Approximating Minimum k-Section in Trees with Linear Diameter.

Minimum k -Section denotes the NP-hard problem to partition the vertex set of a graph into k sets of size as equal as possible while minimizing the cut width, which is the number of edges between these sets. When k is an input parameter and n denotes the number of vertices, it is NP-hard to approximate the width of a minimum k -section within a factor of n c for any c < 1 , even when restricted to trees with constant diameter. Here, we show that every tree T allows a k -section of width at most ( k − 1 ) ( 2 + 16 n / diam ( T ) ) Δ ( T ) . This implies a polynomial time constant factor approximation for the Minimum k -Section Problem when restricted to trees with linear diameter and constant maximum degree. [ABSTRACT FROM AUTHOR]