Graph Bisection Optimization via Packing Lower Bounds
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Solution Overview
Problem
Current software libraries for graph bisection, such as METIS, SCOTCH, and JOSTLE, often produce unsatisfactory solutions due to their inability to prove optimality or provide approximation guarantees, and perform poorly unless graph imbalance is allowed, limiting their effectiveness in solving large instances of the minimum graph bisection problem.
Innovation Solution
The implementation provides a combinatorial lower bound computed in near-linear time, using a packing lower bound technique within a branch-and-bound framework to reduce the number of tree nodes and determine optimum solutions for large graphs by decomposing the input graph into less complex subproblems, which can be solved independently.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Productivity
If known software libraries (METIS, SCOTCH, JOSTLE) are used to solve graph partitioning, then the solutions can be obtained quickly, but the quality of solutions is unacceptable and optimality cannot be proven
Solution Approach 1:
The patent decomposes the input graph into less complex subproblems that can be solved independently. This segmentation allows the branch-and-bound algorithm to work on smaller, more manageable pieces while still guaranteeing optimality for the overall graph bisection problem.
Solution Approach 2:
The patent computes a combinatorial lower bound in near-linear time before the main optimization process. This preliminary computation provides a foundation for the branch-and-bound algorithm to efficiently prune search spaces and prove optimality without sacrificing solution quality.
2Manufacturing precision
If branch-and-bound framework is used to compute exact solutions, then optimality can be proven, but only instances of moderate size can be solved even after hours of processing
Solution Approach 1:
The patent computes combinatorial lower bounds in near-linear time before the branch-and-bound search. This preliminary action provides tight bounds that dramatically reduce the search space, enabling exact solutions for large graphs that would otherwise be intractable.
Solution Approach 2:
By decomposing the graph into subproblems, the patent reduces the complexity of the branch-and-bound tree. Each subproblem can be solved independently with smaller search spaces, yet their combination yields the optimal solution for the entire graph.
3Loss of time
If the branch-and-bound tree size is reduced using packing lower bounds, then computation time decreases, but the complexity of the lower bound computation increases
Solution Approach 1:
The patent uses packing lower bounds that change the parameters used to evaluate partial solutions in the branch-and-bound tree. By computing combinatorial lower bounds based on packing constraints, the algorithm can prune more branches efficiently despite the increased computational complexity of the bound itself.
Data Source
AI summary
Techniques are described for graph partitioning, and in particular, graph bisection. A lower bound is provided that is computed in near-linear time. These bounds may be used to determine optimum solutions to real-world graphs with many vertices (e.g., more than a million for road networks, or tens of thousands for VLSI and mesh instances). A packing lower bound technique determines lower bounds in a branch-and-bound tree, reducing the number of tree nodes. Techniques are employed to assign vertices without branching on them, again reducing the size of the tree. Decomposition is provided to translate an input graph into less complex subproblems. The decomposition boosts performance and determines the optimum solution to an input by solving subproblems independently. The subproblems can be solved independently using a branch-and-bound approach to determine the optimum bisection.