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LLM Routing as Reasoning: A MaxSAT View

Son Nguyen, Xinyuan Liu, Ransalu Senanayake

Abstract

Routing a query through an appropriate LLM is challenging, particularly when user preferences are expressed in natural language and model attributes are only partially observable. We propose a constraint-based interpretation of language-conditioned LLM routing, formulating it as a weighted MaxSAT/MaxSMT problem in which natural language feedback induces hard and soft constraints over model attributes. Under this view, routing corresponds to selecting models that approximately maximize satisfaction of feedback-conditioned clauses. Empirical analysis on a 25-model benchmark shows that language feedback produces near-feasible recommendation sets, while no-feedback scenarios reveal systematic priors. Our results suggest that LLM routing can be understood as structured constraint optimization under language-conditioned preferences.

LLM Routing as Reasoning: A MaxSAT View

Abstract

Routing a query through an appropriate LLM is challenging, particularly when user preferences are expressed in natural language and model attributes are only partially observable. We propose a constraint-based interpretation of language-conditioned LLM routing, formulating it as a weighted MaxSAT/MaxSMT problem in which natural language feedback induces hard and soft constraints over model attributes. Under this view, routing corresponds to selecting models that approximately maximize satisfaction of feedback-conditioned clauses. Empirical analysis on a 25-model benchmark shows that language feedback produces near-feasible recommendation sets, while no-feedback scenarios reveal systematic priors. Our results suggest that LLM routing can be understood as structured constraint optimization under language-conditioned preferences.
Paper Structure (36 sections, 7 equations, 4 figures, 2 tables)

This paper contains 36 sections, 7 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Routing as reasoning under partial observability.
  3. Language feedback as constraints.
  4. A MaxSAT view of LLM routing.
  5. Contributions.
  6. Methodology
  7. Problem setup
  8. Endpoints and partial observability.
  9. Clause injection
  10. Regime-specific constraints.
  11. Postconditions
  12. Neutrality as a hard clause.
  13. Case S.
  14. Failure handling.
  15. Processing Rules.
  16. ...and 21 more sections

Figures (4)

  • Figure 1: Coverage Percentage and Recommendation Precision of both LF groups
  • Figure 2: Percentile and Mean of NF set and model zoo on key objectives.
  • Figure 3: Direction key without language feedback set results for Case S. (a) per-endpoint selection frequency, b) reasoning/cached-input prevalence, (c) set-size statistics.
  • Figure 4: OpenAI Model Pool