A Faster Branching Algorithm for the Maximum $k$-Defective Clique Problem

• URL: http://arxiv.org/abs/2407.16588v2
• Date: Wed, 24 Jul 2024 02:44:58 GMT
• Title: A Faster Branching Algorithm for the Maximum $k$-Defective Clique Problem
• Authors: Chunyu Luo, Yi Zhou, Zhengren Wang, Mingyu Xiao,
• Abstract summary: A $k$-defective clique of an un branching graph $G$ induces a nearly complete graph with a maximum of $k$ missing edges. The maximum $k$-defective clique problem, which asks for the largest $k$-defective clique from the given graph, is important in many applications, such as social and biological network analysis.
• Score: 18.720357905876604
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: A $k$-defective clique of an undirected graph $G$ is a subset of its vertices that induces a nearly complete graph with a maximum of $k$ missing edges. The maximum $k$-defective clique problem, which asks for the largest $k$-defective clique from the given graph, is important in many applications, such as social and biological network analysis. In the paper, we propose a new branching algorithm that takes advantage of the structural properties of the $k$-defective clique and uses the efficient maximum clique algorithm as a subroutine. As a result, the algorithm has a better asymptotic running time than the existing ones. We also investigate upper-bounding techniques and propose a new upper bound utilizing the \textit{conflict relationship} between vertex pairs. Because conflict relationship is common in many graph problems, we believe that this technique can be potentially generalized. Finally, experiments show that our algorithm outperforms state-of-the-art solvers on a wide range of open benchmarks.

Related papers

• Fast Maximum $k$-Plex Algorithms Parameterized by Small Degeneracy Gaps [30.06993761032835]
  The maximum $k$-plex problem is important but computationally challenging in applications such as graph mining and community detection. We present an exact algorithm parameterized by $g_k(G)$, which has the worst-case running time in the size of the input graph and exponential in $g_k(G)$. We further extend our discussion to an even smaller parameter $cg_k(G)$, the gap between the community-degeneracy bound and the size of the maximum $k$-plex.
  arXiv  Detail & Related papers  (2023-06-23T01:28:24Z)
• Online Learning with Feedback Graphs: The True Shape of Regret [82.00098840619847]
  We prove that the minimax regret is proportional to $R*$ for any graph and time horizon $T$. Introducing an intricate exploration strategy, we define the mainAlgorithm algorithm that achieves the minimax optimal regret bound.
  arXiv  Detail & Related papers  (2023-06-05T15:35:00Z)
• Improved Exact and Heuristic Algorithms for Maximum Weight Clique [1.2074552857379273]
  We propose improved exact and labeling algorithms for solving the maximum weight clique problem. Our algorithms interleave successful techniques with novel data reduction rules that use local graph structure. We evaluate our algorithms on a range of synthetic and real-world graphs, and find that they outperform the current state of the art on most inputs.
  arXiv  Detail & Related papers  (2023-02-01T14:02:06Z)
• Differentially-Private Hierarchical Clustering with Provable Approximation Guarantees [79.59010418610625]
  We study differentially private approximation algorithms for hierarchical clustering. We show strong lower bounds for the problem: that any $epsilon$-DP algorithm must exhibit $O(|V|2/ epsilon)$-additive error for an input dataset. We propose a private $1+o(1)$ approximation algorithm which also recovers the blocks exactly.
  arXiv  Detail & Related papers  (2023-01-31T19:14:30Z)
• Correlation Clustering Algorithm for Dynamic Complete Signed Graphs: An Index-based Approach [9.13755431537592]
  In this paper, we reduce the complexity of approximating the correlation clustering problem from $O(mtimesleft( 2+ alpha (G) right)+n)$ to $O(m+n)$ for any given value of $varepsilon$ for a complete signed graph. Our approach gives the same output as the original algorithm and makes it possible to implement the algorithm in a full dynamic setting.
  arXiv  Detail & Related papers  (2023-01-01T10:57:36Z)
• Improved High-Probability Regret for Adversarial Bandits with Time-Varying Feedback Graphs [62.52390282012508]
  We study high-probability regret bounds for adversarial $K$-armed bandits with time-varying feedback graphs over $T$ rounds. We develop an algorithm that achieves the optimal regret $widetildemathcalO((sum_t=1Talpha_t)1/2+max_tin[T]alpha_t]$ with high probability. We also develop the first algorithm that achieves the optimal high-probability regret bound for weakly observable graphs.
  arXiv  Detail & Related papers  (2022-10-04T04:36:15Z)
• Listing Maximal k-Plexes in Large Real-World Graphs [21.79007529359561]
  Listing dense subgraphs in large graphs plays a key task in varieties of network analysis applications like community detection. In this paper, we continue the research line of listing all maximal $k$-plexes and maximal $k$-plexes of prescribed size. Our first contribution is algorithm ListPlex that lists all maximal $k$-plexes in $O*(gammaD)$ time for each constant $k$, where $gamma$ is a value related to $k$ but strictly smaller than 2, and $D$ is the degeneracy of the graph that is far less
  arXiv  Detail & Related papers  (2022-02-17T16:25:56Z)
• Fast Graph Sampling for Short Video Summarization using Gershgorin Disc Alignment [52.577757919003844]
  We study the problem of efficiently summarizing a short video into several paragraphs, leveraging recent progress in fast graph sampling. Experimental results show that our algorithm achieves comparable video summarization as state-of-the-art methods, at a substantially reduced complexity.
  arXiv  Detail & Related papers  (2021-10-21T18:43:00Z)
• Learning Sparse Graph with Minimax Concave Penalty under Gaussian Markov Random Fields [51.07460861448716]
  This paper presents a convex-analytic framework to learn from data. We show that a triangular convexity decomposition is guaranteed by a transform of the corresponding to its upper part.
  arXiv  Detail & Related papers  (2021-09-17T17:46:12Z)
• Random Graph Matching with Improved Noise Robustness [2.294014185517203]
  We propose a new algorithm for graph matching under probabilistic models. Our algorithm recovers the underlying matching with high probability when $alpha le 1 / (log log n)C$. This improves the condition $alpha le 1 / (log n)C$ achieved in previous work.
  arXiv  Detail & Related papers  (2021-01-28T02:39:27Z)
• Online Dense Subgraph Discovery via Blurred-Graph Feedback [87.9850024070244]
  We introduce a novel learning problem for dense subgraph discovery. We first propose a edge-time algorithm that obtains a nearly-optimal solution with high probability. We then design a more scalable algorithm with a theoretical guarantee.
  arXiv  Detail & Related papers  (2020-06-24T11:37:33Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.