Brief Announcement: Classification of Distributed Binary Labeling Problems

Abstract

We present a complete classification of the deterministic distributed time complexity for a family of graph problems: binary labeling problems in trees in the usual LOCAL model of distributed computing. These are locally checkable problems that can be encoded with an alphabet of size two in the edge labeling formalism. Examples of binary labeling problems include sinkless orientation, sinkless and sourceless orientation, 2-vertex coloring, and perfect matching. We show that the complexity of any such problem is in one of the following classes: 𝑂 (1), Θ(log 𝑛), Θ(𝑛), or unsolvable. Furthermore, given the description of any binary labeling problem, we can easily determine in which of the four classes it is and what is an asymptotically optimal algorithm for solving it.

Improved Distributed Lower Bounds for MIS and Bounded (Out-)Degree Dominating Sets in Trees

“…As our main technical contribution, for any ≤ Δ for some constant > 0, we prove an Ω(log Δ)-round lower bound for computing -outdegree dominating sets in Δ-regular trees in the port numbering model. 1 The lower bound for the port numbering model is then lifted to the more powerful general LOCAL model by using standard techniques developed in [3,6,[14][15][16]18], leading to the following main result. Theorem 1.…”

“…• Fixed points: A case limit of the similarity approach is the one in which we try to build a lower bound sequence where there is a non-0-round solvable problem that appears more than once in the sequence (i.e., a fixed point). This directly implies an Ω(log ) deterministic and Ω(log log ) randomized lower bound (see e.g., [3]). • Linear growth: Sometimes, the similarity approach does not seem to work, and if the number of labels grows exponentially we do not get a sufficiently strong lower bound.…”

“…These results were afterwards improved and generalized by Brandt and Olivetti [16]. Then, round elimination was used for giving a complete characterization of locally checkable problems that can be encoded in the edge-formalism of the round elimination framework by using at most two labels [3]. Fraigniaud and Paz showed how to use round elimination with the algebraic topology framework [21].…”

“…As an example, let us see how one can encode the maximal independent set problem in the round elimination framework. For this problem, we define Σ = {M, P, O} (note that it has been shown that we need 3 labels in order to encode MIS in this framework [3]). The node constraint N consists of two configurations: one used by the nodes in the MIS and the other used by nodes not in the MIS.…”

Locally Checkable Problems in Rooted Trees

“…While it is known that questions related to the distributed computational complexity of locally checkable problems are undecidable in general graphs [12,22], there is no known obstacle that would prevent one from completely automating the study of locally checkable problems in trees. Achieving this is one of the major open problems in the field: currently only parts of the complexity landscape are known to be decidable [15], and the general decidability results are primarily of theoretical interest; practical automatic techniques are only known for specific families of problems [3,12,16].…”