We study the Combinatorial Group Testing (CGT) problems in a novel game-theoretic framework, with a solution concept of Adversarial Equilibrium (AE).

In this paper, we focus on computing an NE that optimizes a given objective function.

Hierarchical clustering seeks to uncover nested structures in data by constructing a tree of clusters, where deeper levels reveal finer-grained relationships. Traditional methods, including linkage approaches, face three major limitations: (i) they always return a hierarchy, even if none exists, (ii) they are restricted to binary trees, even if the true hierarchy is non-binary, and (iii) they are highly sensitive to the choice of linkage function. In this paper, we address these issues by introducing the notion of a valid hierarchy and defining a partial order over the set of valid hierarchies. We prove the existence of a finest valid hierarchy, that is, the hierarchy that encodes the maximum information consistent with the similarity structure of the data set. In particular, the finest valid hierarchy is not constrained to binary structures and, when no hierarchical relationships exist, collapses to a star tree. We propose a simple two-step algorithm that first constructs a binary tree via a linkage method and then prunes it to enforce validity. We establish necessary and sufficient conditions on the linkage function under which this procedure exactly recovers the finest valid hierarchy, and we show that all linkage functions satisfying these conditions yield the same hierarchy after pruning. Notably, classical linkage rules such as single, complete, and average satisfy these conditions, whereas Ward's linkage fails to do so.

Decision tree algorithms have been among the most popular algorithms for interpretable (transparent) machine learning since the early 1980's.

There are other approaches (e.g., [ Here, if all team members play strategies according to an NE minimizing the adversary's utility, the Eq.(1c) ensures that binary variable This space is represented by Eq.(1), which involves nonlinear terms in Eq.(1a) Section 3.4 shows that our techniques can significantly reduce the time The procedure of CRM is shown in Algorithm 2, which is illustrated in Appendix A. A collection N of subsets of players is a binary collection if: 1. { i | i N } N ; Eqs.(1b)-(1g), (3), and (4) is the space of NEs. Example 1 provides an example of N .

Ben-David et al. [5] to the online setting and explores its general properties.

In this work, we provide the first continuous relaxation of Dasgupta's discrete optimization problem with provable quality guarantees.