An Exact Algorithm with New Upper Bounds for the Maximum k-Defective Clique Problem in Massive Sparse Graphs

Abstract

The Maximum k-Defective Clique Problem (MDCP), as a clique relaxation model, has been used to solve various problems. Because it is a hard computational task, previous works can hardly solve the MDCP for massive sparse graphs derived from real-world applications. In this work, we propose a novel branch-and-bound algorithm to solve the MDCP based on several new techniques. First, we propose two new upper bounds of the MDCP as well as corresponding reduction rules to remove redundant vertices and edges. The proposed reduction rules are particularly useful for massive graphs. Second, we present another new upper bound by counting missing edges between fixed vertices and an unfixed vertex for cutting branches. We perform extensive computational experiments to evaluate our algorithm. Experimental results show that our reduction rules are very effective for removing redundant vertices and edges so that graphs are reduced greatly. Also, our algorithm can solve benchmark instances efficiently, and it has significantly better performance than state-of-the-art algorithms.

Efficient Maximum k-Defective Clique Computation with Improved Time Complexity

“…RR6. Given a graph 𝐺, for any edge (𝑢, 𝑣) ∈ 𝐸 (𝐺) whose number of common neighbors in 𝐺 is less than 𝑙𝑏 − 𝑘 − 1, we can remove the edge (𝑢, 𝑣) from 𝐺 [16].…”

“…The state-of-the-art time complexity for maximum 𝑘-defective clique computation that beats the trivial O * (2 𝑛 ) time complexity is achieved by the MADEC + algorithm proposed in [11], which runs in O * (𝜎 𝑛 𝑘 ) time where 𝜎 𝑘 < 2 is the largest real root of the equation 𝑥 2𝑘+3 − 2𝑥 2𝑘+2 + 1 = 0. Although a graph coloringbased upper bound as well as other pruning techniques are proposed in [11] aiming to improve the practical performance of MADEC + , it is shown in [16] that the graph coloring-based upper bound proposed in [16] is ineffective and MADEC + is inefficient in practice especially when 𝑘 ≥ 10. For example, for 𝑘 ≥ 15 on the Facebook graphs collection (please refer to Section 4 for the description of the dataset), even the version of MADEC + that is further optimized by the authors of [16] was still not able to find the maximum 𝑘defective clique for any graph instance with a time limit of 3 hours.…”

“…Although a graph coloringbased upper bound as well as other pruning techniques are proposed in [11] aiming to improve the practical performance of MADEC + , it is shown in [16] that the graph coloring-based upper bound proposed in [16] is ineffective and MADEC + is inefficient in practice especially when 𝑘 ≥ 10. For example, for 𝑘 ≥ 15 on the Facebook graphs collection (please refer to Section 4 for the description of the dataset), even the version of MADEC + that is further optimized by the authors of [16] was still not able to find the maximum 𝑘defective clique for any graph instance with a time limit of 3 hours. With the goal of enhancing the practical performance, the KDBB algorithm is designed in [16] which proposes and incorporates preprocessing as well as multiple pruning techniques.…”

“…For example, for 𝑘 ≥ 15 on the Facebook graphs collection (please refer to Section 4 for the description of the dataset), even the version of MADEC + that is further optimized by the authors of [16] was still not able to find the maximum 𝑘defective clique for any graph instance with a time limit of 3 hours. With the goal of enhancing the practical performance, the KDBB algorithm is designed in [16] which proposes and incorporates preprocessing as well as multiple pruning techniques. Nevertheless, KDBB is still inefficient, and moreover, no time complexity better than O * (2 𝑛 ) has been proved for KDBB.…”

“…• We propose a new upper bound UB1 based on graph coloring, which can be computed in linear time and is much tighter than the upper bounds proposed in [11,16]. (Section 3.…”