Planning over Chain Causal Graphs for Variables with Domains of Size 5 Is NP-Hard

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.

💡 Research Summary

The paper investigates the computational boundary of planning problems whose causal graphs are simple directed paths—so‑called chain causal graphs—within the class Cₙ. In Cₙ each operator is unary (affects only a single state variable) and the variables are ordered v₁ → v₂ → … → vₙ, so that the effect of any action can only depend on the value of its immediate predecessor. This structural restriction is one of the most elementary forms of causal interaction, and it has long been conjectured that such simplicity might render planning tractable, at least when the domain size of each variable is bounded.

The authors focus on the parameter k, the maximal size of a variable’s domain, and study the subclass Cₙᵏ. Prior work has shown that Cₙ² (binary domains) is solvable in polynomial time, essentially because the problem reduces to a simple reachability test on a Boolean chain. However, the complexity for larger domains remained open, with only a trivial upper bound (PSPACE) and no matching lower bound for any k ≥ 3.

The main contribution of the paper is a proof that plan existence for Cₙᵏ is NP‑hard as soon as k ≥ 5. The proof proceeds by a polynomial‑time reduction from CNF‑SAT, the canonical NP‑complete problem. The reduction constructs, for any CNF formula Φ, a planning instance P(Φ) in Cₙ⁵ such that Φ is satisfiable iff there exists a plan that reaches the designated goal state of P(Φ).

The construction uses three intertwined gadgets:

1. Variable gadgets – For each propositional variable xᵢ in Φ the reduction creates a contiguous block of planning variables (vᵢ¹,…,vᵢᵐ). Each variable in the block has a domain of five symbols {0,1,⊥,c,d}. The symbols 0 and 1 encode the truth assignment of xᵢ, ⊥ represents “unassigned”, while c and d are auxiliary control values that enforce the correct ordering of actions along the chain. Unary operators are defined so that the only way to move a block from the initial ⊥‑state to a final 0‑ or 1‑state is to apply a “choose‑value” action at the leftmost position, after which the control symbols propagate rightward, guaranteeing that the assignment is made exactly once and cannot be changed later.

2. Clause gadgets – For each clause Cⱼ = (ℓ₁ ∨ ℓ₂ ∨ ℓ₃) a separate clause variable gⱼ is introduced, also with a five‑element domain {init, sat, fail, …}. The operators affecting gⱼ are conditioned on the current values of the relevant variable‑gadget positions that correspond to the literals ℓ₁, ℓ₂, ℓ₃. If at least one literal evaluates to true (i.e., the associated variable gadget is in the 1‑state for a positive literal or 0‑state for a negated literal), an operator can transition gⱼ from init to sat. If none of the literals are satisfied, the only reachable state for gⱼ is fail, which blocks any plan from reaching the overall goal.

3. Goal specification – The global goal requires that every variable gadget ends in either 0 or 1 (i.e., a complete truth assignment) and that every clause gadget is in the sat state. Because the chain structure forces a strict left‑to‑right execution order, any feasible plan must first decide the truth values of all variables and then “activate” each clause gadget. Consequently, a plan exists exactly when the original CNF formula admits a satisfying assignment.

A crucial observation is that five domain values are the minimal number needed to encode both the binary truth choice and the auxiliary control flow without creating unintended shortcuts. With only four values, one cannot simultaneously represent an unassigned state, two truth values, and a control token that forces the sequential propagation required for the reduction. This explains why the hardness proof does not extend to Cₙ⁴ or Cₙ³, leaving those cases as the only open gaps.

By establishing NP‑hardness for all k ≥ 5, the paper narrows the tractability frontier of chain causal graph planning to the two unresolved cases k = 3 and k = 4. It also demonstrates that even the simplest possible causal topology can encode full SAT‑level difficulty once the variable domains are modestly enlarged.

The authors discuss several implications. First, the result cautions against assuming tractability solely from a sparse causal graph; domain size is an equally decisive factor. Second, it suggests a research agenda focused on the remaining k = 3,4 cases, possibly via refined reductions or by identifying additional structural restrictions (e.g., monotonicity of operators) that could yield polynomial‑time algorithms. Third, from a practical standpoint, planners that operate on chain‑structured problems should either enforce binary domains or employ preprocessing techniques that collapse larger domains into binary equivalents, thereby staying within the known tractable regime.

In summary, the paper makes a significant theoretical contribution by pinpointing the exact point at which planning over chain causal graphs transitions from polynomial‑time solvable (k = 2) to NP‑hard (k ≥ 5). This advances our understanding of the delicate interplay between causal graph topology and variable domain size, and it provides a clear direction for future work aimed at closing the remaining complexity gap.