Erratum to: "On the Approximability and Hardness of Minimum Topic Connected Overlay and Its Special Instances"

As was pointed out in [1], Theorem 8 of the paper On the Approximability and Hardness of Minimum Topic Connected Overlay and Its Special Instances[2] is incorrect. This erratum proves a slightly weaker version of this theorem. In Minimum Topic Connected Overlay (Min-TCO), we are given a set T of topics and a collection U of users. Each user is interested in a set of topics. This relation is expressed by the user interest function INT : U → 2 . Our goal is to find a minimum set of edges between users so that, for each topic, the subgraph determined by users interested in this topic is connected, i. e., users interested in the same topic are connected in a network. Although the general problem is LOGAPX -complete, we show in the following theorem a class of instances on which the problem can be solved yet in polynomial time. Theorem 1. An optimal solution of Min-TCO can be computed in polynomial time if |T | ≤ (1 + ε(|U |))−1 · log8 log8 |U |, for a function ε(n) ≥ 3 log8 log8 log8 n log8 log8 n− 3 log8 log8 log8 n . In other words, Min-TCO can be computed in polynomial time if |T | can be IThis research is partly supported by the Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research, 21500013, 21680001, 22650004, 22700010, 23104511, 23310104, Foundation for the Fusion of Science Technology (FOST) and INAMORI FOUNDATION. The research is also partially funded by SNF grant 200021-132510/1. Email addresses: juraj.hromkovic@inf.ethz.ch (Juraj Hromkovič), t-izumi@nitech.ac.jp (Taisuke Izumi), hirotaka@en.kyushu-u.ac.jp (Hirotaka Ono), monika.steinova@inf.ethz.ch (Monika Steinová), wada@hosei.ac.jp (Koichi Wada) Preprint submitted to Elsevier April 26, 2017 bounded from above by a function f(|U |) ≤ log8 log8 |U | − 3 log8 log8 log8 |U | (for all sufficiently large instances). Proof. Let (U, T, INT) be an instance of Min-TCO such that |T | ≤ (1 + ε(|U |))−1 · log8 log8 |U |. Moreover, |T | > 2, otherwise the problem is solvable in polynomial time. We shorten the notation by setting t = |T | and n = |U |. We reduce our instance (U, T, INT) using the reduction from [1] (Theorem 3) to an instance of size no larger than m := tc · 8, for some constant c ∈ N which is fixed by Theorem 3 of [1]. number of users in this instance cannot be larger than m as well. Furthermore the reduction never adds topics and hence the number of topics in the reduced instance cannot be larger than t. On this smaller instance, we exhaustively search over all possible solutions and we pick the one which is minimal. Observe that the optimal solution of our reduced instance cannot have more than t(m − 1) edges—this many edges has a feasible solution which merges together a spanning tree of each topic. Hence, in our exhaustive search, we try all possible sets of 1 ≤ i ≤ t(m− 1) edges and we verify the topic-connectivity requirements for such sets. The verification of the topic-connectivity property can be done in polynomial time per set. Hence, the proof that the exhaustive search is polynomial boils down to a proof that the number of checked sets is polynomial. The number of sets the search checks can be bounded as follows: t(m−1) ∑