The Greedy Algorithm Is Not the Problem

The common criticism leveled against greedy algorithms is that they “fail” by producing results that are not globally optimal. This critique, however, is fundamentally misdirected. It confuses the flawless execution of a command with the flawed formulation of the command itself. The failure lies not with the algorithm, but with the model it is tasked to serve.

The Algorithm’s Domain: Faithful Execution

An algorithm is a tool of execution, not of judgment. It operates on a singular, rigid instruction: at each step, select the choice that is locally optimal according to a user-supplied metric. Its performance, therefore, can only be evaluated against this directive.

If an algorithm is told to optimize for metric A, its success is measured by its adherence to A. If this process results in a suboptimal outcome for a different metric B, this is not an algorithmic failure. It is the correct, faithful consequence of its given instructions. The machine has worked perfectly.

The Modeler’s Domain: Epistemological Integrity

The metrics of total value, v, and value-density, v/w, are not simply different calculations; they are epistemologically isolated. This means knowledge about one cannot be used to form justified conclusions about the other. They represent incommensurable frameworks of value.

Total value is an extensive quantity, concerned with the final, aggregate sum. Value-density is an intensive ratio, concerned with marginal efficiency or quality. Because these two conceptions of “good” are fundamentally disconnected, there is no sound inferential path from optimizing one to achieving an optimal result in the other.

To use v/w as a proxy for v is therefore not a tactical approximation. It is a category error—a mistake in the logical structure of the problem itself.

Conclusion: A Failure of Modeling, Not Mechanics

The supposed failure of a greedy algorithm is, in reality, a misdiagnosis. The breakdown does not occur in the mechanics of the algorithm, which performs its function with perfect fidelity.

The actual failure is epistemological. It is an error made by the modeler who selects a metric that is categorically misaligned with the true, intended objective. This is not a weakness in the algorithm. It is a flaw in the reasoning used to formulate the problem—a substitution of one theory of value for another without logical or structural justification.