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

11/10/2020
by   Sally Dong, et al.
0

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.

READ FULL TEXT
POST COMMENT

Comments

There are no comments yet.

Authors

page 1

page 2

page 3

page 4

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...
02/27/2022

A logic-based algorithmic meta-theorem for mim-width

We introduce a logic called distance neighborhood logic with acyclicity ...
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...
01/18/2014

Computing All-Pairs Shortest Paths by Leveraging Low Treewidth

We present two new and efficient algorithms for computing all-pairs shor...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.