Knapsack in graph groups, HNN-extensions and amalgamated products

It is shown that the knapsack problem, which was introduced by Myasnikov et al. for arbitrary finitely generated groups, can be solved in NP for graph groups. This result even holds if the group elements are represented in a compressed form by SLPs, which generalizes the classical NP-completeness result of the integer knapsack problem. We also prove general transfer results: NP-membership of the knapsack problem is passed on to finite extensions, HNN-extensions over finite associated subgroups, and amalgamated products with finite identified subgroups.

💡 Research Summary

The paper investigates the knapsack problem in the setting of graph groups (also known as right‑angled Artin groups or free partially commutative groups) and establishes that it belongs to the complexity class NP. The authors first formalize exponent equations, a generalization of knapsack equations, of the form
 u₁^{x₁} u₂^{x₂} … uₙ^{xₙ}=v,
where the variables xᵢ are required to be non‑negative integers and the group elements uᵢ, v are given by words over a finite generating set. They prove a crucial “exponential bound” theorem: if such an equation has a solution in a graph group, then there exists a solution where each exponent xᵢ is bounded by an exponential function of the total input length (the sum of the lengths of all words describing the uᵢ and v). This bound is obtained by exploiting trace theory, Levi’s Lemma for traces, and the structure of dependence graphs associated with traces.

With the bound in hand, the authors design an NP algorithm. They nondeterministically guess binary encodings of the exponents (which are of at most exponential size) and verify the equality using the compressed word problem for graph groups. The compressed word problem asks whether two words, each given by a straight‑line program (SLP), represent the same group element. Prior work (Lohrey, Müller, Zetzsche) showed that this problem can be solved in polynomial time for graph groups. Consequently, the knapsack problem—both in its ordinary form (variables must be distinct) and in the more general exponent‑equation form—lies in NP. Moreover, because SLPs can encode binary numbers with linear size overhead, the result subsumes the classical NP‑completeness of integer knapsack (binary encoding) as a special case.

The paper then proves three transfer theorems. First, if a group H is a finite extension of a group G for which compressed exponent equations are in NP, then the same holds for H. Second, for an HNN‑extension G∗{A≅B} where the associated subgroups A and B are finite, the knapsack problem remains in NP. Third, for an amalgamated product G₁∗{C}G₂ with a finite identified subgroup C, NP‑membership is preserved. The proofs rely on the fact that solutions in the base groups can be lifted to the larger constructions without increasing the exponent bound beyond a polynomial factor, and on the semilinearity of solution sets in the base groups.

A significant structural result is that the set of all solutions to an exponent equation over a graph group is semilinear, and an effective semilinear representation can be computed. This aligns with known results for many groups (e.g., co‑context‑free groups). In contrast, the discrete Heisenberg group H₃(ℤ) provides a counterexample: while solvability of exponent equations is decidable, the solution set is not semilinear but defined by a single quadratic Diophantine equation.

Finally, the authors establish NP‑hardness for the direct product of two free groups of rank 2, i.e., F₂ × F₂, which corresponds to the graph group defined by a 4‑cycle. They show that both knapsack and subset‑sum are NP‑complete even when the group elements are given as plain words (or SLPs) and variables may take arbitrary integer values (instead of only non‑negative). This resolves an open problem concerning subset‑sum for this group.

In summary, the paper delivers a comprehensive picture: (1) knapsack and more general exponent equations are in NP for all graph groups, even under compressed input; (2) this NP‑membership transfers to finite extensions, HNN‑extensions over finite associated subgroups, and amalgamated products over finite identified subgroups; (3) solution sets are semilinear in graph groups; and (4) NP‑completeness is shown for a natural non‑abelian graph group (F₂ × F₂). These contributions deepen the understanding of algorithmic problems at the intersection of combinatorial group theory and discrete optimization.