Feedback Arc Set Research Articles

Handling inheritance violation for secure interoperation of heterogeneous systems

In multiple heterogeneous systems, one of key challenges of integrating policies is the conflict detection and resolution while preserving individual policy consistency. In this paper, the inheritance hierarchy of a security policy is formulated with a directed graph. Solving Inheritance Violation Problem (IVP) is formulated as a feedback arc set problem, which is NP-hard. Then some classical approximation algorithms are introduced. The IVP in two interoperating domains is converted into the problem of finding a minimum weight vertex cover problem in a bipartite graph, which is polynomial-time solvable.

Fixed-parameter tractability results for full-degree spanning tree and its dual

We provide first-time fixed-parameter tractability results for the NP-complete problems MAXIMUM FULL-DEGREE SPANNING TREE and MINIMUM-VERTEX FEEDBACK EDGE SET. These problems are dual to each other: In MAXIMUM FULL-DEGREE SPANNING TREE, the task is to find a spanning tree for a given graph that maximizes the number of vertices that preserve their degree. For MINIMUM-VERTEX FEEDBACK EDGE SET the task is to minimize the number of vertices that end up with a reduced degree. Parameterized by the solution size, we exhibit that MINIMUM-VERTEX FEEDBACK EDGE SET is fixed-parameter tractable and has a problem kernel with the number of vertices linearly depending on the parameter k. Our main contribution for MAXIMUM FULL-DEGREE SPANNING TREE, which is W[1]-hard, is a linear-size problem kernel when restricted to planar graphs. Moreover, we present subexponential-time algorithms in the case of planar graphs.

Aggregating inconsistent information

We address optimization problems in which we are given contradictory pieces of input information and the goal is to find a globally consistent solution that minimizes the extent of disagreement with the respective inputs. Specifically, the problems we address are rank aggregation, the feedback arc set problem on tournaments, and correlation and consensus clustering. We show that for all these problems (and various weighted versions of them), we can obtain improved approximation factors using essentially the same remarkably simple algorithm. Additionally, we almost settle a long-standing conjecture of Bang-Jensen and Thomassen and show that unless NP⊆BPP, there is no polynomial time algorithm for the problem of minimum feedback arc set in tournaments.

Capacity inverse minimum cost flow problem

Given a directed graph G = (N,A) with arc capacities u and a minimum cost flow problem defined on G, the capacity inverse minimum cost flow problem is to find a new capacity vector u' for the arc set A such that a given feasible flow x' is optimal with respect to the modified capacities. Among all capacity vectors u' satisfying this condition, we would like to find one with minimum ||u' - u|| value. We consider two distance measures for ||u' - u||, rectilinear and Chebyshev distances. By reduction from the feedback arc set problem we show that the capacity inverse minimum cost flow problem is NP-hard in the rectilinear case. On the other hand, it is polynomially solvable by a greedy algorithm for the Chebyshev norm. In the latter case we propose a heuristic for the bicriteria problem, where we minimize among all optimal solutions the number of affected arcs. We also present computational results for this heuristic.

Consensus Genetic Maps as Median Orders from Inconsistent Sources

A genetic map is an ordering of genetic markers calculated from a population of known lineage. While traditionally a map has been generated from a single population for each species, recently researchers have created maps from multiple populations. In the face of these new data, we address the need to find a consensus map--a map that combines the information from multiple partial and possibly inconsistent input maps. We model each input map as a partial order and formulate the consensus problem as finding a median partial order. Finding the median of multiple total orders (preferences or rankings)is a well studied problem in social choice. We choose to find the median using the weighted symmetric difference distance, a more general version of both the symmetric difference distance and the Kemeny distance. Finding a median order using this distance is NP-hard. We show that for our chosen weight assignment, a median order satisfies the positive responsiveness, extended Condorcet,and unanimity criteria. Our solution involves finding the maximum acyclic subgraph of a weighted directed graph. We present a method that dynamically switches between an exact branch and bound algorithm and a heuristic algorithm, and show that for real data from closely related organisms, an exact median can often be found. We present experimental results using seven populations of the crop plant Zea mays.

Some Parameterized Problems On Digraphs

We survey results and open questions on complexity of parameterized problems on digraphs. The problems include the feedback vertex and arc set problems, induced subdigraph problems and directed k-leaf problems. We also prove some new results on the topic. Most of these new results are on parameterizations of the backward paired comparison problem.

Efficient Exact Algorithms through Enumerating Maximal Independent Sets and Other Techniques

We give substantially improved exact exponential-time algorithms for a number of NP-hard problems. These algorithms are obtained using a variety of techniques. These techniques include: obtaining exact algorithms by enumerating maximal independent sets in a graph, obtaining exact algorithms from parameterized algorithms and a variant of the usual branch-and-bound technique which we call the "colored" branch-and-bound technique. These techniques are simple in that they avoid detailed case analyses and yield algorithms that can be easily implemented. We show the power of these techniques by applying them to several NP-hard problems and obtaining new improved upper bounds on the running time. The specific problems that we tackle are: (1) the Odd Cycle Transversal problem in general undirected graphs, (2) the Feedback Vertex Set problem in directed graphs of maximum degree 4, (3) Feedback Arc Set problem in tournaments, (4) the 4-Hitting Set problem and (5) the Minimum Maximal Matching and the Edge Dominating Set problems. The algorithms that we present for these problems are the best known and are a substantial improvement over previous best results. For example, for the Minimum Maximal Matching we give an O*(1.4425n) algorithm improving the previous best result of O*(1.4422m) [35]. For the Odd Cycle Transversal problem, we give an O*(1.62n) algorithm which improves the previous time bound of O*(1.7724n) [3].

Feedback arc set problem in bipartite tournaments

In this paper we give ratio 4 deterministic and randomized approximation algorithms for the Feedback Arc Set problem in bipartite tournaments. We also generalize these results to give a factor 4 deterministic approximation algorithm for Feedback Arc Set problem in multipartite tournaments.

Improved fixed parameter tractable algorithms for two “edge” problems: MAXCUT and MAXDAG

We give improved parameterized algorithms for two “edge” problems MAXCUT and MAXDAG, where the solution sought is a subset of edges. MAXCUT of a graph is a maximum set of edges forming a bipartite subgraph of the given graph. On the other hand, MAXDAG of a directed graph is a set of arcs of maximum size such that the graph induced on these arcs is acyclic. Our algorithms are obtained through new kernelization and efficient exact algorithms for the optimization versions of the problems. More precisely our results include: (i) a kernel with at most αk vertices and βk edges for MAXCUT. Here 0 < α ⩽ 1 and 1 < β ⩽ 2 . Values of α and β depends on the number of vertices and the edges in the graph; (ii) a kernel with at most 4 k / 3 vertices and 2 k edges for MAXDAG; (iii) an O ∗ ( 1.2418 k ) parameterized algorithm for MAXCUT in undirected graphs. This improves the O ∗ ( 1.4143 k ) 1 1 The O ∗ notation suppresses polynomial terms. Thus we write O ∗ ( T ( x ) ) for a time complexity of the form O ( T ( x ) ) ⋅ poly ( | x | ) where T ( x ) grows exponentially with | x | , the input size. See [G. Woeginger, Exact algorithms for NP-hard problems: A survey, in: Combinatorial Optimization—Eureka! You Shrink! in: Lecture Notes in Comput. Sci., vol. 2570, 2003, pp. 185–207] for a detailed discussion on this. algorithm presented in [E. Prieto, The method of extremal structure on the k-maximum cut problem, in: The Proceedings of Computing: The Australasian Theory Symposium (CATS), 2005, pp. 119–126]; (iv) an O ∗ ( 2 n ) algorithm for optimization version of MAXDAG in directed graphs. This is the first such algorithm to the best of our knowledge; (v) an O ∗ ( 2 k ) parameterized algorithm for MAXDAG in directed graphs. This improves the previous best of O ∗ ( 4 k ) presented in [V. Raman, S. Saurabh, Parameterized algorithms for feedback set problems and their duals in tournaments, Theoretical Computer Science 351 (3) (2006) 446–458]; (vi) an O ∗ ( 16 k ) parameterized algorithm to determine whether an oriented graph having m arcs has an acyclic subgraph with at least m / 2 + k arcs. This improves the O ∗ ( 2 k ) algorithm given in [V. Raman, S. Saurabh, Parameterized algorithms for feedback set problems and their duals in tournaments, Theoretical Computer Science 351 (3) (2006) 446–458]. In addition, we show that if a directed graph has minimum out degree at least f ( n ) (some function of n) then Directed Feedback Arc Set problem is fixed parameter tractable. The parameterized complexity of Directed Feedback Arc Set is a well-known open problem.

Hardness of fully dense problems

In the past decade, there has been a stream of work in designing approximation schemes for dense instances of NP-Hard problems. These include the work of Arora, Karger and Karpinski from 1995 and that of Frieze and Kannan from 1996. We address the problem of proving hardness results for (fully) dense problems, which has been neglected despite the fruitful effort put in upper bounds. In this work, we prove hardness results of dense instances of a broad family of CSP problems, as well as a broad family of ranking problems which we refer to as CSP-Rank. Our techniques involve a construction of a pseudorandom hypergraph coloring, which generalizes the well-known Paley graph, recently used by Alon to prove hardness of feedback arc-set in tournaments.

Feedback arc set in bipartite tournaments is NP-complete

The Feedback Arc Set problem asks whether it is possible to delete at most k arcs to make a directed graph acyclic. We show that Feedback Arc Set is NP-complete for bipartite tournaments, that is, directed graphs that are orientations of complete bipartite graphs.

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.

A branch-and-bound algorithm to solve the linear ordering problem for weighted tournaments

The linear ordering problem consists in finding a linear order at minimum remoteness from a weighted tournament T, the remoteness being the sum of the weights of the arcs that we must reverse in T to transform it into a linear order. This problem, also known as the search of a median order, or of a maximum acyclic subdigraph, or of a maximum consistent set, or of a minimum feedback arc set, is NP-hard; when all the weights of T are equal to 1, the linear ordering problem is the same as Slater's problem. In this paper, we describe the principles and the results of an exact method designed to solve the linear ordering problem for any weighted tournament. This method, of which the corresponding software is freely available at the URL address http://www.enst.fr/~charon/tournament/median.html, is based upon a branch-and-bound search with a Lagrangean relaxation as the evaluation function and a noising method for computing the initial bound. Other components are designed to reduce the BB-search-tree.

On Optimizing Compatible Security Policies in Wireless Networks

This paper deals with finding the maximum number of security policies without conflicts. By doing so we can remove security loophole that causes security violation. We present the problem of maximum compatible security policy and its relationship to the problem of maximum acyclic subgraph, which is proved to be NP-hard. Then we present a polynomial-time approximation algorithm and show that our result has approximation ratio for any integer with complexity .

Ranking Tournaments

A tournament is an oriented complete graph. The feedback arc set problem for tournaments is the optimization problem of determining the minimum possible number of edges of a given input tournament T whose reversal makes T acyclic. Ailon, Charikar, and Newman showed that this problem is NP-hard under randomized reductions. Here we show that it is in fact NP-hard. This settles a conjecture of Bang-Jensen and Thomassen.

Node duplication and routing algorithms for quantum-dot cellular automata circuits

Quantum-dot cellular automata (QCA) is a novel computing mechanism that can represent binary information based on the spatial distribution of electron charge configuration in chemical molecules. QCA circuit layout is currently restricted to a single layer with very limited number of wire crossings permitted. Thus, wire crossing minimisation is crucial in improving the manufacturability of QCA circuits. We present the first QCA node duplication and routing algorithms for wire crossing minimisation. Our duplication algorithm named fan-out tolerance duplication (FTD) explores node duplication in conjunction with node placement using K-layered bipartite graphs (KLBG). FTD successfully removes additional crossings at the cost of increased area and allows flexible tradeoff between area and wire crossing. Our routing algorithm, namely cycle breaker (CB), constructs a modified vertical constraint graph (VCG) to enforce additional vertical relation for wire crossing reduction. We formulate and provide a heuristic solution for the weighted minimum feedback edge set problem to effectively remove cycles from the VCG. As a result, FTD and CB achieve wire crossing results that are very close to theoretical lower bound and outperform the conventional algorithms significantly.

Parameterized algorithms for feedback set problems and their duals in tournaments

The parameterized feedback vertex (arc) set problem is to find whether there are k vertices (arcs) in a given graph whose removal makes the graph acyclic. The parameterized complexity of this problem in general directed graphs is a long standing open problem. We investigate the problems on tournaments, a well studied class of directed graphs. We consider both weighted and unweighted versions. We also address the parametric dual problems which are also natural optimization problems. We show that they are fixed parameter tractable not just in tournaments but in oriented directed graphs (where there is at most one directed arc between a pair of vertices). More specifically, the dual problem we show fixed parameter tractable are: Given an oriented directed graph, is there a subset of k vertices (arcs) that forms an acyclic directed subgraph of the graph? Our main results include: • an O ( ( 2.4143 ) k n ω ) 1 1 ω is the exponent of the best matrix multiplication algorithm. algorithm for weighted feedback vertex set problem, and an O ( ( 2.415 ) k n ω ) algorithm for weighted feedback arc set problem in tournaments; • an O ( ( e 2 k / k ) k k 2 + min { m lg n , n 2 } ) algorithm for the dual of feedback vertex set problem ( maximum vertex induced acyclic graph) in oriented directed graphs, and an O ( 4 k k + m ) algorithm for the dual of feedback arc set problem ( maximum arc induced acyclic graph) in general directed graphs. We also show that the dual of feedback vertex set is W [ 1 ] — hard in general directed graphs and the feedback arc set problem is fixed parameter tractable in dense directed graphs. Our results are the first non-trivial results for these problems.

A classification of tournaments having an acyclic tournament as a minimum feedback arc set

Given a tournament with an acyclic tournament as a feedback arc set we give necessary and sufficient conditions for this feedback arc set to have minimum size.

Two feedback problems for graphs with bounded tree-width

Many difficult (often NP-complete) optimization problems can be solved efficiently on graphs of small tree-width with a given tree-decomposition. In this paper, it is discussed how to solve the minimum feedback vertex set problem and the minimum vertex feedback edge set problem efficiently by using dynamic programming on a tree-decomposition.

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.