On the Isometric Path Partition Problem

This is a part of a proposed research project of Kuwait Uni- versity, Kuwait. The author will appreciate to receive your comments and constructive criticism on this initial draft. The author’s contact address is [email protected]. arXiv:1808.09097v1 [math.CO] 26 Aug 2018

1 On the isometric path partition problem

Paul Manuel

Department of Information Science, College of Computing Science and Engineering, Kuwait University, Kuwait [email protected]

Abstract The isometric path cover (partition) problem of a graph is to ﬁnd a minimum set of isometric paths which cover (partition) the vertex set of the graph. The isometric path cover (partition) number of a graph is the cardinality a minimum isometric path cover (partition). We prove that the isometric path partition problem and the isometric k-path partition problem for k ≥ 3 are NP-complete on general graphs. Fisher and Fitzpatrick [7] have shown that the isometric path cover number of (r × r)-dimensional grid is d2r/3e. We show that the isometric path cover (partition) number of (r × s)-dimensional grid is s when r ≥ s(s − 1). We establish that the isometric path cover (partition) number of (r × r)-dimensional torus is r when r is even and is either r or r + 1 when r is odd. Then, we demonstrate that the isometric path cover (partition) number of an r-dimensional Benes network is 2r. In addition, we provide partial solutions for the isometric path cover (partition) problems for cylinder and multi-dimensional grids.

Keywords: path cover problem; isometric path partition problem; isometric path cover problem; multi-dimensional grids; cylinder; torus; Benes; NP-complete; AMS Subj. Class.: 05C12, 05C70, 68Q17

1 Introduction

An undirected connected graph is represented by G(V,E) where V is the vertex set and E is the edge set. A path means a simple path with distinct vertices. A path P is an induced path in G if the subgraph induced by the vertices of P is a path. A path between two vertices is an isometric path if it induces the shortest distance between the two points. Let us recall that isometric path and geodesic are other names for shortest path. While an isometric path cover is a set of isometric paths

2 which cover the vertex set V , an isometric path partition is a set of isometric paths which partition V . The isometric path cover number which is denoted by ipc(G) is the cardinality of a minimum isometric path cover. The isometric path partition number ipp(G) is defined accordingly. While the isometric path cover problem is to find a minimum isometric path cover, the isometric path partition problem is to find a minimum isometric path partition. A diametral isometric path of a graph is an isometric path whose length is equal to the diameter of the graph. As the path cover (partition) problem is to find a minimum path cover (partition), the induced path cover (partition) problem is to find a minimum induced path cover (partition). Let diam(G) denote the diameter of graph G. Last few decades, the theory of isometric paths has been studied extensively. Aggarwal et al. [1] have illustrated the application of the isometric path cover problem in the design of VLSI layouts. The theory of isometric path problems is the backbone in the design of efficient algorithms in transport networks [20], computer networks [14, 24], parallel architectures [25], social networks [3, 12], VLSI layout design [1], wireless sensor networks [6], multimedia networks [5] and in other networks such as GIS networks [26], large network systems [2] and stochastic networks [23]. Since the Hamiltonian path problem is NP-complete [11], the path cover problem and the path partition problem are NP-complete. Since the Hamiltonian induced path problem is NP-complete [4, 11], the induced path cover problem and the induced path partition problem are NP-complete. However, the complexity status of the isometric path cover problem and the isometric path partition problem are unknown [16]. This fact has been highlighted and emphasized recently [16, 17]. In this paper, we settle the long-standing open problem [16] by proving that the isometric path partition problem is NP-complete on general graphs. The isometric path cover number has been computed for tress, cycles, complete bipartite graphs, the Cartesian product of paths (including hypercubes) under some restricted cases [7, 8, 9, 10]. Fisher and Fitzpatrick [7] have derived a lower bound l |V | m that ipc(G) ≥ diam(G)+1 and have shown that the isometric path cover number of (r×r) grid is d2r/3e. Fitzpatrick et al. [10] have shown that the isometric path cover number of hypercube Qr is at least 2r/(r + 1). In addition, they have also shown r−log2(r+1) that ipc(Qr) = 2 when r + 1 is a power of 2. Pan and Chang have given a linear-time algorithm to solve the isometric path cover problem on block graphs [21], complete r-partite graphs and Cartesian products of 2 or 3 complete graphs [22]. There is no literature on the isometric path partition problem [16]. The readers are suggested to read the survey paper by Manuel [16] for detailed information. In section 2, we show that the isometric path partition problem is NP-complete. We also show that the isometric k-path partition problem is NP-complete on general graphs for k ≥ 3. In section 4, we compute the exact values of the isometric path cover and isometric path partition number for cylinders and multi-dimensional grids

3 under certain conditions. In sections 5 and 6, we also derive the exact values of the isometric path cover number and isometric path partition number for square torus and Benes networks.

2 The isometric path partition problem is NP- complete for general graphs

Given a graph G(V,E), a k-path is a path having at most k vertices. A set S of k-paths is a k-path partition if each vertex of V belongs to exactly one member of S. The k-path partition problem is to ﬁnd a k-path partition of minimum cardinality in G.

Theorem 2.1 ([19]) The 3-path partition problem is NP-complete on bipartite graphs.

Now we will prove that the isometric path partition problem is NP-complete on general graphs. As a ﬁrst step, we provide a polynomial reduction from the 3-path partition problem on bipartite graphs to the isometric path partition problem on general graphs. The key fact in a bipartite graph is that each 3-path in a bipartite graph is an isometric path. Given a graph G(V,E) where V = {1, 2 . . . n}, the reduced graph is denoted by G0(V 0,E0). The vertex set V 0 is V ∪ {x, y, z}. The edge set E0 is E ∪ {xz, zy} ∪ {iz / i ∈ V }. See Figure 1.

Figure 1: (a) G = (V,E) and (b) G0 = (V 0,E0)

Next, we will identify some basic structural properties of G0(V 0,E0).

4 Property 2.2 If G is a bipartite graph, then diam(G0) = 2. For i, j, k ∈ V , a diametral isometric path in G0 is either xzy, izx, izy, ikj or izj when ij is not an edge.

Property 2.3 A bipartite graph G has a 3-path partition S of cardinality k iﬀ G0 has an isometric path partition S0 of cardinality k + 1.

Applying Property 2.2 and 2.3, we state one of the main results of this paper. Theorem 2.4 The isometric path partition problem is NP-complete on general graphs. The isometric k-path partition problem is a generalization of the isometric path partition problem. Using the same logic, one can also prove that Theorem 2.5 The isometric k-path partition problem is NP-complete on general graphs for k ≥ 3.

3 Isometric path partition versus isometric path cover

Our point of discussion in this section is to emphasize that the isometric path cover problem and the isometric path partition problem are two different combinatorial problems. From the perspective of computational complexity, let us see how these two problems differ even on simple architectures such as trees and grids. For the star graph K1,`, while ipc(K1,`) = d`/2e, ipp(K1,`) = ` − 1. Pan and Chang [21] have proved that the isometric path cover number of trees with ` leaves is d`/2e. For the isometric path partition number, this is not true even in complete binary trees. In fact, it is a challenge to find the isometric path partition number for trees and the isometric path partition number for trees is unknown [16]. Fisher and Fitzpatrick [7] have shown that the isometric path cover number of grid Ξ(r, r) is d2r/3e. However, the isometric path partition number of grid Ξ(r, r) remains an open problem [16]. It seems that the isometric path partition number of grid Ξ(r, r) is r. However, it requires a mathematical proof which seems to be a challenge. In the same way, one can apply greedy algorithm or brute-force algorithm to compute the isometric path cover number for some graphs such as interval graph or circular arc graph by starting from its diametral isometric path. On interval graphs , our experiments demonstrate that greedy algorithm provides sharp result for the isometric path cover number but significantly deviates for the isometric path partition number. There are some features which are common to both the problems. One straight- forward lower bound that is common to both the problems is as follows:

5 Theorem 3.1 ([7]) If diam(G) denotes the diameter of a graph G, then ipp(G) ≥ l |V (G)| m ipc(G) ≥ diam(G)+1 . Though the bound in Theorem 3.1 seems to be trivial, it is very effective and useful. This paper truly exploits this lower bound to prove that the isometric path cover number and the isometric path partition number are equal for some networks such as (r × r)-dimensional torus and Benes networks. Monnot and Toulouse [19] have studied an NP-complete problem that is whether a graph on nk vertices can be partitioned into n paths of length k. An isometric path version of this problem is to decide if a graph G(V,E) can be partitioned into diametral isometric paths. Any fixed interconnection network which possesses this feature is considered as a “good” architecture [15, 25] because it is easy to design and implement efficient data communication, message broadcasting and other routing algorithms. In the following sections, we point out that a few fixed interconnection networks such as torus and Benes networks inherit this nice feature.

4 The isometric path cover / partition problem on Cartesian product Pr G Given a graph G, let diam(G) denote the diameter of G. In this section, we consider the Cartesian product Pr G where G is any graph and Pr is a path graph on r vertices. The vertex set V (Pr) of Pr is {1, 2 . . . r} and the vertex set V (Pr G) of Pr G is {(j, v) / j ∈ V (Pr) and v ∈ V (G)}. For each j ∈ V (Pr), the subgraph j induced by the vertices {(j, v) / v ∈ V (G)} of Pr G is denoted by G . In other 1 2 r words, there are r copies of G in Pr G which are represented by G ,G ...G j j respectively. An edge of G in Pr G is called G -edge. A G-edge in Pr G is a Gj-edge for some j = 1, 2 . . . r.

Lemma 4.1 ([13]) Given a graph G and a path graph Pr, an isometric path of the Cartesian product Pr G can have a maximum of diam(G) number of G-edges.

Lemma 4.2 Given a graph G and a path graph Pr, ipc(Pr G) ≤ ipp(Pr G) ≤ |V (G)|.

The following lemma is the key to derive a lower bound on ipc(Pr G)) and ipp(Pr G)):

Lemma 4.3 Given a graph G and a path graph Pr, let S be an isometric path j0 cover (partition) of Cartesian product Pr G. If there exists a G of Pr G for j0 some j0, 1 ≤ j0 ≤ r, such that no isometric path of S contains any G -edge, then |S| ≥ |V (G)|.

6 Lemma 4.4 Given a graph G and a path graph Pr, ipp(Pr G) ≥ ipc(Pr G) ≥ |V (G)| when r ≥ diam(G)|V (G)|.

Following Lemma 4.4 and Lemma 4.2, we state that

Theorem 4.5 Given a graph G and a path graph Pr, ipp(Pr G)) = ipc(Pr G)) = |V (G)| when r ≥ diam(G)|V (G)|.

4.1 The isometric path cover / partition problem on grids and cylinders

In this section, we consider multi-dimensional grids Ξ(d1, d2 . . . dr) which are the

Cartesian product of paths Pd1 ,Pd2 ...Pdr . Fisher and Fitzpatrick [7] have shown that the isometric path cover number of Ξ(r, r) grid is d2r/3e. Fitzpatrick et al. [10] have shown that the lower bound of the isometric path cover number of hypercube r−log2(r+1) Qr is 2r/(r + 1). In addition, they have also shown that ipc(Qr) = 2 when r + 1 is a power of 2. Though there are some results available for the isometric path cover number on grids, there are no literature for the isometric path partition problem on multi-dimensional grids including 2-dimensional grids and cylinders [16].

Theorem 4.6 Given an r-dimensional grid Ξ(d1, d2 . . . dr), ipp(Ξ(d1, d2 . . . dr)) = ipc(Ξ(d1, d2 . . . dr)) = d2d3 . . . dr when d1 ≥ ((d2−1)+(d3−1) ... (dr −1))(d2d3 . . . dr).

Corollary 4.7 The isometric path cover (partition) number of (r × s)-dimensional grid is s when r ≥ s(s − 1).

Theorem 4.8 Given a cylinder Pr Cs, ipp(Pr Cs) = ipc(Pr Cs) = s when r ≥ bs/2cs.

5 The isometric path partition problem on torus

In this section, we study the exact value of the isometric path partition number of (r × r)-dimensional torus. To our knowledge, there is no literature on the isometric path partition problem on torus. In this section, we will show that the isometric path cover (partition) number of (r ×r)-dimensional torus G is r when r is even and is either r or r + 1 when r is odd. An (8 × 8)-dimensional torus is given in Figure 2 and a (9 × 9)-dimensional torus is given in Figure 3.

Theorem 5.1 The isometric path cover (partition) number of (r × r)-dimensional torus is r when r is even and is either r or r + 1 when r is odd.

7 Figure 2: An (8 × 8)-dimensional torus

6 The isometric path partition problem on Benes networks

k Let Zk = {0, 1 . . . k −1} and Z2 = {x0x1 . . . xk−1/ xi = 0 or 1}. When we say i ∈ Zk, it means i mod k. The vertex set of an r-dimensional Butterfly BF (r) is {hw, ii / r r+1 0 0 w ∈ Z2 and i ∈ Z }. Two vertices hw, ii and hw , i i of BF (r) are linked by an edge if i0 = i + 1 and either w = w0 or w and w0 differ only in the bit in position i. An r-dimensional Benes network BN(r) is called back-to-back butterflies and is obtained by merging the r-level vertices of two Butterfly networks BF (r). The r 2r+1 vertex set of BN(r) is {hw, ii / w ∈ Z2 and i ∈ Z }. Thus, the degree of 0-level vertices of BN(r) remains 2. Figure 4 displays BN(3). We apply the lower bound of Theorem 3.1 to find the exact value for ipp(BN(r)) and ipc(BN(r)) of Benes networks.

Theorem 6.1 The isometric path cover (partition) number of an r-dimensional Benes network is 2r.

8 Figure 3: A (9 × 9)-dimensional torus

Acknowledgment

This work is a part of research project proposal which is submitted to Kuwait University, Kuwait.

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