246 2240957909 2073127061 2473549844
Let f be a random Boolean formula that is an instance of 3-SAT. We consider the problem of computing the least real number k such that if the ratio of the number of clauses over the number of variables of f strictly exceeds k , then f is almost certainly unsatisfiable. By a well-known and more or less straightforward argument, it can be shown that kF5.191. This upper bound was improved by to 4.758 by first providing new improved bounds for the occupancy problem. There is strong experimental evidence that the value of k is around 4.2. In this work, we define, in terms of the random formula f, a decreasing sequence of random variables such that, if the expected value of any one of them converges to zero, then f is almost certainly unsatisfiable. By letting the expected value of the first term of the sequence converge to zero, we obtain, by simple and elementary computations, an upper bound for k equal to 4.667. From the expected value of the second term of the sequence, we get the value 4.601q . In general, by letting the U This work was performed while the first author was visiting the School of Computer Science, Carleton Ž University, and was partially supported by NSERC Natural Sciences and Engineering Research Council . of Canada , and by a grant from the University of Patras for sabbatical leaves. The second and third Ž authors were supported in part by grants from NSERC Natural Sciences and Engineering Research . Council of Canada . During the last stages of this research, the first and last authors were also partially Ž . supported by EU ESPRIT Long-Term Research Project ALCOM-IT Project No. 20244 . †An extended abstract of this paper was published in the Proceedings of the Fourth Annual European Ž Symposium on Algorithms, ESA’96, September 25]27, 1996, Barcelona, Spain Springer-Verlag, LNCS, . pp. 27]38 . That extended abstract was coauthored by the first three authors of the present paper. Correspondence to: L. M. Kirousis Q 1998 John Wiley & Sons, Inc. CCC 1042-9832r98r030253-17 253 We present a randomized algorithm for dynamic graph connectivity. With failure probability less than @math (for any constant @math we choose), our solution has worst case running time @math per edge insertion, @math per edge deletion, and @math per query, where @math is the number of vertices. The previous best algorithm has worst case running time @math per edge insertion and @math per edge deletion. The improvement is made by reducing the randomness used in the previous result, so that we save a @math factor in update time. Specifically, kapron2013dynamic uses @math copies of a data structure in order to boost a success probability from @math to @math . We show that, in fact though, because of the special structure of their algorithm, this boosting via repetition is unnecessary. Rather, we can still obtain the same correctness guarantee with high probability by arguing via a new invariant, without repetition. We give a worst-case analysis for two greedy heuristics for the integer programming problem minimize cx , Ax (ge) b , 0 (le) x (le) u , x integer, where the entries in A, b , and c are all nonnegative. The first heuristic is for the case where the entries in A and b are integral, the second only assumes the rows are scaled so that the smallest nonzero entry is at least 1. In both cases we compare the ratio of the value of the greedy solution to that of the integer optimal. The error bound grows logarithmically in the maximum column sum of A for both heuristics. We consider the adversarial convex bandit problem and we build the first @math -time algorithm with @math -regret for this problem. To do so we introduce three new ideas in the derivative-free optimization literature: (i) kernel methods, (ii) a generalization of Bernoulli convolutions, and (iii) a new annealing schedule for exponential weights (with increasing learning rate). The basic version of our algorithm achieves @math -regret, and we show that a simple variant of this algorithm can be run in @math -time per step at the cost of an additional @math factor in the regret. These results improve upon the @math -regret and @math -time result of the first two authors, and the @math -regret and @math -time result of Hazan and Li. Furthermore we conjecture that another variant of the algorithm could achieve @math -regret, and moreover that this regret is unimprovable (the current best lower bound being @math and it is achieved with linear functions). For the simpler situation of zeroth order stochastic convex optimization this corresponds to the conjecture that the optimal query complexity is of order @math .