Deterministic Subgraph Detection in Broadcast CONGEST

We present simple deterministic algorithms for subgraph finding and enumeration in the broadcast CONGEST model of distributed computation: - For any constant k, detecting k-paths and trees on k nodes can be done in O(1) rounds. - For any constant k, detecting k-cycles and pseudotrees on k nodes can be done in O(n) rounds. - On d-degenerate graphs, cliques and 4-cycles can be enumerated in O(d + log n) rounds, and 5-cycles in O(d2 + log n) rounds. In many cases, these bounds are tight up to logarithmic factors. Moreover, we show that the algorithms for d-degenerate graphs can be improved to O(d/logn) and O(d2/logn), respect- ively, in the supported CONGEST model, which can be seen as an intermediate model between CONGEST and the congested clique

Deterministic subgraph detection in broadcast CONGEST

We present simple deterministic algorithms for subgraph finding and enumeration in the broadcast CONGEST model of distributed computation: For any constant k, detecting k-paths and trees on k nodes can be done in O(1) rounds. For any constant k, detecting k-cycles and pseudotrees on k nodes can be done in O(n) rounds. On d-degenerate graphs, cliques and 4-cycles can be enumerated in O(d+log n) rounds, and 5-cycles in O(d2 + log n) rounds. In many cases, these bounds are tight up to logarithmic factors. Moreover, we show that the algorithms for d-degenerate graphs can be improved to O(d/ log n) and O(d2/log n), respectively, in the supported CONGEST model, which can be seen as an intermediate model between CONGEST and the congested clique. © 2017 Janne H. Korhonen and Joel Rybicki.Peer reviewe

Towards a complexity theory for the congested clique

The congested clique model of distributed computing has been receiving attention as a model for densely connected distributed systems. While there has been significant progress on the side of upper bounds, we have very little in terms of lower bounds for the congested clique; indeed, it is now know that proving explicit congested clique lower bounds is as difficult as proving circuit lower bounds. In this work, we use various more traditional complexity-theoretic tools to build a clearer picture of the complexity landscape of the congested clique: -- Nondeterminism and beyond: We introduce the nondeterministic congested clique model (analogous to NP) and show that there is a natural canonical problem family that captures all problems solvable in constant time with nondeterministic algorithms. We further generalise these notions by introducing the constant-round decision hierarchy (analogous to the polynomial hierarchy). -- Non-constructive lower bounds: We lift the prior non-uniform counting arguments to a general technique for proving non-constructive uniform lower bounds for the congested clique. In particular, we prove a time hierarchy theorem for the congested clique, showing that there are decision problems of essentially all complexities, both in the deterministic and nondeterministic settings. -- Fine-grained complexity: We map out relationships between various natural problems in the congested clique model, arguing that a reduction-based complexity theory currently gives us a fairly good picture of the complexity landscape of the congested clique

Detecting Cliques in CONGEST Networks

The problem of detecting network structures plays a central role in distributed computing. One of the fundamental problems studied in this area is to determine whether for a given graph H, the input network contains a subgraph isomorphic to H or not. We investigate this problem for H being a clique K_l in the classical distributed CONGEST model, where the communication topology is the same as the topology of the underlying network, and with limited communication bandwidth on the links. Our first and main result is a lower bound, showing that detecting K_l requires Omega(sqrt{n} / b) communication rounds, for every 4 = sqrt{n}, where b is the bandwidth of the communication links. This result is obtained by using a reduction to the set disjointness problem in the framework of two-party communication complexity. We complement our lower bound with a two-party communication protocol for listing all cliques in the input graph, which up to constant factors communicates the same number of bits as our lower bound for K_4 detection. This demonstrates that our lower bound cannot be improved using the two-party communication framework

Beyond Distributed Subgraph Detection: Induced Subgraphs, Multicolored Problems and Graph Parameters

Subgraph detection has recently been one of the most studied problems in the CONGEST model of distributed computing. In this work, we study the distributed complexity of problems closely related to subgraph detection, mainly focusing on induced subgraph detection. The main line of this work presents lower bounds and parameterized algorithms w.r.t structural parameters of the input graph: - On general graphs, we give unconditional lower bounds for induced detection of cycles and patterns of treewidth 2 in CONGEST. Moreover, by adapting reductions from centralized parameterized complexity, we prove lower bounds in CONGEST for detecting patterns with a 4-clique, and for induced path detection conditional on the hardness of triangle detection in the congested clique. - On graphs of bounded degeneracy, we show that induced paths can be detected fast in CONGEST using techniques from parameterized algorithms, while detecting cycles and patterns of treewidth 2 is hard. - On graphs of bounded vertex cover number, we show that induced subgraph detection is easy in CONGEST for any pattern graph. More specifically, we adapt a centralized parameterized algorithm for a more general maximum common induced subgraph detection problem to the distributed setting. In addition to these induced subgraph detection results, we study various related problems in the CONGEST and congested clique models, including for multicolored versions of subgraph-detection-like problems