A Multi-Parameter Complexity Analysis of Cost-Optimal and Net-Benefit Planning

Aghighi and Bäckström have previously studied cost-optimal planning (COP) and net-benefit planning (NBP) for three action cost domains: the positive integers (Z+), the nonnegative integers (Z0) and the positive rationals (Q+). These were indistinguishable under standard complexity analysis for both problems, but separated for COP using parameterised complexity analysis. With the plan cost, k, as parameter, COP was W[2]-complete for Z+, but para-NP-hard for both Z0 and Q+, i.e. presumably much harder. NBP was para-NPhard for all three domains, thus remaining unseparable. We continue by considering combinations with several additional parameters and also the non-negative rationals (Q0). Examples of new parameters are the plan length, `, and the largest denominator of the action costs, d. Our findings include: (1) COP remains W[2]-hard for all domains, even if combining all parameters; (2) COP for Z0 is in W[2] for the combined parameter {k, `}; (3) COP for Q+ is in W[2] for {k, d} and (4) COP for Q0 is in W[2] for {k, d, `}. For NBP we consider further additional parameters, where the most crucial one for reducing complexity is the sum of variable utilities. Our results help to understand the previous results, eg. the separation between Z+ and Q+ for COP, and to refine the previous connections with empirical findings.