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Continuing the series of articles (... well actually is just the second one) on restricted combinatorial objects constrained by a recursively defined statistic and initiated by [this work](https://papers-gamma.link/paper/38/Dyck%20paths%20with%20a%20first%20return%20decomposition%20constrained%20by%20height) we submitted to review a new paper of subject.

Il y a Lukasuewicz à faire maintenant ... 😀

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Interesting paper around the question: "Can machine learning be used to cheaply detect and exploit structure in practically relevant instances of NP-hard problems that come from the same distribution?".
The problem of listing all maximum cliques in a graph is considered. A binary classifier is trained to discriminate between a node belonging to a maximum clique and a node that do not belong to a maximum clique: logistic regression is used with node features such as the total number of edges, node degree, clustering coefficient and trained on a set of graphs. The classifier is then used to prune the search space of a test graph: nodes that are labeled "not belonging to a maximum clique" by the classifier are removed. Some speedup is observed when comparing the running time of state-of-the-art methods on the original graph (or on a graph pruned using a baseline pruning method (degree pruning)) and on the pruned graph.
# A more challenging baseline pruning
For the baseline pruning: a heuristic is used to find a large clique and then if the found clique has size $k$, all nodes of degree $k-2$ or less are removed. It would have been better to iteratively remove all nodes of degree $k-2$ or less till there is no more such nodes and thus only keeping the $(k-1)$-core of the graph (it seems that only nodes of degree $k-2$ or less in the original graphs are removed). That would lead to much smaller graphs and a more challenging baseline pruning. How good is this k-core pruning?
In table 1, the speedup obtained with degree pruning is not reported for igraph and EmMCE. Is the speedup obtained by the proposed pruning much better than the one obtained by the degree pruning for these two programs?
# Setting the confidence threshold
The confidence threshold used for the logistic regression classifier is arbitrarily fixed to $q=0.55$. If $q$ is higher, then the list of maximum cliques computed on the pruned graph may not be correct (i.e. it may not be the same as in the original graph as nodes belonging to a maximum clique would be pruned). It is not clear to me why all results are exact in table 1: just luck? How about using $q=0.6$ or $0.7$? Parameter sensitivity is not discussed.
Note that if there is a single maximum clique of size $k$ and many cliques of size $k-1$ then removing a node from the maximum clique will completely change the output (which may then be many cliques of size $k-1$ instead of a single clique of size $k$).

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