Multi-Agent Path Finding (MAPF) for car-like robots, addressed by algorithms such as Conflict-Based Search with Continuous Time (CL-CBS), faces significant computational challenges due to expensive kinematic heuristic calculations. Traditional heuristic caching assumes that the heuristic function depends only on the state, which is incorrect in CBS where constraints from conflict resolution make the search space context-dependent. We propose \textbf{CAR-CHASE} (Car-Like Robot Conflict-Aware Heuristic Adaptive Search Enhancement), a novel approach that combines \textbf{conflict-aware heuristic caching} -- which caches heuristic values based on both state and relevant constraint context -- with an \textbf{adaptive hybrid heuristic} that intelligently switches between fast approximate and exact computations. Our key innovations are (1) a compact \emph{conflict fingerprint} that efficiently encodes which constraints affect a state's heuristic, (2) a relevance filter using spatial, temporal, and geometric criteria, and (3) an adaptive switching strategy with theoretical quality bounds. Experimental evaluation on 480 benchmark instances with varying agent counts (10 to 30) and obstacle densities (0\% and 50\%) demonstrates a geometric mean speedup of 2.46$\times$ over the baseline CL-CBS implementation while maintaining solution optimality. The optimizations improve success rate from 77.9\% to 84.8\% (+6.9 percentage points), reduce total runtime by 70.1\%, and enable solving 33 additional instances that previously timed out. Performance gains scale with problem complexity, reaching up to 4.06$\times$ speedup for challenging 30-agent obstacle scenarios. Our techniques are general and applicable to other CBS variants.

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