Cmjv02i06p0804 by Journal of Computer and Mathematical Sciences

J. Comp. & Math. Sci. Vol.2 (6), 804-811 (2011)

Feedback Arc Set In Oriented Graphs INDRA RAJASINGH, BHARATI RAJAN and S. LITTLE JOICE Department of Mathematics, Loyola College, Chennai 600 034, India ABSTRACT A feedback arc set in a directed graph D is a set S of arcs in D such that \ is acyclic. The problem of determining a feedback arc set with least number of arcs is NP - complete for a general digraph. We define the strong feedback arc number ௦ for various orientations of an undirected graph G. The strong feedback arc problem of an undirected graph is to find a strong feedback arc set S of G O for some strongly oriented O G such that | | ୱ . In this paper our focus is on interconnection networks with ௦ 1. Keywords: Oriented graphs, minimum feedback arc set, Hamiltonian graphs, Hexagonal networks, Mesh and honeycomb network.

1. INTRODUCTION AND TERMINOLOGY Given a digraph , , the minimum feedback arc (or vertex) set problem is to find a smallest subset S of E (or V) whose removal induces an acyclic subgraph of D. The feedback arc (or vertex) set problem is to determine the minimum size of a feedback arc (or vertex) set for D. The minimum feedback arc set problem and the vertex set problem are known to be NP-complete if we put no restriction on the input directed graphs1. Recently Jiong et al. have proved that the

minimum feedback arc set problem is NPcomplete for bipartite tournaments12. Motivated by the general hardness result, many subclasses of directed graphs have been considered. It turns out that the minimum feedback arc set problem can be solved in polynomial time in reducible flow graphs20 and in cyclically reducible flow graphs24. The problem was originally formulated in the area of combinatorial circuit design21. Other applications of the problem are connected with resource allocation mechanisms in operating systems that prevent deadlocks, to the constraint

Journal of Computer and Mathematical Sciences Vol. 2, Issue 6, 31 December, 2011 Pages (780-898)

805

Indra Rajasingh, et al., J. Comp. & Math. Sci. Vol.2 (6), 804-811 (2011)

satisfaction problem and Bayesian inference in artificial intelligence, to the study of monopolies in synchronous distributed systems and to converter placement problems in optical networks9,12. Due to its important application in voting systems8, feedback set problems restricted to tournaments received considerable attention. A tournament is a directed graph where there is exactly one arc between each pair of vertices. The best known approximation algorithm for minimum feedback problem is one with an approximation ratio two2. The problem has been studied for some graphs, such as hypercubic graphs, meshes, toroids, butterflies, cube-connected cycles, 2,3,4,9,10,12, hypercubes and directed split-stars 17,18,21,24,25 . Kralovic and Ruzicka15 derived the cardinality of a minimum feedback set of the Kautz undirected graph. Xu et al.26 obtained the feedback numbers of Kautz digraphs. In this paper we define the minimum feedback arc set problem for oriented graphs. An orientation of an undirected graph G is an assignment of exactly one direction to each of the edges of G. Let G be E

an undirected graph. There are 2 orientations for G. Let Ox(G) denote all the orientations of G. For an orientation ௫ , let G(O) denote the directed graph with orientation O and whose underlying graph is G. An orientation O of an undirected graph G is said to be strongly connected if for any two vertices x, y of G(O), there are both (x, y) – path and (y, x) – path in G(O). Let Os(G) denote all the strongly connected orientations of G. We define the feedback arc number ௫ of G as ௫ minை minௌ | |/ is a

feedback arc set for and ௫ . Similarly we define the strong feedback arc number of G as ௦ minை minௌ | |/ is a feedback arc set for and ௦ . The strong feedback arc problem of an undirected graph G is to find a strong feedback arc set S of for some strong orientation such that | | ௦ . The problem is NP- complete since the input is any oriented digraph. Consider the graph G which is the union of two 4-cycles sharing a common vertex as in Figure 1(a). In any strong orientation of G, both the 4-cycles are directed cycles. For example, see Figure 1(b). Thus ௦ 2. For example , consider a chain of n number of 4-cycles such that any two 4-cycles share atmost one vertex. In this paper our focus is on the interconnection networks with ௦ 1. G: C2

Figure 1 (a)

G: Figure 1 (b)

vn v1

Figure 2

Journal of Computer and Mathematical Sciences Vol. 2, Issue 6, 31 December, 2011 Pages (780-898)

Indra Rajasingh, et al., J. Comp. & Math. Sci. Vol.2 (6), 804-811 (2011)

In section 2 we prove that all hamiltonian graphs have ௦ 1. We also observe that many interconnection networks are hamiltonian and hence have ௦ 1. In section 3 we prove that there are nonhamiltonian graphs with ௦ 1. This class includes hexagonal networks, mesh and honeycomb networks. 2. HAMILTONIAN GRAPHS The Hamiltonian property is one of the major requirements of a good interconnection network. This cycle is often used due to its simplicity, expansibility and easiness of implementation. Theorem 1: If G is Hamiltonian then ௦ 1. Proof: Let : ଵ , ଶ , … , ௡ , ଵ be a hamiltonian cycle in . Orient in the anticlockwise sense. Arbitrarily fix the arc ଵ , ଶ Consider the edge ௜ , ௝ . Now one of the ௜ , ௝ segments of contains the arc ଵ , ଶ . Orient ௜ , ௝ in

such a way that the arc ప , ఫ followed by the ௝ , ௜ segment on containing the arc ଵ , ଶ forms a cycle. See Figure 2. Orient every edge of in this manner to obtain a oriented graph . ଵ , ଶ is acyclic.⁫ Now ଵ \

Direct interconnection networks can be modeled by graphs, with nodes and edges corresponding to processors and communication links between them, respectively. A survey of these networks is given in23. Multiprocessor interconnection

806

networks are often required to connect thousands of homogeneously replicated processor-memory pairs, each of which is called a processing node. Design and use of multiprocessor interconnection networks have recently drawn considerable attention due to the availability of inexpensive, powerful microprocessors and memory chips6. We begin with hypercube networks which are hamiltonian. HYPERCUBES An n-dimensional hypercube (r-cube for short) can be represented as an undirected graph !௥ = (V;E) such that V consists of 2r nodes which are labeled as binary numbers of length n from 00…0 to 11…1. E is the set of edges that connects two nodes if and only if they differ in exactly one bit of their labels. Theorem 2: Let G be !௥ , a hypercube of dimension r. Then ௦ !௥ 1. See Figure 3(a).

Figure 3(a): ૜ \ is acyclic

FOLDED HYPERCUBE An n-dimensional folded hypercube (folded n-cube for short) "!௥ is a regular rdimensional hypercube augmented by

Journal of Computer and Mathematical Sciences Vol. 2, Issue 6, 31 December, 2011 Pages (780-898)

807

Indra Rajasingh, et al., J. Comp. & Math. Sci. Vol.2 (6), 804-811 (2011)

adding more links among its nodes. More specifically, a folded r-cube is obtained by adding a link between two nodes whose addresses are complementary to each other in addition to its original n links. Theorem 3: Let G be "!௥ , a folded hypercube of dimension r. Then ௦ 1. See Figure 3(b).

Figure 3(b): ૜ \ is acyclic

PETERSON GRAPH The generalized Petersen graph P (n, ௡ିଵ m), n ≥ 3 and 1 ≤ m ≤ # ଶ $, consists of an outer n-cycle %ଵ , %ଶ , … , %௡ , a set of n spokes %௜ , ௜ , 1 ≤ i ≤ n, and n inner edges ௜ , ௜ା௠ with indices taken modulo n. The Peterson graph P (n, m) with & ' 0 )*+ 4 is hamiltonian. Theorem 4: Let G be a Peterson graph P (n, ௡ିଵ m), n ≥ 3, 1 ≤ m ≤ # ଶ $, & ' 0 )*+ 4 . Then ௦ 1. CIRCULANT GRAPH A circulant undirected graph, denoted by G(n;±S) where S ⊆ {1, 2, ..., .&/2/}, n ≥ 3 is defined as a graph consisting of the vertex set V = {0, 1 ... n − 1} and the edge set E = {(i, j) : |j − i| ≡ s(mod n), s ∈ S}. It is clear that G(n;±1)is

the undirected cycle ௡ and G(n; ±{1, 2 ... .&/2/}) is the complete graph 0௡ . Further G(n; ±{1, 2, ..., j}), 1 ≤ j < .&/2/, n ≥ 3 is a 2j-regular graph. Theorem 5: Let G be &; , a circulant graph. Then ௦ 1. TORUS Consider the mesh 2 ), & . If there are wraparound edge that connect the vertices of first column to those of the last column and wraparound edges that connect the vertices of the first row to those of the last row, then the resulting architecture is called a torus denoted by 3 ), & . Theorem 6: Let G be 3 ), & , a torus. Then ௦ 1. See Figure 4. e

Figure 4: T(4,5) \ e is acyclic

AUGMENTED BUTTERFLY Let r ≥ 1 be an integer. The vertices of the r -dimensional augmented butterfly network are the pairs (l, x) where l is a nonnegative integer 0 ≤ l ≤ n called the level, and x = (x1, x2, x3 …xr) is a binary string of length r. In ABr the vertex (l, x) , 0 ≤ l ≤ r – 1, is adjacent to the vertices (l +1, x), (l+1,

Journal of Computer and Mathematical Sciences Vol. 2, Issue 6, 31 December, 2011 Pages (780-898)

Indra Rajasingh, et al., J. Comp. & Math. Sci. Vol.2 (6), 804-811 (2011)

x1,x2,….xl, xl+1,xl+2,…xr), (l, x1,x2,…xl-1, xl, xl+1…xr ) and (l, x1,x2,….xl ,xl+1,xl+2…xr ). In particular, when l = 0, the vertex (0, x1,x2,x3…xr ) is adjacent to the vertices (1,x1,x2,x3…xr), (1, x1,x2,x3…xr ) and (0, x1,x2,x3…xr ). Also when l = r, (r, x1,x2,x3…xr ) is adjacent to the vertices (r, x1, x2, x3 …xr ), (r – 1 , x1, x2, x3 …xr ) and (r–1, x1,x2,x3…xr-1,xr).Clearly ABr has (r+1)2r vertices and 3r.2r edges. Theorem 7: Let G be ABr, an augmented butterfly of dimension r. Then ௦ 1. See Figure 5. e

808

two ends of 6௜ . Orient the cycle 6଴ and the path 6ଵ , 6ଶ , … , 6௞ in the clockwise direction. Case (i): Let :, :௜ , :௝ 6଴ and ; 6௜ where 6 = : , . . ;, . . : is a path joining the ௜

ఫ

ప

two distinct points :௝ , :௜ . If :௜ < : < :௝ , there is a path in 6 = : , : , , . . :, . . : ଴

ప

పାଵ

ఫ

joining :௜ , :, :௝ . Now, :, ; :, :ఫ 9 : :ఫ , ; and ;, ;, :ప 9 :

ప , : . Clearly, there is a cycle

containing x and y and an edge : ప , :పାଵ is common to both 6଴ and 6଴ °6௜ . Hence the graph is strongly connected and the removal of the common edge : ప , :పାଵ makes the graph acyclic. Thus ௦ 1.

The following cases can be dealt with similarly. Figure 5: AB2 \ e is acyclic

The converse of Theorem 1 is not true. There are non-hamiltonian graphs for which ௦ 1. We prove that the Hexagonal, Mesh and Honeycomb networks belong to this class. Theorem 8: Let G be a block graph with 4 5 1. Then ௦ 1. Proof: Consider an ear decomposition of edges of G. The edge set of G is partitioned into a simple cycle 6଴ and simple paths 6ଵ , 6ଶ , … , 6௞ such that for each 7 5 0, 6௜ is joined to previous paths only at its (2 distint) ends i.e., 6௜ 8 9௜ழ௝ 6௝ consists of the

Case (ii): Let :, :௜ , :௝ 6଴ and ; 6௜ where 6௜ = : ఫ , . . ;, . . :ప is a path joining the two distinct points :௝ , :௜ , if :௝ < : < :௜ . Case (iii): If = 6௝ and ; 6௜ where 6௜ =(:௝ , :௜ ) , 6௝ =(:௞ , :௟ ) are the paths joining the two distinct points :௝ , :௜ and :௞ , :௟ of 6଴ respectively, if :௝ < ; < :௞ < :௜ < = < :௟ or :௝ < ; < :௜ < :௞ < = < :௟ .⁫ We note that Theorem 1 is a corollary to Theorem 8. Since the proof of the Theorem 1 is constructive and most of the interconnection networks are Hamiltonian, Theorem 1 is dealt with separately.

Journal of Computer and Mathematical Sciences Vol. 2, Issue 6, 31 December, 2011 Pages (780-898)

809

Indra Rajasingh, et al., J. Comp. & Math. Sci. Vol.2 (6), 804-811 (2011)

3. HONEYCOMB NETWORKS The honeycomb mesh interconnection network is based on the hexagonal tessellation. Honeycomb meshes can be built from hexagons. One hexagon is a honeycomb mesh of size one, denoted >2ଵ . See Figure 6(a). The honeycomb mesh >2ଶ of size two is obtained by adding six hexagons to the boundary of >2ଵ . See Figure 6(b). Inductively, honeycomb mesh >2௥ of size r is obtained from >2௥ିଵ by adding a layer of hexagons around the boundary of >2௥ିଵ . Alternatively, the size r of >2௥ is determined as the number of hexagons between the center and boundary of >2௥ (inclusive) and the number of vertices and edges of >2௥ are 6@ ଶ and 9@ ଶ B 3@ respectively22.

(a)

(b)

Figure 6 : Honeycomb Network ૚ and ૛

The Honeycomb network can be redrawn using only horizontal and vertical line segments. See Figures 6(b) and 7(a).

Theorem 9: For the Honeycomb network >2௥ of dimension r, ௦ >2௥ 1. See Figure 7(b).

Figure 7(b): ૛

4. HEXAGONAL NETWORK It is known that there exist three regular plane tessellations, composed of the same kind of regular polygons: triangular, square, and hexagonal. They are the basis for the design of direct interconnection networks with highly competitive overall performance. Hexagonal networks are based on triangular plane tessellation, or the partition of a plane into equilateral triangles and are widely studied in6. Hexagonal network >D @ of dimension r has 3@ ଶ B 3@ E 1 vertices and 9@ ଶ B 15@ E 6 edges, where r is the number of vertices on one side of the hexagon22. Theorem 10: Let be >D @ . Then ௦ >D @ 1. 5. MESH

Figure 7(a) : Honeycomb Network of dimension 2 ( ૛

Let 6௡ denote a path on n vertices. For ), & G 2, 6௠ H 6௡ is defined as a two dimensional (2-D) mesh with m rows and n columns.

Journal of Computer and Mathematical Sciences Vol. 2, Issue 6, 31 December, 2011 Pages (780-898)

Indra Rajasingh, et al., J. Comp. & Math. Sci. Vol.2 (6), 804-811 (2011)

Theorem 11: Let be the mesh M (m ,n). Then ௦ 2 ), & 1. 6. SCOPE FOR FURTHER RESEARCH A kernel in a directed graph D is a set K of vertices of D such that no two vertices in K are adjacent and for every vertex u in D\K there is a vertex v in K, such that (u ,v) is an arc of D. The concept of kernels in digraphs was introduced in different ways11,9. Von Neumann and Morgenstern19 were the first to introduce kernels when describing winning positions in 2 person games. They proved that any directed acyclic graph has a unique kernel. Not every digraph has a kernel and if a digraph has a kernel, this kernel is not necessarily unique. All odd length directed cycles and most tournaments have no kernels. If D is finite, the decision problem of the existence of a kernel is NPcomplete for a general digraph5,6, and for a planar digraph with indegrees 2, 7 outdegrees 2 and degrees 3 . It is further known that a finite digraph all of whose cycles have even length has a kernel13 and that the question of the number of kernels is NP- complete even for this restricted class of digraphs14. It is easy to see that it is possible to modify the strong orientations that yield ௦ 1 for various architectures to obtain acyclic orientations of these architectures yielding unique kernels. It would be interesting to study whether feedback arc sets in oriented graphs throw light on the study of kernels in oriented graphs and vice- versa. 7. CONCLUSION In this paper we have solved the strong feedback arc set problem for the

810

oriented hexagonal, mesh and honeycomb networks. It would also be interesting to examine the kernel number problem and the strong feedback arc set problem for other networks like butterfly networks and benes networks. REFERENCES 1.

G. Ausiello, A. D’Atri, and M. Protasi. Structure preserving reductions among convex optimization problems. Journal of Computer and System Sciences, 21, 136–153 (1980). V. Bafna, P. Berman, T. Fujito,A 2approximation algorithm for the undirected feedback vertex set problem, SIAM J. Discrete Math. 12, 289–297 (1999). R. Bar-Yehuda, D. Geiger, J.S. Naor, R.M. Roth, Approximation algorithms for the feedback vertex set problem with applications to constraint satisfaction and Bayesian inference, SIAM J. Comput. 27, 942–959 (1998). S. Bau, L.W. Beineke, Z. Liu, G. Du, R.C. Vandell, Decycling cubes and grids, Utilitas Math. 59,129-137 (2001). C. Berge and P. Duchet, Recent problems and results about kernels in directed graphs, Discrete Mathematics 86 (1990), 27- 31. M.S. Chen, K.G. Shin, D.D. Kandlur, Addressing, routing, and broadcasting in hexagonal mesh multiprocessors, IEEE Transactions on Computers 39, 10–18 (1990). V. Chvatal, On the computational complexity of finding a kernel, Report No. CRM- 300, Centre de Recherches Mathematiques, Universite de Montreal, (1973).

Journal of Computer and Mathematical Sciences Vol. 2, Issue 6, 31 December, 2011 Pages (780-898)

811 8.

10.

11.

12.

13.

14.

15.

16.

Indra Rajasingh, et al., J. Comp. & Math. Sci. Vol.2 (6), 804-811 (2011)

V. Conitzer. Computing Slater rankings using similarities among candidates. In Proc. 21st AAAI. AAAI Press, (2006). P. Festa, P.M. Pardalos, M.G.C. Resende, Feedback set problems, in: D.-Z. Du, P.M. Pardalos (Eds.), Handbook of Combinatorial Optimization, Vol. A, Kluwer Academic Publisher, Dordrecht, 209 (1999). R. Focardi, F.L. Luccio, D. Peleg, Feedback vertex set in hypercubes, Inform. Process. Lett. 76, 1–5 (2000). A. S. Fraenkel, Planar kernel and Grundy with + I 3, + ା I 2, + ି I 2 are NP – complete, Discrete Applied Mathematics 3, 257-262 (1981). Jionh Guo, Falk Huffner, Hannes Moser, Feedback arc set in bipartite tournaments is NP-complete, Information processing letters, 102, 62-65 (2007). JEAN-Marie Le Bars, Counterexample of the 0–1 Law for fragments of existential second order logic: an overview, The Bulletin of Symbolic Logic, 6(1), 67-82 (2000). Jie Wu, Extended Dominating-SetBased Routing in Ad hoc Wireless Networks with Unidirectional Links, IEEE Transactions on Parallel and Distributed Systems, 13 (9), 866-881 (2002). R. Kralovic, P. Ruzicka, Minimum feedback vertex sets in shuffle-based interconnection networks, Inform. Process. Lett. 86 (4),191–196 (2003). M. Kswasnik, The generalization of Richardson theorem, Discussiones Math. IV, 11-14 (1981).

17.

18.

19.

20.

21.

22.

23.

24.

25.

26.

Y.D. Liang, On the feedback vertex set in permutation graphs, Inform. Process. Lett. 52, 123–129 (1994). F.L. Luccio, Almost exact minimum feedback vertex set in meshes and butterflies, Inform. Process. Lett. 66, 59–64 (1998). M. Richardson, Solutions of irreflexive relations, Ann. of Math. 58, 573-590 (1953). A. Shamir. A linear time algorithm for finding cutsets in reduced graphs. SIAM Journal on Computing, 8(4), 645–655 (1979). G.W. Smith Jr., R.B.Walford, The identification of a minimal feedback vertex set of a directed graph, IEEE Trans. Circuits and Systems CAS-22, 9–15 (1975). I. Stojmenovic, Honeycomb networks: topological properties and communication algorithms, IEEE Trans. Parallel Distrib. Systems 8(10), 10361042 (October 1997). I. Stojmenovic, Direct interconnection networks, in: A.Y. Zomaya (Ed.), Paralel and Distributed Computing Handbook, McGraw-Hill, Inc., Tokyo, 537-567 (1996). C.-C.Wang, E.L. Lloyd, M.L. Soffa, Feedback vertex sets and cyclically reducible graphs, J. Assoc. Comput. Mach. 32, 296–313 (1985). F.-H.Wang, C.-J. Hsu, J.-C. Tsai, Minimal feedback vertex sets in directed split-stars, Networks 45 (4), 218–223 (2005). J-M Xu, Ye-Zhou Wu, Jia Huang, Chao Yang, Feedback numbers of Kautz digraphs, Discrete mathematics 307, 1589-1599 (2007).

Journal of Computer and Mathematical Sciences Vol. 2, Issue 6, 31 December, 2011 Pages (780-898)