Abstraet A target is moving on a finite number of cells in discrete time. Kiiowiiig the probabilities of the target's path $election, a searcher is searching for the target in t・his search spa,ce with constraints that he can move firom cell i to one of the adjacent cells, He gains a value V(t) on the detection of the target at timetbut expends cost co(i,t) for the search i]i cell i, at t, In this paper, we prepo$e a method to find an optimal path for the searcher, which maxhnizes the expected reward clefined as the expected vaLue minus the expected cost. We use a branch a.nd bound procedure with an up})er bound estimat・ion given by solving the problem relaxed in continuous search effbrt・,