Planning over Chain Causal Graphs for Variables with Domains of Size 5 Is NP-Hard
This work addresses the boundary between tractable and intractable planning for AI researchers, narrowing the gap to cases with domain sizes 3 and 4, but is incremental as it builds on known results for smaller domains.
The paper tackles the complexity of planning problems with unary operators and directed path causal graphs, showing that plan existence is NP-hard when state variable domains have a size of at least 5, by reduction from CNFSAT.
Recently, considerable focus has been given to the problem of determining the boundary between tractable and intractable planning problems. In this paper, we study the complexity of planning in the class C_n of planning problems, characterized by unary operators and directed path causal graphs. Although this is one of the simplest forms of causal graphs a planning problem can have, we show that planning is intractable for C_n (unless P = NP), even if the domains of state variables have bounded size. In particular, we show that plan existence for C_n^k is NP-hard for k>=5 by reduction from CNFSAT. Here, k denotes the upper bound on the size of the state variable domains. Our result reduces the complexity gap for the class C_n^k to cases k=3 and k=4 only, since C_n^2 is known to be tractable.