DeepAI

# A Nearly-Linear Time Algorithm for Linear Programs with Small Treewidth: A Multiscale Representation of Robust Central Path

Arising from structural graph theory, treewidth has become a focus of study in fixed-parameter tractable algorithms in various communities including combinatorics, integer-linear programming, and numerical analysis. Many NP-hard problems are known to be solvable in O(n · 2^O(tw)) time, where tw is the treewidth of the input graph. Analogously, many problems in P should be solvable in O(n ·tw^O(1)) time; however, due to the lack of appropriate tools, only a few such results are currently known. [Fom+18] conjectured this to hold as broadly as all linear programs; in our paper, we show this is true: Given a linear program of the form min_Ax=b,ℓ≤ x≤ u c^⊤ x, and a width-τ tree decomposition of a graph G_A related to A, we show how to solve it in time O(n ·τ^2 log (1/ε)), where n is the number of variables and ε is the relative accuracy. Combined with existing techniques in vertex separators, this leads to algorithms with O(n ·tw^4 log (1/ε)) and O(n ·tw^2 log (1/ε) + n^1.5) run-times when a tree decomposition is not given. Besides being the first of its kind, our algorithm has run-time nearly matching the fastest run-time for solving the sub-problem Ax=b (under the assumption that no fast matrix multiplication is used). We obtain these results by combining recent techniques in interior-point methods (IPMs), sketching, and a novel representation of the solution under a multiscale basis similar to the wavelet basis. This representation further yields the first IPM with o(rank(A)) time per iteration when the treewidth is small.

11/14/2022

### An Improved Parameterized Algorithm for Treewidth

We give an algorithm that takes as input an n-vertex graph G and an inte...
07/01/2019

### Space-Efficient Vertex Separators for Treewidth

Practical applications that use treewidth algorithms have graphs with tr...
06/02/2020

### Fast Algorithms for Join Operations on Tree Decompositions

Treewidth is a measure of how tree-like a graph is. It has many importan...
11/11/2022

### A Faster Small Treewidth SDP Solver

Semidefinite programming is a fundamental tool in optimization and theor...
07/17/2017

### On Treewidth and Stable Marriage

Stable Marriage is a fundamental problem to both computer science and ec...
11/04/2021

### Finding All Leftmost Separators of Size ≤ k

We define a notion called leftmost separator of size at most k. A leftmo...
12/27/2017

### A Short Note on Parameterized Computation of Network Reliability with respect to Treewidth

We consider the classic problem of network reliability. A network is giv...