The execution-simulation gap (ESG) is a fundamental property of public physical unclonable function (PPUF), which exploits the time gap between direct IC execution and computer simulation. ESG needs to consider both advanced computing scheme, including parallel and approximate computing scheme, and IC physical realization. In this paper, we propose a novel PPUF design, whose execution is equivalent to solving the hard-to-parallel and hard-toapproximate max-flow problem in a complete graph on chip. Thus, max-flow problem can be used as the simulation model to bound the ESG rigorously. To enable an efficient physical realization, we propose a crossbar structure and adopt source degeneration technique to map the graph topology on chip. The di↵erence on asymptotic scaling between execution delay and simulation time is examined in the experimental results. The measurability of output difference is also verified to prove the physical practicality.