Better approximating SONET k-edge partition for small capacity k

We study the SONET edge partition problem that models telecommunication network design to partition the edge set of a given graph into several edge-disjoint subgraphs, such that each subgraph has size no greater than a given capacity k and the sum of the orders of these subgraphs is minimized. The problem is NP-hard when $$k \ge 3$$ k ≥ 3 and admits an $$O(\log k)$$ O ( log k ) -approximation algorithm. For small capacity $$k = 3, 4, 5$$ k = 3 , 4 , 5 , by observing that some subgraph structures are more favorable than the others, we propose modifications to existing algorithms and design novel amortization schemes to prove their improved performance. Our algorithmic results include a $$\frac{4}{3}$$ 4 3 -approximation for $$k = 3$$ k = 3 , improving the previous best $$\frac{13}{9}$$ 13 9 -approximation, a $$\frac{4}{3}$$ 4 3 -approximation for $$k = 4$$ k = 4 , improving the previous best $$(\frac{4}{3} + \epsilon )$$ ( 4 3 + ϵ ) -approximation, and a $$\frac{3}{2}$$ 3 2 -approximation for $$k = 5$$ k = 5 , improving the previous best $$\frac{5}{3}$$ 5 3 -approximation. Besides these improved algorithms, our main contribution is the amortization scheme design, which can be helpful for similar algorithms and problems.