Sum-of-Max Chain Partition of a Tree

TL;DR

The paper addresses the sum-of-max chain partition problem on rooted trees under a knapsack constraint: partition the tree into chains of total weight at most $w_0$ while minimizing the sum of chain costs, defined as the maximum $s_i$ per chain. It introduces a dynamic-programming framework augmented with a novel two-layer heap-over-heap data structure, maintaining candidate costs $cost(v,i)$ via $H_x^{(v)}$ and $\mathbb{H}^{(v)}$, and uses Over-weight and Update-s-maximal procedures to update structures as the root window changes. The main result is an $O(n\log n)$ algorithm, with a generalized version using rank-based dominance ($r$-maximal) that preserves the same time bound. These techniques enable efficient, scalable chain partitioning on trees and extend to broader chain-cost definitions, informing related path-partition and decomposition problems.

Abstract

Path partition problems on trees have found various applications. In this paper, we present an $O(n \log n)$ time algorithm for solving the following variant of path partition problem: given a rooted tree of $n$ nodes $1, \ldots, n$, where vertex $i$ is associated with a weight $w_i$ and a cost $s_i$, partition the tree into several disjoint chains $C_1,\ldots,C_k$, so that the weight of each chain is no more than a threshold $w_0$ and the sum of the largest $s_i$ in each chain is minimized. We also generalize the algorithm to the case where the cost of a chain is determined by the $s_i$ of the vertex with the highest rank in the chain, which can be determined by an arbitrary total order defined on all nodes instead of the value of $s_i$.

Figures (5)

• Figure 1: (a) Illustration for window and outer boundary, where $w_0=7$. The numbers in the vertices are their weights. Squares represent the vertices in the window. (b) Illustration for $\mathsf{sum}(v,i)$. Here, $\mathsf{sum}(v,i)= F[j_1]+F[j_2]+F[j_3]+F[j_4]+F[j_5]$.
• Figure 2: A partition of $\mathsf{win}_v$ into different parts, with each part organized by a heap. In this figure, rhombuses indicate s-maximal vertices and circles indicate other vertices.
• Figure 3: Illustration of the change of the heaps at the stage for computing $F[v]$.
• Figure 4: Example of binomial heaps melding with lazy tag
• Figure 5: Example of binomial heaps pushing up a node $-\infty$. In the first step, 3 is swapped down and becomes $3-1=2$ to counter the $+1$ tag's effect since it was not affected by $+1$ in the first place. Similarly, 2 is swapped down and becomes $2-2=0$ to counter the $+2$ tag.

Theorems & Definitions (33)

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