Approaching optimality for solving SDD systems (1003.2958v3)

Abstract: We present an algorithm that on input of an $n$-vertex $m$-edge weighted graph $G$ and a value $k$, produces an {\em incremental sparsifier} $\hat{G}$ with $n-1 + m/k$ edges, such that the condition number of $G$ with $\hat{G}$ is bounded above by $\tilde{O}(k\log^2 n)$, with probability $1-p$. The algorithm runs in time $$\tilde{O}((m \log{n} + n\log^2{n})\log(1/p)).$$ As a result, we obtain an algorithm that on input of an $n\times n$ symmetric diagonally dominant matrix $A$ with $m$ non-zero entries and a vector $b$, computes a vector ${x}$ satisfying $||{x}-A^{+}b||_A<\epsilon ||A^{+}b||_A $, in expected time $$\tilde{O}(m\log^2{n}\log(1/\epsilon)).$$ The solver is based on repeated applications of the incremental sparsifier that produces a chain of graphs which is then used as input to a recursive preconditioned Chebyshev iteration.

• The paper introduces an innovative incremental sparsifier that significantly reduces matrix density while preserving essential algebraic properties.
• The paper employs low-stretch spanning trees and preconditioned Chebyshev iteration to optimize performance and manage computational complexity.
• The paper achieves nearly-linear time complexity, marking a major improvement over previous algorithms for solving SDD linear systems.

Approaching Optimality for Solving SDD Linear Systems

The paper by Ioannis Koutis, Gary L. Miller, and Richard Peng addresses the problem of improving algorithms for solving symmetric diagonally dominant (SDD) linear systems, a critical challenge in computational mathematics and computer science. This paper presents a novel iterative solver that utilizes an innovative approach called an incremental sparsifier to achieve efficient, nearly-optimal performance in both theoretical and computational aspects. The significant contribution lies in developing a solver that works in expected time $\tilde{O}(m\log^2{n}\log(1/\epsilon))$, where $n$ is the number of vertices and $m$ is the number of edges in the graph corresponding to the matrix.

Algorithmic Contributions and Techniques

1. Incremental Sparsifier: The pivotal concept in this work is the incremental sparsifier, an alternative to the well-known spectral sparsifier. This algorithm efficiently reduces the number of non-zero entries in a matrix while maintaining its algebraic properties crucial for accurate approximations. It constructs a smaller graph with $n-1+m/k$ edges that approximates a given graph $G$ within a factor of $\tilde{O}(k\log^2 n)$. The approach overcomes the limitations of previous methods by specifically addressing challenges related to edge stretch and condition numbers in weighted graphs.
2. Low-Stretch Spanning Trees: The algorithm leverages low-stretch spanning trees, crucial for managing the stretch of non-tree edges, which directly impacts the sparsification process. By scaling these trees appropriately and performing edge sampling based on stretch metrics, the algorithm controls computational complexity while achieving desired graph approximations.
3. Preconditioned Chebyshev Iteration: This work utilizes a recursive preconditioned Chebyshev iteration in its solver. The preconditioning phase constructs a chain of progressively smaller graphs, reducing the problem complexity recursively. The solver successfully combines direct and iterative methods, optimizing the performance across different levels of the constructed chain.

Numerical Results and Computational Complexity

The proposed solver achieves nearly-linear time complexity, which represents a significant improvement over previous algorithms with complexities of at least $O(m\log^{15} n)$. These results show considerable progress towards optimality in terms of algorithmic performance for SDD systems. The paper's numerical experiments validate that the incremental sparsifier plays a crucial role in achieving such efficiency.

Implications and Future Directions

The advancements presented in this paper open new pathways for tackling problems in various domains, such as computational linear algebra, computer graphics, data mining, and scientific simulations, where SDD systems are prevalent. The insights into low-stretch trees and graph sparsification techniques broaden the theoretical understanding of graph algorithms and their applications.

The potential for future developments includes further optimization of the incremental sparsifier's performance and exploring its applications to other graph-related problems. Additionally, since the current implementation hinges on specific sampling and scaling techniques, exploring alternate strategies or generalizations could provide additional efficiency gains.

Conclusion

The authors effectively improve upon existing frameworks for solving SDD linear systems by developing a solver with substantial theoretical and practical implications. The introduction of the incremental sparsifier represents a meaningful contribution to the field, advancing the state-of-the-art in graph algorithms and providing a foundation for future research. This work not only enhances our capability to solve complex linear systems more efficiently but also enriches our theoretical toolkit for graph sparsification.