A Distributed Constraint Optimization Problem (DCOP) is a fundamental problem that can formalize various applications related to multi-agent cooperation. Since it is NP-hard, considering faster incomplete algorithms is necessary for large-scale applications. Most incomplete algorithms generally do not provide any guarantees on the quality of solutions. Some notable exceptions are DALO, the bounded max-sum algorithm, and ADPOP. In this paper, we develop a new solution criterion called p-optimality and an incomplete algorithm for obtaining a p-optimal solution. The characteristics of this algorithm are as follows: (i) it can provide the upper bounds of the absolute/relative errors of the solution, which can be obtained a priori/a posteriori, respectively, (ii) it is based on a pseudo-tree, which is a widely used graph structure in complete DCOP algorithms, (iii) it is a one-shot type algorithm, which runs in polynomial-time in the number of agents n, and (iv) it has adjustable parameter ...