Modern automated heuristic discovery (AHD) systems increasingly rival hand-crafted heuristics by leveraging Large Language Models (LLMs) to iteratively mutate and refine solutions. A natural assumption is that more powerful and more expensive models consistently yield better heuristics. We challenge this empirically: across multiple NP-hard combinatorial optimization problems, we find no consistent advantage for larger, costlier models. This raises a deeper question: given a pool of LLMs spanning frontier and freely hosted or open-weight models, how should one adaptively select which model to invoke at each step of the search? The challenge is non-trivial. The optimal LLM may shift over the course of a run, and because one LLM's output seeds the context for subsequent calls, each model's reward signal is non-stationary and entangled across the heuristic population, thus violating the independence assumptions of classical online decision-making settings. We address this through Ballad (Bandit-based LLM Routing for Automated Heuristic Discovery), an online bandit algorithm that learns to orchestrate a pool of LLMs into an ensemble that matches or exceeds what any soloist LLM achieves alone. This orchestration is powered by an LLM selection policy that favors rare, high-scoring heuristics over stable averages, a DAG-based credit propagation mechanism through the genealogy of the heuristic population, and discounting of stale observations to remain attuned to each model's current utility. Across five NP-hard combinatorial optimization problems and a pool of four models of varying capability and cost, Ballad not only matches the best individual LLM in hindsight but, in several cases, surpasses every individual model while reducing API cost by up to 65.6% relative to always using the strongest model.
PageRank is widely used to quantify the importance of nodes in directed networks. In applications such as recommendation systems, it is increasingly important that the PageRank distribution provides a fair representation of node exposure. This motivates the problem of making minimal changes to a graph so that the PageRank values of a given set of target nodes $T$ match a desired vector $\boldsymbol{t}$. In this paper, we study this problem under an edge-addition budget $B$, where exactly $B$ edges may be added to the graph. We show that deciding whether a multiset of exactly $B$ added edges can make the target nodes attain their desired absolute PageRank scores is NP-complete. After this, we define the $\ell_1$ Target PageRank problem in which we seek to minimize the $\ell_1$ distance between the target vector and resultant vector. We show that this optimization problem admits no polynomial-time multiplicative constant-factor approximation unless $P=NP$. Motivated by these hardness results, we propose two greedy heuristics for approaching the target PageRank vector. We additionally propose a Random Walk Hungarian (RWH) heuristic that addresses the addition of edges between dissimilar nodes by incorporating a penalty for such edges. Finally, we evaluate the proposed heuristics on several real-world and synthetic datasets and compare their performance in terms of PageRank accuracy, computational efficiency, and the relevance of the added edges.