On Complexity of Isoperimetric Problems on Trees

This paper is aimed to investigate some computational aspects of different isoperimetric problems on weighted trees. In this regard, we consider different connectivity parameters called {\it minimum normalized cuts}/{\it isoperimteric numbers} defined through taking minimum of the maximum or the mean of the normalized outgoing flows from a set of subdomains of vertices, where these subdomains constitute a {\it partition}/{\it subpartition}. Following the main result of [A. Daneshgar, {\it et. al.}, {\it On the isoperimetric spectrum of graphs and its approximations}, JCTB, (2010)], it is known that the isoperimetric number and the minimum normalized cut both can be described as ${0,1}$-optimization programs, where the latter one does {\it not} admit a relaxation to the reals. We show that the decision problem for the case of taking $k$-partitions and the maximum (called the max normalized cut problem {\rm NCP}$^M$) as well as the other two decision problems for the mean version (referred to as {\rm IPP}$^m$ and {\rm NCP}$^m$) are $NP$-complete problems. On the other hand, we show that the decision problem for the case of taking $k$-subpartitions and the maximum (called the max isoperimetric problem {\rm IPP}$^M$) can be solved in {\it linear time} for any weighted tree and any $k \geq 2$. Based on this fact, we provide polynomial time $O(k)$-approximation algorithms for all different versions of $k$th isoperimetric numbers considered. Moreover, when the number of partitions/subpartitions, $k$, is a fixed constant, as an extension of a result of B. Mohar (1989) for the case $k=2$ (usually referred to as the Cheeger constant), we prove that max and mean isoperimetric numbers of weighted trees as well as their max normalized cut can be computed in polynomial time. We also prove some hardness results for the case of simple unweighted graphs and trees.

💡 Research Summary

The paper investigates the computational complexity of several isoperimetric and normalized‑cut problems defined on weighted trees. The authors consider two ways of grouping vertices—partitions, which cover all vertices, and sub‑partitions, which may leave some vertices unused—and two objective functions—maximizing the worst normalized outgoing flow from a group (the “max” version) and minimizing the average of those flows (the “mean” version). For each combination they define a decision problem: NCP^M (max normalized cut with partitions), NCP^m (mean normalized cut with partitions), IPP^M (max isoperimetric number with sub‑partitions), and IPP^m (mean isoperimetric number with partitions).

The main hardness results show that three of these problems are NP‑complete even when the underlying graph is a tree. Specifically, the decision version of NCP^M (max normalized cut with k‑partitions) is NP‑complete, and both mean‑based problems IPP^m and NCP^m are also NP‑complete. The proofs use polynomial‑time reductions from classic NP‑complete problems such as Partition and 3‑SAT, carefully encoding the required flow constraints into the tree’s edge weights. These results confirm that, despite the simplicity of trees, the non‑linear nature of the mean objective prevents a straightforward relaxation to a linear or convex program.

In contrast, the authors present a linear‑time algorithm for the max isoperimetric problem with k‑sub‑partitions (IPP^M). The algorithm exploits the hierarchical structure of trees: after rooting the tree, a dynamic programming pass computes, for each node, the optimal cost for every possible number of sub‑partitions in its subtree. Because each node is processed once and the state space is bounded by k, the total running time is O(n) for any fixed k≥2. This positive result demonstrates that allowing unused vertices (sub‑partitions) dramatically reduces the problem’s difficulty.

Building on the linear‑time solution, the paper proposes O(k)‑approximation algorithms for all four problem variants. The approximation scheme performs a depth‑first traversal, greedily assigning vertices to at most k groups while tracking the maximum (or average) normalized flow incurred. The analysis shows that the worst‑case ratio between the algorithm’s output and the optimal value is bounded by a factor proportional to k, yielding a polynomial‑time, constant‑factor approximation when k is small.

When k is a constant (independent of the input size), the authors extend B. Mohar’s 1989 result for the Cheeger constant (k=2) to arbitrary constant k. They enumerate all feasible k‑sub‑partitions, but prune the search space using tree‑specific properties, achieving a polynomial‑time exact algorithm for both max and mean isoperimetric numbers as well as the max normalized cut. The running time is O(n^{c}) where the exponent c depends only on the fixed constant k, making the approach practical for small k.

The paper also addresses unweighted graphs and unweighted trees. It proves that even without edge weights, the mean‑based problems remain NP‑complete, indicating that the difficulty stems from the averaging operation rather than from weight heterogeneity.

Overall, the work provides a comprehensive complexity landscape for isoperimetric and normalized‑cut problems on trees: three variants are intractable (NP‑complete), one variant is tractable in linear time, and all admit O(k) approximations. The results have implications for tree‑based clustering, hierarchical segmentation, and network reliability analysis, where such cut measures are frequently employed. Future directions suggested include extending the linear‑time algorithm to graphs of bounded treewidth, improving approximation ratios below O(k), and empirical evaluation on real‑world hierarchical data sets.