Minimum Linear Arrangement of Series-Parallel Graphs

Abstract

We present a factor 14D 2 approximation algorithm for the minimum linear arrangement problem on series-parallel graphs, where D is the maximum degree in the graph. Given a suitable decomposition of the graph, our algorithm runs in time O(|E|) and is very easy to implement. Its divide-andconquer approach allows for an effective parallelization. Note that a suitable decomposition can also be computed in time O(|E| log |E|) (or even O(log |E| log * |E|) on an EREW PRAM using O(|E|) processors). For the proof of the approximation ratio, we use a sophisticated charging method that uses techniques similar to amortized analysis in advanced data structures.On general graphs, the minimum linear arrangement problem is known to be NP-hard. To the best of our knowledge, the minimum linear arrangement problem on series-parallel graphs has not been studied before.

Arrow Matrix Decomposition: A Novel Approach for Communication-Efficient Sparse Matrix Multiplication

“…We show how to construct such an arrow matrix decomposition for several sparsity structures, as characterized by the graphs they represent. The main idea is to use the relationship with minimum linear arrangement [14,20,41]. Moreover, we prove that the pruning of high-degree vertices enabled by the arrow shape provides a polynomial improvement in the communication volume in power law graphs.…”

“…If the graph 𝐺 is clear from the context, we omit 𝐺 from the notation. A linear arrangement of 𝐺 with the smallest cost is a minimum linear arrangement (MLA) [14,20,41]. Computing a minimum linear arrangement is NP-hard, however, it can be approximated in polynomial time within a 𝑂 ( √︁ log 𝑛 log log 𝑛) factor [14] and solved exactly in polynomial time on trees [4] and chordal graphs [42].…”

Ordering a Sparse Graph to Minimize the Sum of Right Ends of Edges

“…The minimum linear arrangement problem is a classic NP-complete problem [12] and has been intensively studied. Approximation algorithms and inapproximability results are known [1,2,8,14], as well as exact exponential and parameterized algorithms [3,[9][10][11], and efficient algorithms for special graph classes [5][6][7]13]. MinSumEnds is much less explored.…”