## Abstract

In this paper we consider the following coin weighing problem: Given n coins for which some of them are counterfeit with the same weight. The problem is: given the weights of the counterfeit coin and the authentic coin, detect the counterfeit coins a with minimal number of weighings. This problem has many applications in computational learning theory, compressed sensing and multiple access adder channels. An old optimal non-adaptive polynomial time algorithm of Lindstrom can detect the counterfeit coins with O(n/logn) weighings. An information theoretic proof shows that Lindstrom's algorithm is optimal. In this paper we study non-adaptive algorithms for this problem when some of the answers of the weighings received are incorrect or unknown. We show that no coin weighing algorithm exists that can detect the counterfeit coins when the number of incorrect weighings is more than 1/4 fraction of the number of weighings. We also give the tight bound Θ(n/log n) for the number of weighings when the number of incorrect answers is less than 1/4 fraction of the number of weighings. We then give a non-adaptive polynomial time algorithm that detects the counterfeit coins with k = O(n/log log n) weighings even if some constant fraction of the answers of the weighings received are incorrect. This improves Bshouty and Mazzawi's algorithm [7] that uses O(n) weighings. This is the first sublinear algorithm for this problem.

Original language | English |
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Title of host publication | Approximation, Randomization, and Combinatorial Optimization |

Subtitle of host publication | Algorithms and Techniques - 15th International Workshop, APPROX 2012, and 16th International Workshop, RANDOM 2012, Proceedings |

Pages | 471-482 |

Number of pages | 12 |

DOIs | |

State | Published - 2012 |

Event | 15th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2012 and the 16th International Workshop on Randomization and Computation, RANDOM 2012 - Cambridge, MA, United States Duration: 15 Aug 2012 → 17 Aug 2012 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 7408 LNCS |

### Conference

Conference | 15th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2012 and the 16th International Workshop on Randomization and Computation, RANDOM 2012 |
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Country/Territory | United States |

City | Cambridge, MA |

Period | 15/08/12 → 17/08/12 |

## All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- General Computer Science