# Constraint Optimization over Semirings

## TL;DR

This paper studies the complexity of constraint optimization problems over various semirings, generalizing satisfiability, and provides complexity bounds, algorithms, and inapproximability results for different semiring-based formulas.

## Contribution

It introduces the first complexity analysis of constraint optimization over semirings, including bounds, algorithms, and hardness results for various semiring interpretations.

## Key findings

- optConfVal and optConf are in FP^NP for propositional formulas in negation normal form.
- For CNF formulas, optConfVal is at most 1/4^{m-r}, where r is the maximum satisfiable clauses.
- optConfVal for CNF formulas is hard for FP^NP[log], with polynomial-time approximation algorithms and inapproximability results.

## Abstract

Interpretations of logical formulas over semirings have applications in various areas of computer science including logic, AI, databases, and security. Such interpretations provide richer information beyond the truth or falsity of a statement. Examples of such semirings include Viterbi semiring, min-max or access control semiring, tropical semiring, and fuzzy semiring. The present work investigates the complexity of constraint optimization problems over semirings. The generic optimization problem we study is the following: Given a propositional formula $\varphi$ over $n$ variable and a semiring $(K,+,\cdot,0,1)$, find the maximum value over all possible interpretations of $\varphi$ over $K$. This can be seen as a generalization of the well-known satisfiability problem. A related problem is to find an interpretation that achieves the maximum value. In this work, we first focus on these optimization problems over the Viterbi semiring, which we call optConfVal and optConf. We show that for general propositional formulas in negation normal form, optConfVal and optConf are in ${\mathrm{FP}}^{\mathrm{NP}}$. We investigate optConf when the input formula $\varphi$ is represented as a CNF. For CNF formulae, we first derive an upper bound on optConfVal as a function of the number of maximum satisfiable clauses. In particular, we show that if $r$ is the maximum number of satisfiable clauses in a CNF formula with $m$ clauses, then its optConfVal is at most $1/4^{m-r}$. Building on this we establish that optConfVal for CNF formulae is hard for the complexity class ${\mathrm{FP}}^{\mathrm{NP}[\log]}$. We also design polynomial-time approximation algorithms and establish an inapproximability for optConfVal. We establish similar complexity results for these optimization problems over other semirings including tropical, fuzzy, and access control semirings.