Same-locality simulation of Π1^{LP} by LD

Determine whether every Π1^{LP} local decision algorithm (with certificate sizes bounded polynomially in each node’s local view) can be simulated by an LD local decision algorithm without increasing the locality radius, i.e., using the same radius r rather than the currently proven doubling to 2r. This addresses whether the locality overhead in the known simulation from Π1^{LP} to LD is necessary.

Background

The paper studies relations between distributed decision classes under different bounds on local computation time and certificate sizes. For universal-certificate classes based on local-view-bounded certificates (Π1^{LP}), the authors prove that Π1^{LP} is strictly contained in LD and provide a simulation that converts any Π1^{LP} algorithm of radius r into an LD algorithm of radius 2r.

While this establishes containment, it leaves open whether the overhead in locality radius is essential. Resolving this would sharpen the structural relationship between Π1^{LP} and LD, clarifying whether locality can be preserved exactly when transitioning from certificate-bounded universal verification to deterministic local decision.