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 nc 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+16n/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.