Finding the Minimum-Weight k-Path

TL;DR

This paper presents new algorithms for finding minimum-weight k-paths and k-trees in weighted graphs, achieving faster exact and approximate solutions with randomized methods under certain weight restrictions.

Contribution

It introduces improved randomized algorithms for minimum-weight k-path and k-tree problems with better time complexities for both exact and approximate solutions.

Findings

Exact k-path can be found in time O*(2^k M) for integer weights.

Approximate solutions run in time involving loglog M and 1/ε factors.

Exact k-tree solutions run in time O*(2^k n^3) with similar approximation enhancements.

Abstract

Given a weighted $n$-vertex graph $G$ with integer edge-weights taken from a range $[-M,M]$, we show that the minimum-weight simple path visiting $k$ vertices can be found in time $\tilde{O}(2^k \poly(k) M n^\omega) = O^*(2^k M)$. If the weights are reals in $[1,M]$, we provide a $(1+\varepsilon)$-approximation which has a running time of $\tilde{O}(2^k \poly(k) n^\omega(\log\log M + 1/\varepsilon))$. For the more general problem of $k$-tree, in which we wish to find a minimum-weight copy of a $k$-node tree $T$ in a given weighted graph $G$, under the same restrictions on edge weights respectively, we give an exact solution of running time $\tilde{O}(2^k \poly(k) M n^3) $ and a $(1+\varepsilon)$-approximate solution of running time $\tilde{O}(2^k \poly(k) n^3(\log\log M + 1/\varepsilon))$. All of the above algorithms are randomized with a polynomially-small error probability.