Minimal Subset Identification via Dichotomic Constraint Removal
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Solution Overview
Problem
Existing methods for finding a minimal subset of constraints that satisfy an upward-closed property are inefficient, particularly due to their recursive or iterative approaches, which are costly in terms of property checks and often require computation of multiple or all Minimal Correction Sets, making them unsuitable for finding a single unsatisfiable subset.
Innovation Solution
A computer-implemented method that iterates over a list of constraints, determining which constraints to add to a set of unsatisfiable constraints by removing adjacent constraints in increasing numbers until the set becomes satisfiable, using dichotomic search and lazy checks to optimize the process, thereby identifying a minimal subset efficiently.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Adaptability or versatility
If recursive methods (e.g., QuickXplain) are used to find a minimal subset of constraints, then the method can handle complex problems, but the number of property checks increases significantly
Solution Approach 1:
The algorithm segments the set of constraints into two parts: a working set and a remaining set. It iteratively processes the working set by removing constraints in groups of increasing sizes (1, 2, 4, 8, ...) to determine minimality, while maintaining the ability to handle complex problems through systematic division of the constraint space.
Solution Approach 2:
The algorithm performs preliminary actions by first identifying a candidate minimal subset, then systematically removing constraints in groups of increasing sizes to verify minimality. This preliminary identification followed by systematic verification reduces the total number of property checks compared to recursive methods that repeatedly search from scratch.
2Reliability
If iterative methods compute multiple or all Minimal Correction Sets, then completeness is achieved, but the computation cost increases
Solution Approach 1:
The algorithm extracts only the necessary information to find a single minimal subset by removing constraints in groups from the constraint set. Instead of computing all Minimal Correction Sets, it extracts one minimal subset by systematically removing constraints and checking satisfiability, thereby reducing computation cost while maintaining reliability for the specific goal of finding one minimal subset.
Solution Approach 2:
The algorithm performs partial action by removing constraints in groups of increasing sizes (1, 2, 4, 8, ...) rather than checking each constraint individually. This excessive removal approach (removing more constraints at once) allows the algorithm to quickly determine minimality with fewer total property checks compared to exhaustive methods.
3Measurement precision
If individual constraint removal is performed to verify minimality, then accuracy is ensured, but the number of property checks becomes excessive
Solution Approach 1:
The algorithm merges multiple constraint removal operations into groups of increasing sizes (1, 2, 4, 8, ...). Instead of removing and checking each constraint individually, it combines constraints into groups and removes them together, thereby maintaining accuracy of minimality verification while significantly reducing the number of property checks required.
Solution Approach 2:
The algorithm uses excessive action by removing constraints in groups larger than one at a time. This approach removes multiple constraints simultaneously (groups of 2, 4, 8, ...) to verify minimality, which maintains accuracy by ensuring that no unnecessary constraints remain while reducing the total number of property checks through batch operations.
Data Source
AI summary
According to an aspect, a method for providing a minimal explanation to a set of unsatisfiable constraints involves retrieving a minimal subset of constraints that remain together unsatisfiable. The method includes iterating over a list of n constraints, and building a minimal explanation to a set of unsatisfiable constraints by determining which constraint to add to the set of unsatisfiable constraints. Building includes accelerating by removing an increasing number of constraints until removed further constraints makes the set of constraints satisfiable. A dichotomic search is performed on the removed further constraints. The average observed distance is identified between successive constraints in the set of unsatisfiable constraints. A plurality of 2k further constraints located in the list of constraints is removed at the average observed distance from the most recently added constraint. Testing whether a current selected subset is unsatisfiable is performed for the first log2(n) added constraints.