Minimum Feedback Vertex Set Research Articles

Minimum feedback vertex set and acyclic coloring

In the feedback vertex set problem, the aim is to minimize, in a connected graph G=( V, E), the cardinality of the set V(G)⊆V , whose removal induces an acyclic subgraph. In this paper, we show an interesting relationship between the minimum feedback vertex set problem and the acyclic coloring problem (which consists in coloring vertices of a graph G such that no two colors induce a cycle in G). Then, using results from acyclic coloring, as well as other techniques, we are able to derive new lower and upper bounds on the cardinality of a minimum feedback vertex set in large families of graphs, such as graphs of maximum degree 3, of maximum degree 4, planar graphs, outerplanar graphs, 1-planar graphs, k-trees, etc. Some of these bounds are tight (outerplanar graphs, k-trees), all the others differ by a multiplicative constant never exceeding 2.

New bounds on the size of the minimum feedback vertex set in meshes and butterflies

Given a graph G = (V;E), the minimum feedback vertex set S is a subset of vertices of minimum size, whose removal induces an acyclic subgraph G0 = (VnS;E 0). The problem of finding S is NP–complete in general graphs, although polynomial time solutions exist for particular classes of graphs. this paper we present upper and lower bounds on the size of the minimum feedback vertex set in meshes and butterflies improving results of Luccio [10].

On enumerating all minimal solutions of feedback problems

We present an algorithm that generates all (inclusion-wise) minimal feedback vertex sets of a directed graph G =( V , E ). The feedback vertex sets of G are generated with a polynomial delay of O (|V| 2 (|V|+|E|)) . We further show that the underlying technique can be tailored to generate all minimal solutions for the undirected case and the directed feedback arc set problem, both with a polynomial delay of O (|V| |E| (|V|+|E|) . Finally, we prove that computing the number of minimal feedback arc sets is #P-hard.

On the Hardness of Approximating Some NP-optimization Problems Related to Minimum Linear Ordering Problem

We study hardness of approximating several minimaximal and maximinimal NP-optimization problems related to the minimum linear ordering problem (MINLOP). MINLOP is to find a minimum weight acyclic tournament in a given arc-weighted complete digraph. MINLOP is APX-hard but its unweighted version is polynomial time solvable. We prove that MIN-MAX-SUBDAG problem, which is a generalization of MINLOP and requires to find a minimum cardinality maximal acyclic subdigraph of a given digraph, is, however, APX-hard. Using results of Hastad concerning hardness of approximating independence number of a graph we then prove similar results concerning MAX-MIN-VC (respectively, MAX-MIN-FVS) which requires to find a maximum cardinality minimal vertex cover in a given graph (respectively, a maximum cardinality minimal feedback vertex set in a given digraph). We also prove APX-hardness of these and several related problems on various degree bounded graphs and digraphs.

Feedback vertex set in hypercubes

Given a graph G=(V,E), the minimum feedback vertex set V is a subset of vertices of minimum size whose removal induces an acyclic subgraph G′=(V\\ V,E′) . The problem of finding V is NP -hard for general networks but interesting polynomial solutions have been found for particular graph classes. In this paper we find close upper and lower bounds to the size of V in a k-dimensional hypercube.

Approximating minimum feedback vertex sets in hypergraphs

The feedback vertex set problem for hypergraphs is considered and an efficient approximation algorithm is presented. It is shown that an approximation factor of k is guaranteed when the cardinality of every hyperedge is bounded by an integer k, generalizing the existing result for ordinary graphs.

On computing the minimum feedback vertex set of a directed graph by contraction operations

Finding the minimum feedback vertex set (MFVS) in a graph is an important problem for a variety of computer-aided design (CAD) applications and graph reduction plays an important role in solving this intractable problem. This paper is largely concerned with three new and powerful reduction operations. Each of these operations defines a new class of graphs, strictly larger than the class of contractible graphs [Levy and Low (1988)] in which the MFVS can be found in polynomial-time complexity. Based on these operations, an exact algorithm run on branch and bound manner is developed. This exact algorithm uses a good heuristic to find out an initial solution and a good bounding strategy to prune the solution space. To demonstrate the efficiency of our algorithms, we have implemented our algorithms and applied them to solving the partial scan problem in ISCAS'89 benchmarks. The experimental results show that if our three new contraction operations are applied, 27 out of 31 circuits in ISCAS'89 benchmarks can be fully reduced. Otherwise, only 12 out of 31 can be fully reduced. Furthermore, for all ISCAS'89 benchmarks our exact algorithm can find the exact cutsets in less than 3 s (CPU time) on a SUN-UltraII workstation. Therefore, the new contraction operations and our algorithms are demonstrated to be very effective in the partial scan application.

An 8-Approximation Algorithm for the Subset Feedback Vertex Set Problem

We present an 8-approximation algorithm for the problem of finding a minimum weight subset feedback vertex set (or subset-fvs, in short). The input in this problem consists of an undirected graph G=(V,E) with vertex weights c(v) and a subset of vertices S called special vertices. A cycle is called interesting if it contains at least one special vertex. A subset of vertices is called a subset-fvs with respect to S if it intersects every interesting cycle. The goal is to find a minimum weight subset-fvs. The best previous algorithm for the general case provided only a logarithmic approximation factor. The minimum weight subset-fvs problem generalizes two NP-complete problems: the minimum weight feedback vertex set problem in undirected graphs and the minimum weight multiway vertex cut problem. The main tool that we use in our algorithm and its analysis is a new version of multicommodity flow, which we call relaxed multicommodity flow. Relaxed multicommodity flow is a hybrid of multicommodity flow and multiterminal flow.

A POLYNOMIAL ALGORITHM FOR FINDING THE MINIMUM FEEDBACK VERTEX SET OF A 3-REGULAR SIMPLE GRAPH 1

A subset of the vertex set of a graph is a feedback vertex set of the graph if the resulting graph is a forest after removed the vertex subset from the graph. A polynomial algorithm for finding a minimum feedback vertex set of a 3-regular simple graph is provided.

Resynthesis and retiming for optimum partial scan

An effective partial scan approach selects flip-flops (FPs) in the minimum feedback vertex set (MFVS) of the FF dependency graph, so that all loops, except self-loops, are broken. However, the MFVS of the circuit (the minimum number of gates whose removal makes the circuit acyclic) is a lower bound and in many cases, significantly smaller than the MFVS of the FF dependency graph. Since only FFs can be considered for scan, this paper investigates the possibility of repositioning FFs so that, in the modified circuit, every circuit MFVS gate drives at least one FF that can be scanned. We show that resynthesis and retiming can always transform any circuit into an equivalent circuit whose FF dependency graph MFVS is equal to the MFVS of the original circuit. Therefore, the MVFS of a circuit is a tight lower bound on the number of scan FFs needed. We first identify the necessary and sufficient conditions under which legal retiming can produce the desired FF repositioning. We show that circuits that do not satisfy these conditions can always be suitably modified using two new resynthesis transformations. The modified circuit can always be retimed to achieve the desired FF repositioning. Experimental results for several large sequential benchmarks show that the number of scan FFs required for the resynthesized and retimed circuit is significantly smaller than that required for the original circuit.

A 2-Approximation Algorithm for the Undirected Feedback Vertex Set Problem

A feedback vertex set of a graph is a subset of vertices that contains at least one vertex from every cycle in the graph. The problem considered is that of finding a minimum feedback vertex set given a weighted and undirected graph. We present a simple and efficient approximation algorithm with performance ratio of at most 2, improving previous best bounds for either weighted or unweighted cases of the problem. Any further improvement on this bound, matching the best constant factor known for the vertex cover problem, is deemed challenging. The approximation principle, underlying the algorithm, is based on a generalized form of the classical local ratio theorem, originally developed for approximation of the vertex cover problem, and a more flexible style of its application.

The Complexity of Planar Counting Problems

We prove the #P-hardness of the counting problems associated with various satisfiability, graph, and combinatorial problems, when restricted to planar instances. These problems include 3Sat, 1-3Sat, 1-Ex3Sat, Minimum Vertex Cover, Minimum Dominating Set, Minimum Feedback Vertex Set, X3C, Partition Into Triangles, and Clique Cover. We also prove the NP-completeness of the Ambiguous Satisfiability} problems [J. B. Saxe, Two Papers on Graph Embedding Problems, Tech. Report CMU-CS-80-102, Dept. of Computer Science, Carnegie Mellon Univ., Pittsburgh, PA, 1980] and the DP-completeness (with respect to random polynomial reducibility) of the unique satisfiability problems [L. G. Valiant and V. V. Vazirani, NP is as easy as detecting unique solutions, in Proc. 17th ACM Symp. on Theory of Computing, 1985, pp. 458--463] associated with several of the above problems, when restricted to planar instances. Previously, very few #P}-hardness results, no {\sf NP}-hardness results, and no DP-completeness results were known for counting problems, ambiguous satisfiability problems, and unique satisfiability problems, respectively, when restricted to planar instances. Assuming {\sf P \neq $ NP}, one corollary of the above results is that there are no $\epsilon$-approximation algorithms for the problems of maximizing or minimizing a linear objective function subject to a planar system of linear inequality constraints over the integers.

Hardware-optimal test register insertion

Implementing a built-in self-test by a "test per clock" scheme offers advantages concerning fault coverage, detection of delay faults, and test application time. Such a scheme is implemented by test registers, for instance built-in logic block observers (BILBO's) and concurrent BILBO's (CBILBO's), which are inserted into the circuit structure at appropriate places. An algorithm is presented which is able to find the cost optimal placement of test registers for nearly all the ISCAS'89 sequential benchmark circuits, and a suboptimal solution with slightly higher costs is obtained for all the circuits within a few minutes of computing time. The algorithm can also be applied to the Minimum Feedback Vertex Set problem in partial scan design, and an optimal solution is found for all the benchmark circuits. The provably optimal solutions for the benchmark circuits mainly use CBILBO's which can simultaneously generate test patterns and compact test responses. Hence, test scheduling is not required, test control is simplified, and test application time is reduced.

Minimum feedback vertex sets in cocomparability graphs and convex bipartite graphs

Polynomial-time algorithms for the feedback vertex set problem in cocomparability graphs and convex bipartite graphs are presented.

Parallel Algorithms for Reducible Flow Graphs

We present parallel NC algorithms for recognizing a reducible flow graph (rfg) and for finding dominators, minimum feedback vertex sets, and a depth first search tree in an rfg. On ann-node rfg, all of these algorithms run inpolylog(n) time usingM(n) processors, whereM(n) is the number of processors needed to multiply twon×nmatrices in polylog time. We show that finding a minimum feedback vertex set in vertex-weighted rfg's or finding a minimum feedback arc set in arc-weighted rfg's is P-complete. For arc or vertex weights in unary, we present RNC algorithms for these problems and show that the problems are in NC if and only if the problem of finding a minimum cut in a flow network is in NC.

Minimizing the number of switchboxes for region definition and ordering assignment

For a building block placement, the routing space will be fully defined into channels and switchboxes because the definition of switchboxes releases all the cycles in a channel precedence graph and further yields a safe routing ordering. However, switchbox routing is more difficult than channel routing. Due to the requirements in the routing phase, it is important for us to minimize the number of switchboxes in the definition of channels and switchboxes. In this paper, a region definition and ordering assignment (RDAOA) algorithm on minimizing the number of switchboxes is proposed. First, the routing space is modeled as a floorplan graph. According to the routing precedence in one "T" type junction, the graph is further transformed into a channel precedence graph. Hence, the problem of minimizing the number of switchboxes in region definition will correspond to the minimum feedback vertex set (MFVS) problem in the channel precedence graph. Second, based on the cyclic properties in a channel precedence graph, the MFVS problem is solved by the minimal-cycle phase and the long-cycle phase. All the minimal cycles and most of the long cycles are broken in the minimal-cycle phase, and the remaining long cycles are further broken in the long-cycle phase. As a result, fewer switchboxes are defined, and these channels and switchboxes are assigned to guarantee a safe routing ordering. The time complexity of the algorithm is proved to be in O(n) time, where n is the number of line segments in a given floorplan graph. Finally, several examples have been tested on the proposed algorithm and other published algorithms, and the experimental results show that our algorithm defines fewer switchboxes than other algorithms.

An optimal algorithm for cycle breaking in directed graphs

This paper describes an exact algorithm for the identification of a minimal feedback vertex set in digital circuits. The proposed algorithm makes use of graph reduction and efficient graph partitioning methods based on local properties of digital circuits. It has been implemented and applied to ISCAS-89 benchmark circuits. Previously, non-optimum solutions were found. In other cases, the optimality of the solution could not be established for all circuits. By using the proposed algorithm we obtained the optimum results for all the circuits in a relatively short CPU time.

On the feedback vertex set problem in permutation graphs

Brandstädt and Kratsch (1987) presented an O( n 6) time algorithm for the minimum weight feedback vertex set problem in a permutation graph of n vertices and m edges. This result was recently improved to O( n.m.m̄) by Brandstät (1993), where m̄ is the number of edges in the complement of G. In this paper, we employ a different dynamic programming scheme to reduce the time complexity to O( nm) for this problem in permutation graphs.

On locating minimum feedback vertex sets

In this paper we study the problem of locating minimum feedback vertex sets in directed graphs. First, we introduce three new transformation-based classes of graphs for which minimum feedback vertex sets can be computed in polynomial time. Second, we delineate an inclusion hierarchy among all of the classes of graphs for which polynomial time feedback vertex set algorithms presently exist. Among the classes of graphs included in the hierarchy are: the reducible flow graphs, the cyclically reducible graphs, the three transformation-based classes that we introduce, and an infinite sequence of classes based on an algorithm of Smith and Walford for finding minimum feedback vertex sets in arbitrary graphs. It follows from our results that one of our new classes, as well as each “Smith/Walford” class, properly includes both the reducible flow graphs and the cyclically reducible graphs. The results presented here serve to unify and focus the work on locating minimum feedback vertex sets in polynomial time.

Finding a minimum feedback arc set in reducible flow graphs

We present a polynomial-time algorithm for finding a minimum weight feedback arc set (FAS) in arc-weighted reducible flow graphs and for finding a minimum weight feedback vertex set (FVS) in vertex-weighted reducible flow graphs. The algorithm has time complexity O(mn 2 log( n 2 m )) , where n is the number of vertices and m is the number of arcs in the reducible flow graph. For unweighted reducible flow graphs, the algorithm for FAS has time complexity O( min(mn 5 3 , m 2)) . We also show that any algorithm that solves the FAS problem or the vertex-weighted FVS problem on reducible flow graphs has time complexity at least that of finding a minimum cut in a flow network, for which the best algorithm currently known has time complexity Θ(mn log( n 2 m )) for networks with arbitrary capacities, and Θ( min(mn 2 3 , m 3 2 )) for networks with unit capacities. Our results establish several connections between the FAS problem on reducible flow graphs and the minimum cut-maximum flow problem on flow networks.