Minimum Feedback Arc Set Problem Research Articles

Finding small feedback arc sets on large graphs

The minimum feedback arc set problem (FASP), which seeks to remove a minimum set of arcs from a directed graph to make the remaining graph acyclic, is fundamental in graph algorithms with many applications in social network analysis, circuit testing, deadlock resolution, algorithmic games, and other fields. However, in theory, FASP is NP-Hard and cannot be approximated within any constant under the Unique-Games Conjecture. In practice, the performance of existing algorithms on large graphs is still unsatisfactory. To solve FASP on large graphs, we propose two novel methods in this paper. First, we systematically study the reduction rules for FASP that can quickly reduce the scale of input graphs without losing optimality. We integrate the reduction rules into an algorithm that can be used as a preprocessor for all existing FASP algorithms to simplify the input graphs. Second, we provide a scalable heuristic algorithm based on the divide-and-conquer method for better solution quality. Extensive experiments on a wide range of real-world graphs show that the reduction algorithm effectively reduces the size of input instances. Furthermore, for half of the circuit benchmark graphs, our final algorithm improves the quality of the existing heuristic solution by 24% to 40% within an acceptable running time.

Optimized Reduced Jacobian Formulation for Simultaneous Solution of Control Systems in Electromagnetic Transient Simulations

This paper proposes a new optimized and reduced order Jacobian formulation for the simultaneous solution of control system block-diagrams with feedback loops in the simulation of electromagnetic transients. A reduced order Jacobian matrix formulation has been previously introduced for block-diagrams by solving a minimum Feedback Arc Set (FAS) problem. This paper enhances the latter method by proposing a faster formulation and exploring efficient heuristics for the FAS problem. This optimized reduced Jacobian formulation is implemented in a generic tool for the simulation of electromagnetic transients and demonstrated for large-scale and practical power systems.

Comments on “An Exact Method for the Minimum Feedback Arc Set Problem”

We comment on the ACM Journal of Experimental Algorithmics article “An Exact Method for the Minimum Feedback Arc Set Problem” by Ali Baharev, Hermann Schichl, Arnold Neumaier, and Tobias Achterberg, and we point out that a straightforward implementation of an algorithm we published in 1985, with the same modern technology as used in the article we address, is competitive and mostly superior.

An Exact Method for the Minimum Feedback Arc Set Problem

A feedback arc set of a directed graph G is a subset of its arcs containing at least one arc of every cycle in G . Finding a feedback arc set of minimum cardinality is an NP-hard problem called the minimum feedback arc set problem . Numerically, the minimum set cover formulation of the minimum feedback arc set problem is appropriate as long as all simple cycles in G can be enumerated. Unfortunately, even those sparse graphs that are important for practical applications often have Ω (2 n ) simple cycles. Here we address precisely such situations: An exact method is proposed for sparse graphs that enumerates simple cycles in a lazy fashion and iteratively extends an incomplete cycle matrix. In all cases encountered so far, only a tractable number of cycles has to be enumerated until a minimum feedback arc set is found. The practical limits of the new method are evaluated on a test set containing computationally challenging sparse graphs, relevant for industrial applications. The 4,468 test graphs are of varying size and density and suitable for testing the scalability of exact algorithms over a wide range.

Ranking tournaments with no errors I: Structural description

In this series of two papers we examine the classical problem of ranking a set of players on the basis of a set of pairwise comparisons arising from a sports tournament, with the objective of minimizing the total number of upsets, where an upset occurs if a higher ranked player was actually defeated by a lower ranked player. This problem can be rephrased as the so-called minimum feedback arc set problem on tournaments, which arises in a rich variety of applications and has been a subject of extensive research. In this series we study this NP-hard problem using structure-driven and linear programming approaches. Let T=(V,A) be a tournament with a nonnegative integral weight w(e) on each arc e. A subset F of arcs is called a feedback arc set if T\\F contains no cycles (directed). A collection C of cycles (with repetition allowed) is called a cycle packing if each arc e is used at most w(e) times by members of C. We call T cycle Mengerian (CM) if, for every nonnegative integral function w defined on A, the minimum total weight of a feedback arc set is equal to the maximum size of a cycle packing. The purpose of these two papers is to show that a tournament is CM iff it contains none of four Möbius ladders as a subgraph; such a tournament is referred to as Möbius-free. In this first paper we present a structural description of all Möbius-free tournaments, which relies heavily on a chain theorem concerning internally 2-strong tournaments.

Optimal Segmentation of Directed Graph and the Minimum Number of Feedback Arcs

The minimum feedback arc set problem asks to delete a minimum number of arcs (directed edges) from a digraph (directed graph) to make it free of any directed cycles. In this work we approach this fundamental cycle-constrained optimization problem by considering a generalized task of dividing the digraph into D layers of equal size. We solve the D-segmentation problem by the replica-symmetric mean field theory and belief-propagation heuristic algorithms. The minimum feedback arc density of a given random digraph ensemble is then obtained by extrapolating the theoretical results to the limit of large D. A divide-and-conquer algorithm (nested-BPR) is devised to solve the minimum feedback arc set problem with very good performance and high efficiency.

The minimum feedback arc set problem and the acyclic disconnection for graphs

A minimum feedback arc set of a digraph D is a minimum set of arcs which removal leaves the resultant graph free of directed cycles; its cardinality is denoted by τ1(D). The acyclic disconnection of D, ω⃗(D), is defined as the maximum number of colors in a vertex coloring of D such that every directed cycle of D contains at least one monochromatic arc. In this article we study the relationship between the minimum feedback arc set and the acyclic disconnection of a digraph, we prove that the acyclic disconnection problem is NP-complete. We define the acyclic disconnection and the minimum feedback for graphs. We also prove that ω⃗(G)+τ1(G)=|V(G)| if G is a wheel, a grid or an outerplanar graph.

Efficient computation of feedback arc set at web-scale

The minimum feedback arc set problem is an NP-hard problem on graphs that seeks a minimum set of arcs which, when removed from the graph, leave it acyclic. In this work, we investigate several approximations for computing a minimum feedback arc set with the goal of comparing the quality of the solutions and the running times. Our investigation is motivated by applications in Social Network Analysis such as misinformation removal and label propagation. We present careful algorithmic engineering for multiple algorithms to improve the scalability of each approach. In particular, two approaches we optimize (one greedy and one randomized) provide a nice balance between feedback arc set size and running time complexity. We experimentally compare the performance of a wide range of algorithms on a broad selection of large online networks including Twitter, LiveJournal, and the Clueweb12 dataset. The experiments reveal that our greedy and randomized implementations outperform the other approaches by simultaneously computing a feedback arc set of competitive size and scaling to web-scale graphs with billions of vertices and tens of billions of arcs. Finally, we extend the algorithms considered to the probabilistic case in which arcs are realized with some fixed probability and provide detailed experimental comparisons.

Feedback Arc Set Problem and NP-Hardness of Minimum Recurrent Configuration Problem of Chip-Firing Game on Directed Graphs

In this paper we present further studies of recurrent configurations of chip-firing games on Eulerian directed graphs (simple digraphs), a class on the way from undirected graphs to general directed graphs. A computational problem that arises naturally from this model is to find the minimum number of chips of a recurrent configuration, which we call the minimum recurrent configuration (MINREC) problem. We point out a close relationship between MINREC and the minimum feedback arc set (MINFAS) problem on Eulerian directed graphs, and prove that both problems are NP-hard.

The Feedback Arc Set Problem with Triangle Inequality Is a Vertex Cover Problem

We consider the (precedence constrained) Minimum Feedback Arc Set problem with triangle inequalities on the weights, which finds important applications in problems of ranking with inconsistent information. We present a surprising structural insight showing that the problem is a special case of the minimum vertex cover in hypergraphs with edges of size at most 3.

The Left and Right Context of a Word

Since a Chinese syllable can correspond to many characters (homophones), the syllable-to-character conversion task is quite challenging for Chinese phonetic input methods (CPIM). There are usually two stages in a CPIM: 1. segment the syllable sequence into syllable words, and 2. select the most likely character words for each syllable word. A CPIM usually assumes that the input is a complete sentence, and evaluates the performance based on a well-formed corpus. However, in practice, most Pinyin users prefer progressive text entry in several short chunks, mainly in one or two words each (most Chinese words consist of two or more characters). Short chunks do not provide enough contexts to perform the best possible syllable-to-character conversion, especially when a chunk consists of overlapping syllable words. In such cases, a conversion system often selects the boundary of a word with the highest frequency. Short chunk input is even more popular on platforms with limited computing power, such as mobile phones. Based on the observation that the relative strength of a word can be quite different when calculated leftwards or rightwards, we propose a simple division of the word context into the left context and the right context. Furthermore, we design a double ranking strategy for each word to reduce the number of errors in Step 1. Our strategy is modeled as the minimum feedback arc set problem on bipartite tournament with approximate solutions derived from genetic algorithm. Experiments show that, compared to the frequency-based method (FBM) (low memory and fast) and the conditional random fields (CRF) model (larger memory and slower), our double ranking strategy has the benefits of less memory and low power requirement with competitive performance. We believe a similar strategy could also be adopted to disambiguate conflicting linguistic patterns effectively.

AN IMPLICIT COVER PROBLEM IN WILD POPULATION STUDY

In an implicit combinatorial optimization problem, the constraints are not enumerated explicitly but rather stated implicitly through equations, other constraints or auxiliary algorithms. An important subclass of such problems is the implicit set cover (or, equivalently, hitting set) problem in which the sets are not given explicitly but rather defined implicitly. For example, the well-known minimum feedback arc set problem is such a problem. In this paper, we consider such a cover problem that arises in the study of wild populations in biology in which the sets are defined implicitly via the Mendelian constraints and prove approximability results for this problem.

The Minimum Feedback Arc Set Problem is NP-Hard for Tournaments

Answering a question of Bang-Jensen and Thomassen [4], we prove that the minimum feedback arc set problem is NP-hard for tournaments.

The Minimal Feedback Arc Set Problems

Given a directed network G(N;A), the minimum feedback arc set problem is to find a minimal weighted subset of arcs A' ⊂ A so that G(N;A\\A′) is acyclic. We first provide a mixed 0-1 integer programming formulation whose LP relaxation can be solved by a strongly polynomial algorithm of complexity O(|N|3 log |N|). An enumerative framework is provided which makes use of this relaxation repeatedly. We also provide a result which gives rise to a heuristic algorithm.

Combinatorial algorithms for feedback problems in directed graphs

Given a weighted directed graph G=( V, A), the minimum feedback arc set problem consists of finding a minimum weight set of arcs A′⊆ A such that the directed graph ( V, A⧹ A′) is acyclic. Similarly, the minimum feedback vertex set problem consists of finding a minimum weight set of vertices containing at least one vertex for each directed cycle. Both problems are NP-complete. We present simple combinatorial algorithms for these problems that achieve an approximation ratio bounded by the length, in terms of number of arcs, of a longest simple cycle of the digraph.

On Integrality, Stability and Composition of Dicycle Packings and Covers

Given a digraph D, the minimum integral dicycle cover problem (known also as the minimum feedback arc set problem) is to find a minimum set of arcs that intersects every dicycle; the maximum integral dicycle packing problem is to find a maximum set of pairwise arc disjoint dicycles. These two problems are NP-complete.