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 assuming p is fixed, and (iv) it has adjustable parameter p, so that agents can trade-off better solution quality against computational overhead. The evaluation results illustrate that this algorithm can obtain better quality solutions and bounds compared to existing bounded incomplete algorithms, while the run time of this algorithm is shorter.