Knapsack in Graph Groups, HNN-Extensions and Amalgamated Products

Knapsack in Graph Groups, HNN-Extensions and Amalgamated Products Markus Lohrey1 and Georg Zetzsche∗2 1 2 Universität Siegen, Germany LSV, CNRS & ENS Cachan, Université Paris-Saclay, France Abstract 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. 1998 ACM Subject Classification F.2.2 Nonnumerical Algorithms and Problems Keywords and phrases Graph groups, HNN-extensions, amalgamated products, knapsack Digital Object Identifier 10.4230/LIPIcs.STACS.2016.50 1 Introduction In their paper [36], Myasnikov, Nikolaev, and Ushakov started the investigation of classical discrete optimization problems, which are classically formulated over the integers, for arbitrary in general non-commutative groups. Among other problems, they introduced for a finitely generated 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 (specified by finite words over the generators of G) and it is asked whether there exists a solution (x1 , . . . , xk ) ∈ Nk of the equation g1x1 · · · gkxk = g. For the subset sum problem one restricts the solution to {0, 1}k . For the particular case G = Z (where the additive notation x1 · g1 + · · · + xk · gk = g is usually preferred) these problems are NP-complete if the numbers g1 , . . . , gk , g are encoded in binary representation. For subset sum, this is a classical result from Karp’s seminal paper [24] on NP-completeness. Knapsack for integers is usually formulated in a more general form in the literature; NP-completeness of the above form (for binary encoded integers) was shown in [17], where the problem was called multisubset sum.1 Interestingly, if we consider subset sum for the group G = Z, but encode the input numbers g1 , . . . , gk , g in unary notation, then the problem is in DLOGTIME-uniform TC0 (a small subclass of polynomial time and even of logarithmic space that captures the complexity of multiplication of binary encoded numbers) [14], and the same holds for knapsack, since the instance x1 · g1 + · · · + xk · gk = g has a ∗ This author is supported by a fellowship within the Postdoc-Program of the German Academic Exchange Service (DAAD). 1 Note that if we ask for a solution (x1 , . . . , xk ) in Zk , then knapsack can be solved in polynomial time (even for binary encoded integers) by checking whether gcd(g1 , . . . , gk ) divides g. © Markus Lohrey and Georg Zetzsche; licensed under Creative Commons License CC-BY 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016). Editors: Nicolas Ollinger and Heribert Vollmer; Article No. 50; pp. 50:1–50:14 Leibniz International Proceedings in Informatics Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany 50:2 Knapsack in Graph Groups, HNN-Extensions and Amalgamated Products solution if and only if it has a solution with xi ≤ k · (max{g1 , . . . , gk , g})3 [37]. This allows to reduce unary knapsack to unary subset sum. See [21] for related results. In [36] the authors encode 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. Among others, the following results were shown in [36]: Subset sum and knapsack can be solved in polynomial time for every hyperbolic group. Subset sum for a virtually nilpotent group (a finite extension of a nilpotent group) can be solved in polynomial time. For the following groups, subset sum is NP-complete (whereas the word problem can be solved in polynomial time): free metabelian non-abelian groups of finite rank, the wreath product Z o Z, Thompson’s group F , and the Baumslag-Solitar group BS(1, 2). Further results on knapsack and subset sum have been recently obtained in [27]: For a virtually nilpotent group, subset sum belongs to NL (nondeterministic logspace). There is a nilpotent group of class 2 (in fact, a direct product of sufficiently many copies of the discrete Heisenberg group H3 (Z)), for which knapsack is undecidable. The knapsack problem for the discrete Heisenberg group H3 (Z) is decidable. In particular, together with the previous point it follows that decidability of knapsack is not preserved under direct products. There is a polycyclic group with an NP-complete subset sum problem. The knapsack problem is decidable for every co-context-free group. The focus of this paper will be on the knapsack problem. We will prove that this problem can be solved in NP for every graph group. Graph groups are also known as right-angled Artin groups or free partially commutative groups. A graph group is specified by a finite simple graph. The vertices are the generators of the group, and two generators a and b are allowed to commute if and only if a and b are adjacent. Graph groups somehow interpolate between free groups and free abelian groups and can be seen as a group counterpart of trace monoids (free partially commutative monoids), which have been used for the specification of concurrent behavior. In combinatorial group theory, graph groups are currently an active area of research, mainly because of their rich subgroup structure (see e.g. [5, 8, 16]). To prove that knapsack belongs to NP for a graph group, we proceed in two steps: We first show that if an instance g1x1 · · · gkxk = g has a solution in a graph group, then it has a solution where every xi is bounded exponentially in the input length (the total length of all words representing the group elements g1 , . . . , gk , g). We then guess the binary encodings of numbers n1 , . . . , nk that are bounded by the exponential bound from the previous point and verify in polynomial time the identity g1n1 · · · gknk = g. The latter problem is an instance of the so-called compressed word problem for a graph group. This is the classical word problem, where the input group element is given succinctly by a so-called straight-line program (SLP), which is a context-free grammar that produces a single word (here, a word over the group generators and their inverses). An SLP with n productions in Chomsky normal form can produce a string of length 2n . Nevertheless, the compressed word problem for a fixed graph group can be solved in polynomial time (see [29] for details). In fact, our proof yields a stronger result: First, it yields an NP procedure for solving knapsack-like equations h0 g1x1 h1 · · · hk−1 gkxk hk = 1 where some of the variables x1 , . . . , xk are (...truncated)