Knapsack and subset sum problems in nilpotent, polycyclic, and co-context-free groups

In their paper [22], Myasnikov, Nikolaev, and Ushakov started the investigation of classical discrete integer optimization problems in general non-commutative groups. Among other problems, they introduced for a finitely generated (f.g.) group G the knapsack problem and the subset sum problem. The input for the knapsack problem is a sequence of group elements g1, . . . , gk, g ∈ G and it is asked whether there exists a solution (x1, . . . , xk) ∈ N of the equation g1 1 · · · gk k = g. For the subset sum problem one restricts the solution to {0, 1}k. For the particular caseG = Z (where the additive notation x1·g1+· · ·+xk ·gk = g is usually prefered) these problems are NP-complete if the numbers g1, . . . , gk, g are encoded in binary representation. For subset sum this is shown in Karp’s classical paper [13]. The statement for knapsack (in the above version) can be found in [10]. In [22] the authors enocde elements of the finitely generated group G by words over the group generators and their inverses. For G = Z this representation corresponds to the unary encoding of integers. It is known that for unary encoded integers, knapsack and subset sum over Z can be both solved in polynomial time, and the precise complexity is DLOGTIME-uniform TC [6], which is a very small complexity class that roughly speaking captures the complexity of multiplying binary coded integers. In [22], Myasnikov et al. proved the following new results: