Two Batch Search With Lie Cost

We consider the problem of searching for an unknown number in the search space U={0,...,M-1}. q-ary questions can be asked and some of the answers may be wrong. An arbi- trary integer weighted bipartite graph Gamma is given, stipulating the cost Gamma(i,j) of each answer j \neq i when the correct answer is i, i.e., the cost of a wrong answer. Correct answers are supposed to be cost-less. It is assumed that a maximum cost e for the sum of the cost of all wrong answers can be afforded by the responder during the whole search. We provide tight upper and lower bounds for the largest size M=M(q,e,Gamma,n) for which it is possible to find an unknown number x* in U with n q-ary questions and maximum lie cost . Our results improve the bounds of Cicalese et al. (2004) and Ahlswede et al. (2008). The questions in our strategies can be asked in two batches of nonadaptive questions. Finally, we remark that our results can be further generalized to a wider class of error models including also unidirectional errors.