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Relaxation-based coarsening and multiscale graph organization
Dorit Ron, Ilya Safro, Achi Brandt
TL;DR
Coarsening requires a useful measure of variable closeness, while classical coupling-based definitions can fail for some systems and graph structures. The paper measures closeness from relaxed-value correlations and tests relaxation-based coarsening; most tested minimum 2-sum graphs benefit over AMG-based coarsening.
Problem
Coarsening needs a measure of whether one variable’s value after relaxation implies another’s, while classical coupling-based closeness can fail for some systems and graph structures.
Source: §1 · p. 2Method
The paper defines variable closeness from correlations in relaxed vectors and extends the distance to best-fitted interpolation from a subset of variables.
Source: §1 · pp. 2–3Results
For minimum 2-sum, most tested graphs benefit from relaxation-based coarsening, with AMG-to-relaxation average-cost ratios greater than 1; introducing algebraic distance did not significantly increase typical running-time estimates.
Source: §4.1 · p. 10, §4.3 · p. 12Takeaways & Limitations
At large distances, detailed individual graph distances are usually not of interest, and distance in a general graph is a fuzzy notion.
Source: §5 · p. 13The computational implementation was nonparallel and unoptimized, and results are qualitative rather than best-known results for the test sets.
Source: §4 · p. 10
Abstract
from arXiv · showhide
In this paper we generalize and improve the multiscale organization of graphs by introducing a new measure that quantifies the "closeness" between two nodes. The calculation of the measure is linear in the number of edges in the graph and involves just a small number of relaxation sweeps. A similar notion of distance is then calculated and used at each coarser level. We demonstrate the use of this measure in multiscale methods for several important combinatorial optimization problems and discuss the multiscale graph organization.
1. Introduction.
The paper introduces algebraic distance to guide multilevel coarsening by measuring how well variables’ relaxed values are correlated. It extends the measure to interpolation from variable subsets and treats distance as local and multiscale.
1. Introduction.
- Coarsening needs a measure of whether one variable’s value after relaxation implies another’s, since sufficiently close variables can be aggregated.Coarse variables and explicit interpolation are chosen together within the multilevel process.Source: §1 · p. 2
- Classical coupling-based closeness can be ineffective without sufficient diagonal dominance and can mis-coarsen even dominant-diagonal graph Laplacians with nonlocal edges.The proposed measure instead evaluates relationships between variables after relaxation.Source: §1 · p. 2
- Algebraic distance measures closeness through correlations between variables across relaxed vectors, obtained by relaxing the homogeneous system and normalizing the results.The distance is the reciprocal of closeness, and p ≥ 2 weights larger differences more strongly.Source: §1 · pp. 2–3
- For a node and a subset of nodes, distance is the minimum L2 deviation across relaxed vectors of the best-fitted interpolation from that subset.The least-square interpolation is the one introduced in bootstrap AMG (BAMG) for coarse-to-fine explicit interpolation.Source: §1 · p. 3
- Algebraic distance is a crude local measure: it is meaningful between neighbors, while its exact value for non-neighbors carries no further meaning.A similar distance is calculated at each coarser level.Source: §1 · p. 3
- The authors argue that meaningful distances in general graphs should be defined through a multiscale process rather than only calculated at one scale.They advocate replacing spectral methods with AMG-like multilevel algorithms for fuzzy graph problems.Source: §1 · p. 3
2. Notation and problem definitions.
The section introduces linear-ordering objectives and graph partitioning as problems addressed by multilevel methods. It describes volume-weighted ordering and the multilevel partitioning framework.
2.1. Linear ordering.
- Minimum p-sum ordering minimizes a functional over permutations; in the multilevel version, vertices have volumes and are positioned at their centers of mass.The original problem is the equal-volume special case; minimum linear arrangement uses p = 1 and M2SP uses p = 2.Source: §2.1 · p. 4
2.2. Partitioning.
- Graph partitioning is NP-hard and has applications including VLSI design, parallel-computing data distribution, and task scheduling.The multilevel framework recursively coarsens the graph, solves the coarsest level, and uncoarsens the solution; α is the given imbalance factor.Source: §2.2 · p. 4, §3 · p. 4
3. The coarsening algorithm.
The section presents relaxation-based graph coarsening that replaces direct edge weights with a neighborhood-sensitive algebraic coupling. It derives the coupling from relaxed test vectors and uses it to select seeds, form aggregates, and construct coarse graphs.
3. The coarsening algorithm.
- Coarsening chooses seed nodes, assigns each nonseed among aggregates, then computes couplings between the resulting coarse nodes.The interpolation matrix P encodes the fractions of fine nodes belonging to aggregates.Source: §3 · pp. 5, 6, 9
- The method replaces direct edge weight with algebraic coupling, reciprocal of an algebraic distance that accounts for connections between the nodes’ neighborhoods.The measure weakens an ij edge when i’s neighbors are not connected to j’s neighbors.Source: §3 · pp. 5–6
- Algebraic distance compares node values across relaxed test vectors; vectors come from relaxing Ax=0, typically with 10 Jacobi under-relaxation sweeps and K=20 in tests.The test vectors collectively represent algebraically smooth error.Source: §3 · pp. 5–6
- For K=10 and r=20 with wij=1, dij is three times bigger than the minimum nearest-neighbor distance; stronger ij may attract i, but i’s neighbors favor others.The reported example also finds that i and j do not tend to join unless wij is comparable to the sum of i’s other couplings.Source: §3 · p. 7, Table 3.1 · p. 8, Table 3.2 · p. 8
- Seed selection first promotes nodes whose future volume exceeds ν times the average, with ν=2 in the tests.Further seed insertion depends on a threshold for strength of coupling to the existing seed set; Q=0.5 in the tests.Source: §3 · pp. 7–9
- Each nonseed’s coarse neighborhood retains strongly algebraically coupled seeds, capped by caliber l and ranked by graph-edge weights when necessary.The reported experiments use l=1 and l=2.Source: §3 · pp. 8–9
4. Computational results.
The experiments compare relaxation-based coarsening with classical AMG for graph ordering and partitioning. They report improvements on tested problems, with no significant increase in estimated running time.
4. Computational results.
- The evaluation used a nonparallel, unoptimized implementation, so the authors characterize results as qualitative rather than best-achievable comparisons.The p-sum tests used 150 graphs, small calibers 1 and 2, and left adaptive BAMG interpolation for higher calibers beyond scope.Source: §4 · p. 10, §4.1 · p. 10
4.1. The minimum p-sum problem.
- Most tested minimum 2-sum graphs favored relaxation-based coarsening, although local optimization narrowed the gap between the methods.The reported ratio was AMG-based cost divided by relaxation-based cost; ratios greater than 1 favor relaxation-based coarsening.Source: §4.1 · p. 10
- Minimum linear arrangement showed more significant improvement with relaxation-based coarsening, with VLSI and general-optimization graphs among the most beneficial cases.The authors associate these cases with irregular graphs and report that caliber 1 achieved best-known results in several examples where classical AMG required larger calibers.Source: §4.1 · pp. 10–11
- The mixed scheme improved results on about 25 more graphs than algebraic-coupling-only coarsening, with an average improvement of 1.5%.The comparison was for minimum 2-sum with local-optimization postprocessing.Source: §4.1 · p. 12
4.2. The minimum 2-partitioning problem.
- For minimum 2-partitioning, the four best ratios came from graphs with power-law degree distributions.The comparison covered 15 graphs of differing nature and size.Source: §4.2 · p. 12
4.3. Running time.
- Introducing algebraic distance did not significantly increase the cited running-time estimates for AMG-based graph-ordering frameworks.Detailed implementation and running-time analysis are outside the paper's scope.Source: §4.3 · p. 12
5. Multiscale distance definition and hierarchical organization.
The paper treats graph distance as a multiscale quantity, defining useful local distances at each coarsening level rather than detailed pairwise distances throughout the graph. It argues that this is both cheaper and more adaptable to the properties of larger aggregates.
5. Multiscale distance definition and hierarchical organization.
- Algebraic distance is a fast, fuzzy local measure for coarsening; at coarser levels, distances apply between aggregates rather than every pair of distant nodes.The paper argues that distances at larger scales should incorporate properties of increasingly large aggregates.Source: §5 · p. 13
- Coarse-level distance definitions can be tuned to the task, such as using average edge weights for pose similarity or maximum weights for transitive face clustering.The paper gives these alternatives for aggregates formed from pairs of fine-level nodes.Source: §5 · p. 14
- Algebraic distance is presented as a fast, crude approximation to diffusion distance at small diffusion time, while computing diffusion maps for large graphs can be expensive.The paper notes that diffusion-map calculation may require many graph-Laplacian eigenvectors.Source: §5 · p. 14
- The paper argues that AMG-style multilevel organization can address graph problems directly instead of first computing spectral representations.It connects this approach to hierarchical graph organization, multiscale clustering, image segmentation, and other graph problems.Source: §5 · pp. 14–15
- The demonstrated coarsening improves results on minimum (1,2)-sum ordering and minimum 2-partitioning, while preventing nonlocal vertices from sharing a coarse aggregate.The authors attribute the improvement to detecting important graph couplings.Source: §6 · p. 15
6. Conclusions.
The supplied passage records funding acknowledgments and thanks for assistance with the minimum 2-partitioning experiments.
6. Conclusions.
- The work acknowledges CSCAPES and DOE support, and thanks Cédric Chevalier for a Scotch modification and help designing the partitioning experiments.The passage also names DOE Contract DE-AC02-06CH11357.Source: Acknowledgments · p. 15