Approximation Algorithms for Budget Constrained Network Upgradeable Problems

TL;DR

This paper introduces approximation algorithms for budget-constrained network upgrade problems, including maximum spanning trees and longest paths in DAGs, improving upon previous models and providing probabilistic guarantees.

Contribution

It presents new randomized approximation algorithms for upgradeable network problems with budget constraints, extending previous work and offering improved performance guarantees.

Findings

Probabilistic approximation for maximum spanning tree within budget

Polynomial-time algorithms for longest path with improvements in DAGs

Extension of results to shortest path problems in DAGs

Abstract

We study budget constrained network upgradeable problems. We are given an undirected edge weighted graph $G=(V,E)$ where the weight an edge $e \in E$ can be upgraded for a cost $c(e)$. Given a budget $B$ for improvement, the goal is to find a subset of edges to be upgraded so that the resulting network is optimum for $B$. The results obtained in this paper include the following. Maximum Weight Constrained Spanning Tree We present a randomized algorithm for the problem of weight upgradeable budget constrained maximum spanning tree on a general graph. This returns a spanning tree $\mathcal{T}^{'}$ which is feasible within the budget $B$, such that $\Pr [ l(\mathcal{T}^{'}) \geq (1-\epsilon)\text{OPT}\text{ , } c(\mathcal{T}^{'} ) \leq B] \ge 1-\frac{1}{n}$ (where $l$ and $c$ denote the length and cost of the tree respectively), for any fixed $\epsilon >0$, in time polynomial in $|V|=n$, $|E|=m$. Our results extend to the minimization version also. Previously Krumke et. al. \cite{krumke} presented a$(1+\frac{1}{\gamma}, 1+ \gamma)$ bicriteria approximation algorithm for any fixed $\gamma >0$ for this problem in general graphs for a more general cost upgrade function. The result in this paper improves their 0/1 cost upgrade model. Longest Path in a DAG We consider the problem of weight improvable longest path in a $n$ vertex DAG and give a $O(n^3)$ algorithm for the problem when there is a bound on the number of improvements allowed. We also give a $(1-\epsilon)$-approximation which runs in $O(\frac{n^4}{\epsilon})$ time for the budget constrained version. Similar results can be achieved also for the problem of shortest paths in a DAG.