[Paper Review] Computing Dense and Sparse Subgraphs of Weakly Closed Graphs

Tomohiro Koana, Christian Komusiewicz+1 more|arXiv (Cornell University)|Jan 1, 2020

Advanced Graph Theory Research2 citations

TL;DR

This paper introduces fixed-parameter tractable (FPT) algorithms for finding dense and sparse subgraphs in weakly γ-closed graphs, a class of graphs that generalizes degeneracy and closure. It shows that problems like clique enumeration, biclique detection, s-plexes, and dominating set variants are FPT when parameterized by γ, with efficient kernelization and improved running times, while also proving kernel lower bounds for Independent Dominating Set.

ABSTRACT

A graph G is weakly γ-closed if every induced subgraph of G contains one vertex v such that for each non-neighbor u of v it holds that |N(u)∩ N(v)| < γ. The weak closure γ(G) of a graph, recently introduced by Fox et al. [SIAM J. Comp. 2020], is the smallest number such that G is weakly γ-closed. This graph parameter is never larger than the degeneracy (plus one) and can be significantly smaller. Extending the work of Fox et al. [SIAM J. Comp. 2020] on clique enumeration, we show that several problems related to finding dense subgraphs, such as the enumeration of bicliques and s-plexes, are fixed-parameter tractable with respect to γ(G). Moreover, we show that the problem of determining whether a weakly γ-closed graph G has a subgraph on at least k vertices that belongs to a graph class 𝒢 which is closed under taking subgraphs admits a kernel with at most γ k² vertices. Finally, we provide fixed-parameter algorithms for Independent Dominating Set and Dominating Clique when parameterized by γ+k where k is the solution size.

Motivation & Objective

• To extend the algorithmic utility of weakly γ-closed graphs beyond clique enumeration to broader classes of dense and sparse subgraph problems.
• To establish fixed-parameter tractability for key NP-hard problems such as Independent Dominating Set, Dominating Clique, and biclique enumeration under the γ parameter.
• To provide kernelization results with tight bounds for subgraph problems closed under induced subgraphs, parameterized by γ and solution size k.
• To explore the limits of kernelization, showing that Independent Dominating Set does not admit polynomial kernels for constant γ under standard assumptions.
• To investigate the complexity of clique relaxations like s-Club in weakly closed graphs, particularly for s ≥ 2.

Proposed method

• Propose a weak closure ordering σ that ensures every induced subgraph has a vertex with fewer than γ common neighbors with any non-neighbor.
• Design a recursive search tree algorithm for Dominating Clique with branching factor at most γ−1 per node, leading to O*( (γ−1)^k ) running time.
• Use a depth-bounded search tree with pruning based on weak closure to enumerate dense subgraphs like cliques and s-plexes efficiently.
• Apply kernelization techniques to reduce instances of subgraph problems closed under induced subgraphs to at most γk² vertices.
• Leverage reductions from λ-Hitting Set to prove kernel lower bounds for Independent Dominating Set under standard complexity assumptions.
• Analyze the complexity of s-Club problems in weakly closed graphs, showing NP-hardness for 2-Club even in 4-closed graphs.

Experimental results

Research questions

• RQ1 Can problems related to finding dense or sparse subgraphs be solved efficiently in weakly γ-closed graphs, and what is the parameterized complexity with respect to γ?
• RQ2 Does the weak closure parameter γ enable kernelization for problems like Independent Dominating Set and Dominating Clique, and what are the limits of such kernels?
• RQ3 Are clique relaxations such as s-Club fixed-parameter tractable in weakly γ-closed graphs, particularly for small s?
• RQ4 Can the weak closure parameter γ be used to design faster algorithms for biclique and s-plex enumeration compared to degeneracy or closure number?
• RQ5 What are the tightest possible kernel sizes for subgraph problems closed under induced subgraphs in weakly γ-closed graphs?

Key findings

• The problem of finding a subgraph on at least k vertices belonging to a hereditary graph class G is kernelizable to at most γk² vertices when parameterized by γ and k.
• Dominating Clique admits an FPT algorithm with running time O*( (γ−1)^k ), which is unlikely to be significantly improved under the Exponential Time Hypothesis.
• Independent Dominating Set does not admit a polynomial kernel for constant γ unless coNP ⊆ NP/poly, indicating inherent limits in kernelization.
• 2-Club is NP-hard even in 4-closed graphs, suggesting that clique relaxations remain hard in weakly closed graphs.
• The weak closure number γ is never larger than degeneracy d+1 or closure number c, making it a potentially superior parameter for algorithm design.
• For split graphs and graphs with bounded clique size, almost tight kernels of size k^O(γ) and k^O(γ²) respectively are known, indicating potential for further kernelization results.