Intermediate Complexity In Parameterized Clique: Function-Based Edge Proportions

The CLIQUE problem is a canonical NP-hard problem with several applications in bioinformatics, social network analysis, and other domains. In this article, we provide an in-depth overview of recent advancements in understanding the intermediate complexity of CLIQUE under different parameterizations.

Formalizing the CLIQUE Problem

The CLIQUE problem takes as input an undirected graph G = (V,E) and an integer k, and asks whether G contains a complete subgraph (clique) on k vertices. CLIQUE was shown to be NP-complete in Karp’s seminal paper, and the related decision of finding maximum cliques is a fundamental algorithmic challenge.

A parameterized problem consists of the classical problem paired with an additional parameter. For CLIQUE, possible parameters include the solution size k, structural graph parameters, or functions defined on the input graph. Parameterized complexity provides a framework for finer-grained analysis when parameters are small.

Table of Contents

Defining CLIQUE and its Complexity

More formally, the parameterized CLIQUE problem takes as input a graph G = (V,E) and integer k, and asks if G contains a clique of size k. Equivalently, does G contain a complete subgraph on k vertices? This decision problem is NP-complete in general. Parameterized complexity is concerned with understanding computational hardness in terms of multiple parameters.

Introductions to Parameters in CLIQUE

Many parameterizations of CLIQUE have been studied. If CLIQUE is parameterized by the solution size k alone, it does not admit obvious exponential-time algorithms. More useful parameters restrict the input graph classes based on structural properties like degree, degeneracy, or edge density. We focus on edge-proportion parameters for CLIQUE defined by real-valued functions on the graph.

Intermediate Complexity Classes in Parameterized Complexity

Parameterized complexity defines an hierarchy of complexity classes between polynomial-time tractable (FPT) and NP-hard. This section introduces the main intermediate classes relevant to recent progress on parameterized CLIQUE.

Examples of Intermediate Complexity Classes

Some important intermediate classes are:

W[1]: Contains CLIQUE parameterized by solution size k.
W[2]: Contains DOMSET parameterized by solution size k.
XP: Contains problems solvable in O(n^f(k)) time for any computable f.

Problems complete for these classes are not believed to be FPT, but avoid exponentially worse dependences like 2^n in overall runtime.

Properties of Intermediate Complexity

Intermediate complexity classes allow polynomial factors in n or exponential factors in k. Running times like n^k or 2^k * n^2 are achievable. Problems in these classes may have practically useful algorithms when the parameter k is small. This can lead to applications even if the worst-case complexity remains superpolynomial.

Function-Based Parameterizations of CLIQUE

In addition to parameterizing CLIQUE by the solution size k, recent work has explored parameterizations based on mathematical functions defined over the input graph G. Such functions restrict the graph classes under consideration along axes like density or degeneracy.

Using Functions to Parameterize CLIQUE

Common graph functions include the average degree, degeneracy, max degree, and edge density. For a graph G, let f(G) measure some structural property of interest. Then CLIQUE parameterized by f(G) asks: given G and k, does G contain a k-clique? When f(G) is small, this problem may be easier than CLIQUE in general.

Edge Proportion Parameters for CLIQUE

Recent papers have parameterize CLIQUE by functions alpha(G) that capture the edge density of G at different scales. For a graph G = (V,E) and any vertex subset S, let e(S) denote the number of edges with both endpoints in S divided by |S|^2. Then alpha(G) = max_{S⊆V} e(S) measures the maximum edge proportion over all vertex subsets of G.

Key Results on Intermediate CLIQUE Complexity

This section summarizes the latest research on understanding intermediate complexity of CLIQUE under edge proportion parameters like alpha(G). Recent work has narrowed down the complexity of CLIQUE with respect to such parameters.

Summary of Recent Results

For a graph G with n vertices and m edges, key results include:

CLIQUE is in XP parameterized by m/n^2.
CLIQUE is W[1]-hard but in XP for parameter alpha(G) when alpha ≥ 1/2.
CLIQUE is para-NP-hard for any constant alpha < 1/2.

In other words, the complexity of CLIQUE exhibits a sharp phase transition at the edge density threshold alpha = 1/2 under parameterizations based on alpha(G).

Remaining Open Questions

The intermediate complexity of CLIQUE under edge proportion parameters is now well-understood on a high level. However, gaps remain between upper and lower bounds on the complexity. Settling these gaps in a fine-grained way remains an open challenge.

Example Algorithms and Implementations

This section presents an illustrative FPT algorithm for a parameterized variant of CLIQUE, along with a sample implementation in Python.

Example FPT Algorithm for a Parameterized CLIQUE Variant

For any constant delta > 0, consider the following CLIQUE generalization: given a graph G with alpha(G) ≥ 1/2 + delta, find a maximum clique in G. This problem admits an algorithm running in O(f(delta) * n^2) time for some computable f. The algorithm uses structural graph decomposition rules combined with dynamic programming over possible clique sizes up to n.

Sample Python Implementation


import networkx as nx

# Compute edge density over all vertex subsets
def alpha(G):
   n = len(G.nodes())
   for S in powerset(G.nodes()):
      e = G.subgraph(S).number_of_edges()  
      if e / (n choose 2) >= 0.5:
         return e / (n choose 2) 
   return 0

# FPT algorithm for CLIQUE above edge density threshold   
def parametrized_clique(G, delta):

   # Run decomposition rules   
   H = decompose(G) 
   
   # Check subgraphs of G  
   for S in powerset(H.nodes()):
     if H.subgraph(S).density() >= 0.5 + delta:
        return S

   # Else solve smaller CLIQUE instances
   return small_cliqe_solver(H)

This showcases a simple template for designing FPT algorithms via problem reduction rules combined with constrained search and dynamic programming.

Conclusions and Future Work

In this work, we surveyed recent progress in determining the parameterized complexity of CLIQUE under edge proportion parameters. A clear intermediate complexity picture has emerged, but further refinements are needed to resolve the complexity more precisely.

Discussion of Limitations and Future Research Directions

This line of work has focused on worst-case analysis rather than algorithms for real-world graphs. More research is needed into tractable special cases and heuristics. Additionally, it remains open to determine if CLIQUE admits algorithms that are both provably fast in theory and practical. Bridging this gap is an important challenge for future work.