Minimum Fill-In: Inapproximability and Almost Tight Lower Bounds

TL;DR

This paper investigates the computational complexity of the minimum fill-in problem, establishing new inapproximability results, lower bounds, and a novel reduction from vertex cover, advancing understanding of its hardness.

Contribution

It provides the first strong inapproximability and parameterized lower bounds for the minimum fill-in problem, using a new reduction from vertex cover.

Findings

Excludes PTASs assuming P≠NP.

Rules out $2^{O(n^{1-\delta})}$-time approximation schemes assuming ETH.

Establishes a new reduction from vertex cover for hardness proofs.

Abstract

Given an $n*n$ sparse symmetric matrix with $m$ nonzero entries, performing Gaussian elimination may turn some zeroes into nonzero values. To maintain the matrix sparse, we would like to minimize the number $k$ of these changes, hence called the minimum fill-in problem. Agrawal et al.~[FOCS'90] developed the first approximation algorithm, based on early heuristics by George [SIAM J Numer Anal 10] and by Lipton et al.~[SIAM J Numer Anal 16]. The objective function they used is $m+k$, the number of nonzero elements after elimination. An approximation algorithm using $k$ as the objective function was presented by Natanzon et al.~[STOC'98]. These two versions are incomparable in terms of approximation. Parameterized algorithms for the problem was first studied by Kaplan et al.~[FOCS'94]. Fomin & Villanger [SODA'12] recently gave an algorithm running in time $2^{O(\sqrt{k} \log k)}+n^{O(1)}$. Hardness results of this problem are surprisingly scarce, and the few known ones are either weak or have to use nonstandard complexity conjectures. The only inapproximability result by Wu et al.~[IJCAI'15] applies to only the objective function $m+k$, and is grounded on the Small Set Expansion Conjecture. The only nontrivial parameterized lower bounds, by Bliznets et al.~[SODA'16], include a very weak one based on ETH, and a strong one based on hardness of subexponential-time approximation of the minimum bisection problem on regular graphs. For both versions of the problem, we exclude the existence of PTASs, assuming P$\ne$NP, and the existence of $2^{O(n^{1-\delta})}$-time approximation schemes for any positive $\delta$, assuming ETH. It also implies a $2^{O(k^{1/2-\delta})} n^{O(1)}$ 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.