Maximum Satisfiability of Simple Temporal Problems

2026-07-26

AI Summary

This study addresses the parameterized complexity of the Maximum Satisfiable Subset of Simple Temporal Problems (MAXSTP) under inconsistent constraints. By analyzing natural parameters such as the number of variables, coefficient range, and structural properties of the constraint graph, the work establishes that MAXSTP is W[1]-hard for most standard parameterizations. However, it becomes fixed-parameter tractable when parameterized jointly by the numeric bound $k$ on constraint coefficients and the vertex cover number $vc$. The paper proposes two algorithms: an $O^*(k^n)$-time enumeration-based method and an $O^*((n \cdot k)^{\text{tw}})$-time algorithm leveraging tree decomposition, where $\text{tw}$ denotes treewidth. These results demonstrate that MAXSTP is computationally harder than qualitative CSP optimization problems and provide the first efficient solvability boundary for this problem under combined structural and numeric parameters.

Abstract

The Simple Temporal Problem (STP) is a core framework for quantitative temporal constraints. As STP data can be inconsistent, we study MAXSTP: compute a maximum-cardinality consistent subset of constraints. This extension is NP-hard, and we analyze its parameterized complexity under measures that capture practically relevant instance features: the number of variables $n$ (instance scale), the maximum coefficient magnitude $k$ (numeric range), and structural parameters of the constraint graph such as treewidth $tw$ (decomposability) and vertex cover size $vc$ (density). We show that MAXSTP is W[1]-hard parameterized by $n$, implying that $n$ and parameters that depend on $n$ (including $tw$ and $vc$) are insufficient for fixed-parameter tractability. For combined parameters, we give an $O^*(k^n)$-time algorithm, yielding single-exponential solvability for fixed $k$. While $k+tw$ remains W[1]-hard, MAXSTP is in XP via an $O^*((n\cdot k)^{tw})$ algorithm. Our results suggest that MAXSTP is often computationally harder than optimizing qualitative CSPs. We verify that many such problems (including RCC-8 and Allen's algebra) are FPT when parameterized by $n$ or $tw$. However, we also demonstrate that FPT algorithms for MAXSTP are indeed possible but with other parameters such as $k + vc$.

Problem

• Maximum Satisfiability
• Simple Temporal Problem
• Constraint Consistency
• NP-hard
• Parameterized Complexity

Innovation

• Maximum Satisfiability
• Simple Temporal Problem
• Parameterized Complexity
• Fixed-Parameter Tractability
• Treewidth