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Binary weights spanning trees and the $k$-red spanning tree problem in linear time

Dorit S. Hochbaum

TL;DR

This paper studies spanning tree problems on graphs with binary edge weights. It shows that both minimum/maximum spanning trees and the $k$-red spanning tree can be solved in linear time in $m$, contrasting with super-linear algorithms for general weights. The approach partitions edges into red and blue and uses linear-time BFS-based component analysis to verify existence and construct the desired trees, including a constructive $k$-red tree when feasible. It also discusses extensions to the general-weight case, linking the target-sum problem to subset sum and conjecturing strong NP-hardness even for small weight sets.

Abstract

We address here spanning tree problems on a graph with binary edge weights. For a general weighted graph the minimum spanning tree is solved in super-linear running time, even when the edges of the graph are pre-sorted. A related problem, of finding a spanning tree with a pre-specified sum of weights, is NP-hard. In contrast, for a graph with binary weights associated with the edges, it is shown that the minimum spanning tree and finding a spanning tree with a given total sum, are solvable in linear time with simple algorithms.

Binary weights spanning trees and the $k$-red spanning tree problem in linear time

TL;DR

This paper studies spanning tree problems on graphs with binary edge weights. It shows that both minimum/maximum spanning trees and the -red spanning tree can be solved in linear time in , contrasting with super-linear algorithms for general weights. The approach partitions edges into red and blue and uses linear-time BFS-based component analysis to verify existence and construct the desired trees, including a constructive -red tree when feasible. It also discusses extensions to the general-weight case, linking the target-sum problem to subset sum and conjecturing strong NP-hardness even for small weight sets.

Abstract

We address here spanning tree problems on a graph with binary edge weights. For a general weighted graph the minimum spanning tree is solved in super-linear running time, even when the edges of the graph are pre-sorted. A related problem, of finding a spanning tree with a pre-specified sum of weights, is NP-hard. In contrast, for a graph with binary weights associated with the edges, it is shown that the minimum spanning tree and finding a spanning tree with a given total sum, are solvable in linear time with simple algorithms.
Paper Structure (4 sections, 1 theorem)

This paper contains 4 sections, 1 theorem.

Table of Contents

  1. Introduction
  2. An approach for finding a $k$-red tree and why it is wasteful
  3. Existence and construction of a $k$-red tree in linear time
  4. Discussion and extensions

Key Result

Lemma 3.1

There is a $k$-red spanning tree in $G$ if and only if conditions 1 and 2 are satisfied; or equivalently: $\sum _{i=1}^p m_i(b) \geq q-k \geq 0$.

Theorems & Definitions (4)

  • Claim 2.1
  • proof
  • Lemma 3.1
  • proof