# $K$-Best Solutions of MSO Problems on Tree-Decomposable Graphs

## TL;DR

This paper presents an efficient algorithm to find the top k solutions for monadic second-order logic expressible optimization problems on graphs with bounded treewidth, significantly improving speed for shortest path problems.

## Contribution

It introduces a method to compute the k best solutions for MSO problems on bounded treewidth graphs in near-linear time, including shortest path problems with exponential speedup.

## Key findings

- Achieves A(n + k \u2212 g n) time complexity
- Applies to shortest simple path problems with exponential speedup
- Extends to a broad class of MSO-expressible optimization problems

## Abstract

We show that, for any graph optimization problem in which the feasible solutions can be expressed by a formula in monadic second-order logic describing sets of vertices or edges and in which the goal is to minimize the sum of the weights in the selected sets, we can find the $k$ best solutions for $n$-vertex graphs of bounded treewidth in time $\mathcal O(n+k\log n)$. In particular, this applies to the problem of finding the $k$ shortest simple paths between given vertices in directed graphs of bounded treewidth, giving an exponential speedup in the per-path cost over previous algorithms.