Minimum Fill-in Problem Research Articles

Minimum fill-in: Inapproximability and almost tight lower bounds

Given an n×n sparse symmetric matrix with m nonzero entries, performing Gaussian elimination may turn some zeroes into nonzero values, so called fill-ins. The minimum fill-in problem asks whether it is possible to perform the elimination with at most k fill-ins. We exclude the existence of polynomial time approximation schemes for this problem, assuming P≠NP, and the existence of 2O(n1−δ)-time approximation schemes for any positive δ, assuming the Exponential Time Hypothesis. We also give a 2O(k1/2−δ)⋅nO(1) parameterized lower bound. All these results come as corollaries of a new reduction from vertex cover to the minimum fill-in problem, which might be of its own interest: All previous reductions for similar problems start from some kind of graph layout problems, and hence have limited use in understanding their fine-grained complexity.

Solving Graph Problems via Potential Maximal Cliques

The BT algorithm of Bouchitté and Todinca based on enumerating potential maximal cliques, originally proposed for the treewidth and minimum fill-in problems, yields improved exact exponential-time algorithms for various graph optimization problems related to optimal triangulations. While the BT algorithm has received significant attention in terms of theoretical analysis, less attention has been paid on engineering efficient implementations of the algorithm for different problems and thereby on empirical studies on its effectiveness in practice. In this work, we provide an experimental evaluation of an implementation of the BT algorithm, based on our second-place winning entry in the 2nd Parameterized Algorithms and Computational Experiments Challenge (PACE 2017), extended to several related graph problems: treewidth, minimum fill-in, generalized and fractional hypertreewidth, and the total table size problem. Instead of focusing on problem-specific optimization of BT for a particular problem, our focus in this work is on studying the applicability of BT more generally to a range of problems. Based on the results, we conclude that an efficient implementation of the BT algorithm yields an empirically competitive approach to each of the considered problems when compared to available implementations of alternative problem-specific algorithmic approaches.

Searching for better fill-in

Minimum Fill-in is a fundamental and classical problem arising in sparse matrix computations. In terms of graphs it can be formulated as a problem of finding a triangulation of a given graph with the minimum number of edges. In this paper, we study the parameterized complexity of local search for the Minimum Fill-in problem in the following form: Given a triangulation H of a graph G, is there a better triangulation, i.e. triangulation with less edges than H, within a given distance from H? We prove that this problem is fixed-parameter tractable (FPT) being parameterized by the distance from the initial triangulation, by providing an algorithm that in time f(k)|G|O(1) decides if a better triangulation of G can be obtained by swapping at most k edges of H. Our result adds Minimum Fill-in to the list of very few problems for which local search is known to be FPT.

Minimum Fill-in of Sparse Graphs: Kernelization and Approximation

The Minimum Fill-in problem is to decide if a graph can be triangulated by adding at most k edges. The problem has important applications in numerical algebra, in particular in sparse matrix computations. We develop kernelization algorithms for the problem on several classes of sparse graphs. We obtain linear kernels on planar graphs, and kernels of size $\mathcal {O}(k^{3/2})$ in graphs excluding some fixed graph as a minor and in graphs of bounded degeneracy. As a byproduct of our results, we obtain approximation algorithms with approximation ratios $\mathcal {O}(\log{k})$ on planar graphs and $\mathcal {O}(\sqrt{k}\log{k})$ on H-minor-free graphs. These results significantly improve the previously known kernelization and approximation results for Minimum Fill-in on sparse graphs.

Subexponential Parameterized Algorithm for Minimum Fill-In

The Minimum Fill-in problem is used to decide if a graph can be triangulated by adding at most $k$ edges. In 1994, Kaplan, Shamir, and Tarjan showed that the problem is solvable in time $\mathcal{O}(2^{\mathcal{O}({k})}+k^2 nm)$ on graphs with $n$ vertices and $m$ edges and thus is fixed parameter tractable. Here, we give the first subexponential parameterized algorithm solving Minimum Fill-in in time $\mathcal{O}(2^{\mathcal{O}(\sqrt{k}\log{k})} +k^2 nm)$. This substantially lowers the complexity of the problem. Techniques developed for Minimum Fill-in can be used to obtain subexponential parameterized algorithms for several related problems, including Minimum Chain Completion, Chordal Graph Sandwich, and Triangulating Colored Graph.

Editorial: ISAAC 2008 Special Issue

This special issue contains a selection of six papers from the 19th Annual International Symposium on Algorithms and Computation (ISAAC 2008), which was held in Gold Coast, Australia on December 15–17, 2008. The ISAAC 2008 program committee accepted 78 papers among 229 high-quality submissions from 40 countries, after a rigorous review process. Among those accepted papers, the following six papers were selected and invited to this special issue, based on their evaluation by the program committee. These six papers all went through the standard refereeing process of Algorithmica before being accepted for publication. The papers in this issue show recent advances in various areas of algorithms and theory of computation research. The first paper, “Signature Theory in Holographic Algorithms” by Cai and Lu, develops the signature theory in holographic algorithms, in terms of d-realizability and d-admissibility. For the class of 2-realizable signatures, they prove a Birkhoff-type theorem which determines this class. This is followed by characterization theorems for 1-realizability and 1-admissibility. Finally, they present some counting problems solvable in polynomial time by holographic algorithms. The second paper, “Faster Parameterized Algorithms for MINIMUM FILL-IN” by Bodlaender, Heggernes and Villanger, presents two parameterized algorithms for the MINIMUM FILL-IN problem, also known as the CHORDAL COMPLETION problem. The first algorithm requires polynomial space, which improves the exponential part

Faster Parameterized Algorithms for Minimum Fill-in

We present two parameterized algorithms for the Minimum Fill-in problem, also known as Chordal Completion: given an arbitrary graph G and integer k, can we add at most k edges to G to obtain a chordal graph? Our first algorithm has running time $\mathcal {O}(k^{2}nm+3.0793^{k})$ , and requires polynomial space. This improves the base of the exponential part of the best known parameterized algorithm time for this problem so far. We are able to improve this running time even further, at the cost of more space. Our second algorithm has running time $\mathcal {O}(k^{2}nm+2.35965^{k})$ and requires $\mathcal {O}^{\ast}(1.7549^{k})$ space. To achieve these results, we present a new lemma describing the edges that can safely be added to achieve a chordal completion with the minimum number of edges, regardless of k.

Minimum fill-in and treewidth of [formula omitted] and [formula omitted] graphs

In this paper we investigate how graph problems that are NP-hard in general, but polynomial time solvable on split graphs, behave on input graphs that are close to being split. For this purpose we define split + k e and split + k v graphs to be the graphs that can be made split by removing at most k edges and at most k vertices, respectively. We show that problems like treewidth and minimum fill-in are fixed parameter tractable with parameter k on split + k e graphs. Along with positive results of fixed parameter tractability of several problems on split + k e and split + k v graphs, we also show a surprising hardness result. We prove that computing the minimum fill-in of split + k v graphs is NP-hard even for k = 1 . This implies that also minimum fill-in of chordal + k v graphs is NP-hard for every k . In contrast, we show that the treewidth of split + 1 v graphs can be computed in linear time. This gives probably the first graph class for which the treewidth and the minimum fill-in problems have different computational complexity.

Treewidth and Minimum Fill-in: Grouping the Minimal Separators

We use the notion of potential maximal clique to characterize the maximal cliques appearing in minimal triangulations of a graph. We show that if these objects can be listed in polynomial time for a class of graphs, the treewidth and the minimum fill-in are polynomially tractable for these graphs. We prove that for all classes of graphs for which polynomial algorithms computing the treewidth and the minimum fill-in exist, we can list their potential maximal cliques in polynomial time. Our approach unifies these algorithms. Finally we show how to compute in polynomial time the potential maximal cliques of weakly triangulated graphs for which the treewidth and the minimum fill-in problems were open.

A Polynomial Approximation Algorithm for the Minimum Fill-In Problem

In the minimum fill-in problem, one wishes to find a set of edges of smallest size, whose addition to a given graph will make it chordal. The problem has important applications in numerical algebra and has been studied intensively since the 1970s. We give the first polynomial approximation algorithm for the problem. Our algorithm constructs a triangulation whose size is at most eight times the optimum size squared. The algorithm builds on the recent parameterized algorithm of Kaplan, Shamir, and Tarjan for the same problem. For bounded degree graphs we give a polynomial approximation algorithm with a polylogarithmic approximation ratio. We also improve the parameterized algorithm.

A linear time algorithm for minimum fill-in and treewidth for distance hereditary graphs

A graph is distance hereditary if it preserves distances in all its connected induced subgraphs. The MINIMUM FILL-IN problem is the problem of finding a chordal supergraph with the smallest possible number of edges. The TREEWIDTH problem is the problem of finding a chordal embedding of the graph with the smallest possible clique number. In this paper we show that both problems are solvable in linear time for distance hereditary graphs.

Tractability of Parameterized Completion Problems on Chordal, Strongly Chordal, and Proper Interval Graphs

We study the parameterized complexity of three NP-hard graph completion problems. The minimum fill-in problem asks if a graph can be triangulated by adding at most k edges. We develop O(ck m) and O(k2mn+f(k)) algorithms for this problem on a graph with n vertices and m edges. Here f(k) is exponential in k and the constants hidden by the big-O notation are small and do not depend on k. In particular, this implies that the problem is fixed-parameter tractable (FPT). The proper interval graph completion problem, motivated by molecular biology, asks if a graph can be made proper interval by adding no more than k edges. We show that the problem is FPT by providing a simple search-tree-based algorithm that solves it in O(ck m)-time. Similarly, we show that the parameterized version of the strongly chordal graph completion problem is FPT by giving an O(ck m log n)-time algorithm for it. All of our algorithms can actually enumerate all possible k-completions within the same time bounds.

Minimum Fill-in on Circle and Circular-Arc Graphs

We present two algorithms solving the minimum fill-in problem on circle graphs and on circular-arc graphs in timeO(n3).

Triangulating multitolerance graphs

In this paper we introduce a new class of graphs which generalize both the tolerance and the trapezoid graphs, the multitolerance graphs. We show that the difference between the pathwidth and the treewidth of a multitolerance graph is at most one, and we develop an algorithm which solves the minimum fill-in problem for a multitolerance graph with a given representation in polynomial time. These results complement the recent results of Habib and Möhring [18, 25] about the treewidth and pathwidth of cocomparability graphs and graphs without asteroidal triples, and those of Kloks et al. [21] about the minimum fill-in problem.

A local reductive elimination for the fill-in of graphs

Denote by G the complementary graph of a graph G. F ⊆ E( G) is called a fill-in of G, if G + F is a chordal graph. A minimum fill-in of G is a fill-in of G with minimum cardinality. In this paper we show that, for a k-connected graph G, if v is a vertex of G with degree k, and there exists a ( k − 1)-clique of G contained in N( v), then for every minimum fill-in F of G − v − E o, F⌣ E o is a minimum fill-in of G, where E o = {xy ϵ E( G) | x, y ϵ N(v)} . Using this local reductive elimination we design a linear time algorithm to solve the minimum fill-in problem for Halin graphs. We also show that the cardinality of the minimum fill-in of a Halin graph can be denoted by a formula.

Optimal pivot strategies for load-flow calculation

The determination of optimal sequences of pivots for a matrix M is a well-known but open problem in load-flow calculation. Optimal means that the system Mx = b can be solved by Gauss's algorithm, where the number of arithmetic operations is minimal; the space to store the triangular factors of M should also be minimal. This problem is commonly studied by considering the equivalent (general) minimum fill-in problem. This graph-theoretical optimization problem is the subject of this paper. A valuable tool to attack it is the Initial Theorem due to Bertele and Brioschi. We demonstrate the feasibility of this method by calculating optimal orderings for the well-known AEP test networks. The Initial Theorem can be generalized to a special class of graphs which represents the zero, non-zero pattern of the Jacobian matrix, derived from the nonlinear network equations. Additionally, a separation approach is presented, producing (if necessary assumptions are satisfied) optimal orderings of the global network which are composed of optimal orderings of the (local) sub-networks.