Counting Subgraphs and Induced Subgraphs in Somewhere Dense Graphs

Grunnleggende konsepter

The complexity of counting subgraphs and induced subgraphs of a small pattern graph H in a large host graph G is fully classified, with dichotomies between fixed-parameter tractable and #W[1]-hard cases, when the host graph G is restricted to be somewhere dense.

Sammendrag

The paper studies the problems of counting copies (#Sub(H→G)) and induced copies (#IndSub(H→G)) of a small pattern graph H in a large host graph G. The main results present exhaustive and explicit complexity classifications for these problems when the host graph G is restricted to be a somewhere dense graph class.
Key highlights:

The problems of counting k-matchings (#Match(G)) and k-independent sets (#IndSet(G)) are identified as the minimal hard cases for #Sub(H→G) and #IndSub(H→G) respectively, when G is a monotone and somewhere dense graph class.

Dichotomy theorems are proved, showing that #Match(G) and #IndSet(G) are fixed-parameter tractable if and only if G is nowhere dense. Otherwise, they are #W[1]-hard and cannot be solved in time f(k) · |G|o(k/log k) for any function f.

These results subsume and significantly strengthen previous hardness results for counting subgraphs and induced subgraphs in restricted graph classes like bipartite graphs, F-colorable graphs, and degenerate graphs.

The proofs use a novel approach based on graph fractures and colorful tensor products, which allows for simpler and more general hardness proofs compared to previous techniques.

For homomorphism counting #Hom(H→G), a dichotomy is proved showing that the problem is FPT if G is nowhere dense or if the treewidth of H is bounded, and #W[1]-hard otherwise.

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Counting Subgraphs in Somewhere Dense Graphs

by Marco Bressa... klokken arxiv.org 04-15-2024

https://arxiv.org/pdf/2209.03402.pdf

Dypere Spørsmål

Can the techniques developed in this work be extended to obtain complexity classifications for other parameterized counting problems beyond subgraph and induced subgraph counting

The techniques developed in the work can potentially be extended to obtain complexity classifications for other parameterized counting problems beyond subgraph and induced subgraph counting. The key lies in adapting the interpolation technique used in the study to other counting problems. By formulating the problem in a parameterized framework and identifying suitable graph invariants, it may be possible to apply similar reduction strategies to analyze the complexity of different counting tasks. However, the applicability of these techniques would depend on the specific problem at hand and the availability of appropriate invariants and reduction methods.

Are there natural graph invariants, beyond treewidth, that can characterize the tractability of #Hom(H→G) when the host graph class G is not necessarily monotone or hereditary

While treewidth is a crucial graph invariant that plays a significant role in characterizing the tractability of #Hom(H→G), there are other natural graph invariants that could potentially provide insights into the complexity of the problem. For instance, parameters such as the vertex cover number, the chromatic number, or the girth of the graph could be explored to understand the computational complexity of counting homomorphisms in a broader context. By investigating how these invariants interact with the structure of the host graph class G, it may be possible to identify additional criteria that determine the tractability of #Hom(H→G) in various settings.

What are the implications of these results for practical applications of subgraph counting, such as in network analysis, biology, or data mining

The results of the study have significant implications for practical applications of subgraph counting in various fields such as network analysis, biology, and data mining. By understanding the complexity of counting subgraphs in different graph classes, researchers and practitioners can make informed decisions about the feasibility of solving these problems efficiently. The insights gained from the complexity classifications can guide the development of more efficient algorithms for subgraph counting tasks, leading to improved performance in real-world applications. For example, in network analysis, the ability to determine when counting subgraphs is tractable can help in identifying patterns and structures in large networks more effectively. Similarly, in biology and data mining, efficient subgraph counting algorithms can aid in analyzing complex data sets and extracting valuable information for research and decision-making purposes. Overall, the findings of the study provide a theoretical foundation for optimizing subgraph counting algorithms in practical scenarios.