Efficient Dynamic Algorithms for Feedback Problems in Tournaments

Core Concepts

This paper presents the first dynamic algorithms for the Feedback Arc Set problem and the Feedback Vertex Set problem in tournaments. The algorithms maintain a dynamic tournament under arc reversals and efficiently answer queries about the existence of feedback sets of bounded size.

Abstract

The paper studies feedback problems in dynamic tournaments. The two main problems considered are the Feedback Arc Set problem (FAST) and the Feedback Vertex Set problem (FVST).

The authors first introduce two data structures for dynamically maintaining a tournament and efficiently detecting triangles, which are the key building blocks for their algorithms. The first data structure, DTP[n], supports arc reversals in O(√ADT(T)) time and triangle detection in O(ADT(T)√ADT(T)) time, where ADT(T) is the maximum number of arc-disjoint triangles in the tournament T. The second data structure, DT[n], has a polylogarithmic update time of O(log^2 n) but a query time of O(ADT(T)log^2 n).

For the dynamic FAST problem, the authors provide two algorithms. In the promise model, where the size of the feedback arc set is bounded by a computable function g(K) of the parameter K, they give an O(√g(K)) update and O(3^K K√K) query algorithm. In the general setting, they offer an O(log^2 n) update and O(3^K K log^2 n) query time algorithm.

For the dynamic FVST problem, the authors provide an algorithm in the promise model with O(g^5(K)) update and O(3^K K^3 g(K)) query time, where g(K) is a computable function bounding the size of the feedback vertex set.

The key ideas behind the algorithms are: (1) maintaining a partition of the vertices based on their indegrees, (2) keeping track of empty indegree buckets and back arcs, and (3) reducing the number of long back arcs by removing a small set of "heavy" vertices.

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Key Insights Distilled From

Dynamic Parameterized Feedback Problems in Tournaments

by Anna... at arxiv.org 04-22-2024

https://arxiv.org/pdf/2404.12907.pdf

Deeper Inquiries

How can the dynamic algorithms be extended to handle more general graph classes beyond tournaments

To extend the dynamic algorithms to handle more general graph classes beyond tournaments, we can consider adapting the data structures and algorithms to accommodate the specific properties of the new graph classes. For example, for general directed graphs, we may need to modify the triangle detection data structures to account for cycles and directed paths that are not present in tournaments. Additionally, the update and query operations may need to be adjusted to handle the different connectivity patterns and structural characteristics of the new graph classes. By analyzing the unique features of the target graph class, we can tailor the dynamic algorithms to effectively address the feedback problems within those graphs.

Are there any lower bounds on the update and query times for dynamic feedback problems in tournaments, and how do the presented algorithms compare to these lower bounds

In the context of dynamic feedback problems in tournaments, there are lower bounds on the update and query times that provide a theoretical limit on the efficiency of dynamic algorithms. The presented algorithms can be compared to these lower bounds to assess their performance. If the algorithms achieve a time complexity that matches or improves upon the lower bounds, it indicates that they are optimal or near-optimal in terms of computational efficiency. On the other hand, if the algorithms fall short of the lower bounds, it suggests that there may be room for further optimization or algorithmic improvements to enhance their efficiency.

What other applications or extensions of the triangle detection data structures could be explored beyond the feedback problems considered in this paper

The triangle detection data structures introduced in the paper have applications beyond feedback problems in tournaments. One potential extension could be in the field of social network analysis, where identifying triangles (cycles of length three) can reveal important structural patterns such as clustering and community detection. By leveraging the efficient triangle detection capabilities of these data structures, researchers could develop algorithms for community detection, identifying closely connected groups of nodes within a network. Additionally, the data structures could be applied to problems in computational biology, such as analyzing protein-protein interaction networks to identify functional modules or protein complexes based on triangle motifs. Expanding the use cases of these data structures to diverse fields can unlock new possibilities for graph analysis and optimization.