{'fraction_non_alphanumeric': 0.02001906577693041, 'fraction_numerical': 0.019065776930409915, 'mean_word_length': 4.769230769230769, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Integer models and upper bounds for the 3‐club problem

Given an undirected graph, the k ‐club problem seeks a maximum cardinality subset of nodes that induces a subgraph with diameter at most k. We present two new formulations for the 3‐club problem: one is compact and the other has a nonpolynomial number of constraints. By defining an integer compact relaxation of the second formulation, we obtain a new upper bound on the 3‐club optimum that improves on the 3‐clique number bound. We derive new families of valid inequalities for the 3‐club polytope and use them to strengthen the LP relaxations of the new models. The computational study is performed on 120 graphs with up to 200 nodes and edge densities reported in the literature to produce difficult instances of the 3‐club problem. The results show that the new compact formulation is competitive with the exact solution methods reported in the literature, and that a large proportion of the LP gap is bridged with the new valid inequalities. © 2012 Wiley Periodicals, Inc. NETWORKS, 2012