Minimum Fill-In: Inapproximability and Almost Tight Lower Bounds

Abstract

Performing Gaussian elimination to a sparse matrix may turn some zeroes into nonzero values, so called fill-ins, which we want to minimize to keep the matrix sparse. Let n denote the rows of the matrix and k the number of fill-ins. For the minimum fill-in problem, we exclude the existence of polynomial time approximation schemes, assuming P =NP, and the existence of 2 O(n 1−δ ) -time approximation schemes for any positive δ, assuming the Exponential Time Hypothesis. Also implied is a 2 O(k 1/2−δ ) · n O(1) parameterized lower bound. Behind these results is a new reduction from vertex cover, which might be of its own interest: All previous reductions for similar problems are from some kind of graph layout problems.

Approximation algorithms in combinatorial scientific computing

“…A polynomial time approximation scheme (PTAS) is a polynomial time algorithm that takes a parameter and produces an approximate solution for a minimization problem with approximation ratio . Recently, Cao and Sandeep (2017) proved that if the problems of minimizing the fill or the total number of edges in the filled graph has a PTAS, then .…”

Inapproximability of Rank, Clique, Boolean, and Maximum Induced Matching-Widths under Small Set Expansion Hypothesis

“…Recently, Manurangsi showed inapproximability results for maximum biclique problems, minimum k-cut, and densest at-least-k-subgraph [8]. In [9], Wu et al (2014) showed under SSEH that there are no constant factor approximation algorithms for cut, path, tree-widths, minimum fill-in (it has recently been shown that minimum fill-in has no polynomial time approximation scheme unless P = NP and that assuming ETH, there is some positive ε such that no algorithm can find a (1 + ε) approximation in time 2 O(n 1−δ ) for any positive constant δ [10]), one-shot black pebbling costs, and other problems. Those were the first results showing the hardness of constant factor approximation for these graph parameters.…”