Solving the 2-Disjoint Paths Problem in Nearly Linear Time

Given four distinct vertices s₁,s₂,t₁, and t₂ of a graph G, the 2-disjoint paths problem is to determine two disjoint paths, p₁ from s₁ to t₁ and p₂ from s₂ to t₂, if such paths exist. Disjoint can mean vertex- or edge-disjoint. Both, the edge- and the vertex-disjoint version of the problem, are NP-hard in the case of directed graphs. For undirected graphs, we show that the O(mn)-time algorithm of Shiloach can be modified to solve the 2-vertex-disjoint paths problem in only O(n + mα(m,n)) time, where m is the number of edges in G, n is the number of vertices in G, and where α denotes the inverse of the Ackermann function. Our result also improves the running time for the 2-edge-disjoint paths problem on undirected graphs as well as the running times for the 2-vertex- and the 2-edge-disjoint paths problem on dags.     

The k-in-a-Path Problem for Claw-free Graphs

The k-in-a-Path problem is to test whether a graph contains an induced path spanning k given vertices. This problem is NP-complete in general graphs, already when k=3. We show how to solve it in polynomial time on claw-free graphs, when k is an arbitrary fixed integer not part of the input. As a consequence, also the k-Induced Disjoint Paths and the k-in-a-Cycle problem are solvable in polynomial time on claw-free graphs for any fixed k. The first problem has as input a graph G and k pairs of specified vertices (s _i ,t _i ) for i=1,…,k and is to test whether G contain k mutually induced paths P _i such that P _i connects s _i and t _i for i=1,…,k. The second problem is to test whether a graph contains an induced cycle spanning k given vertices. When k is part of the input, we show that all three problems are NP-complete, even for the class of line graphs, which form a subclass of the class of claw-free graphs.     

Approximate k-Steiner Forests via the Lagrangian Relaxation Technique with Internal Preprocessing

An instance of the k -Steiner forest problem consists of an undirected graph G=(V,E), the edges of which are associated with non-negative costs, and a collection $\mathcal{D}=\{(s_{1},t_{1}),\ldots,(s_{d},t_{d})\}An instance of the k -Steiner forest problem consists of an undirected graph G=(V,E), the edges of which are associated with non-negative costs, and a collection D={(s₁,t₁),?,(s_d,t_d)}\mathcal{D}=\{(s_{1},t_{1}),\ldots,(s_{d},t_{d})\} of distinct pairs of vertices, interchangeably referred to as demands. We say that a forest ℱ⊆G connects a demand (s _i ,t _i ) when it contains an s _i -t _i path. Given a profit k _i for each demand (s _i ,t _i ) and a requirement parameter k, the goal is to find a minimum cost forest that connects a subset of demands whose combined profit is at least k. This problem has recently been studied by Hajiaghayi and Jain (SODA ’06), whose main contribution in this context was to relate the inapproximability of k-Steiner forest to that of the dense k -subgraph problem. However, Hajiaghayi and Jain did not provide any algorithmic result for the respective settings, and posed this objective as an important direction for future research. In this paper, we present the first non-trivial approximation algorithm for the k-Steiner forest problem, which is based on a novel extension of the Lagrangian relaxation technique. Specifically, our algorithm constructs a feasible forest whose cost is within a factor of O(min{n^2/3,?d}·logd)O(\min \{n^{2/3},\sqrt{d}\}\cdot \log d) of optimal, where n is the number of vertices in the input graph and d is the number of demands. We believe that the approach illustrated in the current writing is of independent interest, and may be applicable in other settings as well.     

On the locatic number of graphs

A vertex v of a connected graph G distinguishes a pair u, w of vertices of G if d(v, u)≠d(v, w), where d(·,·) denotes the length of a shortest path between two vertices in G. A k-partition Π={S ₁, S ₂, …, S _k } of the vertex set of G is said to be a locatic partition if for every pair of distinct vertices v and w of G, there exists a vertex s∈S _i for all 1≤i≤k that distinguishes v and w. The cardinality of a largest locatic partition is called the locatic number of G. In this paper, we study the locatic number of paths, cycles and characterize all the connected graphs of order n having locatic number n, n?1 and n?2. Some realizable results are also given in this paper.     

An Efficient Algorithm for the k-Pairwise Disjoint Paths Problem in Hypercubes

A graph G(V, E) (|V|⩾2k) satisfies property A_k if, given k pairs of distinct nodes (s₁, t₁), …, (s_k, t_k) of V(G), there are k mutually node-disjoint paths, one connecting s_i and t_i for each i, 1⩽i⩽k. A necessary condition for any graph to satisfy A_k is that it is (2k−1)-connected. Hypercubes are important interconnection topologies for parallel computation and communication networks. It has been known that hypercubes of dimension n (which are n-connected) satisfy A_⌈n/2⌉. In this paper we give an algorithm which, given k=⌈n/2⌉ pairs of distinct nodes (s₁, t₁), …, (s_k, t_k) in the n-dimensional hypercube, finds the k disjoint paths of length at most n+⌈log n⌉+1 in O(n² log* n) time.     

Length 3 Edge-Disjoint Paths Is NP-Hard

In 2003, it was claimed that the following problem was solvable in polynomial time: do there exist k edge-disjoint paths of length exactly 3 between vertices s and t in a given graph? The proof was flawed, and in this note we show that this problem is NP-hard. We use a reduction from Partial Orientation, a problem recently shown by Pálv?lgyi to be NP-hard.     

Parallel Algorithm for Shortest Pairs of Edge-Disjoint Paths

LetG= (V,E) be a directed graph having a nonnegative cost associated with each edge. LetsVbe a special vertex called the source andWVbe a set of other vertices called sinks inG. In this paper, a parallel algorithm is proposed for finding a pair of edge-disjoint paths fromsto each possible sinktWsuch that the sum of the costs of the two paths is minimized. This algorithm has processor and time complexities same as those needed to find shortest paths fromsto all sinkstW, i.e.,n³/lognprocessors andO(log²n) time.     

Linear-Time Algorithms for Partial \boldmathk -Tree Complements

It is known that a graph decision problem can be solved in linear time over partial k -trees if the problem can be defined in Monadic Second-order (or MS) logic. MS logic allows quantification of vertex and edge subsets, with respect to which logical sentences can encode many different conditions that an input graph must satisfy. It is not always clear, however, which graph problems can be expressed in such a way. In this paper we consider problems stated as logical conditions on subsets of the vertices and nonedges of the input graph. If such a problem can be defined in MS logic (i.e., in terms of the vertices and edges of the input graph), then there is a linear-time algorithm to solve the problem over partial k -trees. This algorithm also provides a solution to some problem over the graph-theoretic complements of partial k -trees. We study several examples of these ``complement-problems.' We introduce a variation of MS logic in which, if a graph-problem can be defined over the class of partial k -tree complements, then there is a linear-time algorithm to solve that problem over partial k -tree complements, and (equivalently) a linear-time algorithm to solve its complement-problem over partial k -trees.     

The 1-Fixed-Endpoint Path Cover Problem is Polynomial on Interval Graphs

We consider a variant of the path cover problem, namely, the k-fixed-endpoint path cover problem, or kPC for short, on interval graphs. Given a graph G and a subset T\mathcal{T} of k vertices of V(G), a k-fixed-endpoint path cover of G with respect to T\mathcal{T} is a set of vertex-disjoint paths ℘ that covers the vertices of G such that the k vertices of T\mathcal{T} are all endpoints of the paths in ℘. The kPC problem is to find a k-fixed-endpoint path cover of G of minimum cardinality; note that, if T\mathcal{T} is empty the stated problem coincides with the classical path cover problem. In this paper, we study the 1-fixed-endpoint path cover problem on interval graphs, or 1PC for short, generalizing the 1HP problem which has been proved to be NP-complete even for small classes of graphs. Motivated by a work of Damaschke (Discrete Math. 112:49–64, 1993), where he left both 1HP and 2HP problems open for the class of interval graphs, we show that the 1PC problem can be solved in polynomial time on the class of interval graphs. We propose a polynomial-time algorithm for the problem, which also enables us to solve the 1HP problem on interval graphs within the same time and space complexity.     

An Efficient Parallel Strategy for ComputingK-Terminal Reliability and Finding Most Vital Edges in 2-Trees and Partial 2-Trees

In this paper, we first develop a parallel algorithm for computingK-terminal reliability, denoted byR(G_K), in 2-trees. Based on this result, we can also computeR(G_K) in partial 2-trees using a method that transforms, in parallel, a given partial 2-tree into a 2-tree. Finally, we solve the problem of finding most vital edges with respect toK-terminal reliability in partial 2-trees. Our algorithms takeO(log n) time withC(m, n) processors on a CRCW PRAM, whereC(m, n) is the number of processors required to find the connected components of a graph withmedges andnvertices in logarithmic time.     

Embedding cycles and paths on solid grid graphs

Despite many algorithms for embedding graphs on unbounded grids, only a few results on embedding graphs on restricted grids have been published. In this paper, we study the problem of embedding paths and cycles on solid grid graphs. We show that a cycle of length k is unit-length embeddable on a solid grid graph G if k is an even integer between four and the length of the longest cycle of G. In addition, our result shows that a path of length k is unit-length embeddable on G, between its two given vertices s and t, if \(k\le L\) and \(k\equiv L (\mathrm{mod}\ 2)\), in which L is the length of the longest path between s and t. Our presented two algorithms show that such embeddings can be found in linear time for cycles and quadratic time for paths, with respect to the size of graph G. In the case of rectangular grid graphs, the running time of the algorithms can be improved to O(k) and O\((k^2)\), respectively. In addition, we extend our results to \(m\times n\times o\) 3D grids. A application of our result is in the interconnection network mapping in parallel processing.     

Constant ratio fixed-parameter approximation of the edge multicut problem

The input of the Edge Multicut problem consists of an undirected graph G and pairs of terminals {s₁,t₁},…,{sm,tm}; the task is to remove a minimum set of edges such that si and ti are disconnected for every 1?i?m. The parameterized complexity of the problem, parameterized by the maximum number k of edges that are allowed to be removed, is currently open. The main result of the paper is a parameterized 2-approximation algorithm: in time f(k)⋅nO(1), we can either find a solution of size 2k or correctly conclude that there is no solution of size k.The proposed algorithm is based on a transformation of the Edge Multicut problem into a variant of the parameterized Max-2SAT problem, where the parameter is related to the number of clauses that are not satisfied. It follows from previous results that the latter problem can be 2-approximated in a fixed-parameter time; on the other hand, we show here that it is W[1]-hard. Thus the additional contribution of the present paper is introducing the first natural W[1]-hard problem that is constant-ratio fixed-parameter approximable.     

Chain packing in graphs

This paper discusses the complexity of packingk-chains (simple paths of lengthk) into an undirected graph; the chains packed must be either vertex-disjoint or edge-disjoint. Linear-time algorithms are given for both problems when the graph is a tree, and for the edge-disjoint packing problem when the graph is general andk = 2. The vertex-disjoint packing problem for general graphs is shown to be NP-complete even when the graph has maximum degree three andk = 2. Similarly the edge-disjoint packing problem is NP-complete even when the graph has maximum degree four andk = 3.This is a revised version of the technical report [15].     

Output-Sensitive Reporting of Disjoint Paths

A k -path query on a graph consists of computing k vertex-disjoint paths between two given vertices of the graph, whenever they exist. In this paper we study the problem of performing k -path queries, with , in a graph G with n vertices. We denote with the total length of the reported paths. For , we present an optimal data structure for G that uses O(n) space and executes k -path queries in output-sensitive time. For triconnected planar graphs, our results make use of a new combinatorial structure that plays the same role as bipolar (st ) orientations for biconnected planar graphs. This combinatorial structure also yields an alternative construction of convex grid drawings of triconnected planar graphs. Received August 24, 1996; revised April 8, 1997.     

Finding multiple induced disjoint paths in general graphs

In an undirected graph, paths P₁,P₂,…,Pk are induced disjoint if each one of them is chordless (i.e., is an induced path) and any two of them have neither common nodes nor adjacent nodes. This paper investigates the Maximum Induced Disjoint Paths (MIDP) problem: in an undirected graph G=(V,E), given k node pairs {s₁,t₁},…,{sk,tk}, connect maximum number of these node pairs via induced disjoint paths. Till now, the only things known about MIDP are: i) it is NP-hard; ii) it is NP-hard even when k=2; iii) it can be solved in polynomial time when k is a fixed constant and the given graph is a directed planar graph (Kobayashi, 2009 [9]). This paper proves that for general k and any ?>0, it is NP-hard to approximate MIDP within m1/2−?, where m=|E|. Two algorithms for MIDP are given by this paper: a greedy algorithm whose approximation ratio is and an on-line algorithm which has a good lower bound.     

An Algorithm for Vertex-pair Connectivity

An algorithm is developed for finding the minimum number of vertices ω _s,t whose removal from a directed graph G breaks all directed paths from a given vertex s to a given vertex t. The algorithm is performed on G without any modification of its structure and entails examining each vertex at most r times with 2ω _s,t  ≥ r and, in practice, almost always with ω _s,t  ≥ r.     

On Short Paths Interdiction Problems: Total and Node-Wise Limited Interdiction

Given a directed graph G=(V,A) with a non-negative weight (length) function on its arcs w:A→ℝ₊ and two terminals s,t∈V, our goal is to destroy all short directed paths from s to t in G by eliminating some arcs of A. This is known as the short paths interdiction problem. We consider several versions of it, and in each case analyze two subcases: total limited interdiction, when a fixed number k of arcs can be removed, and node-wise limited interdiction, when for each node v∈V a fixed number k(v) of out-going arcs can be removed. Our results indicate that the latter subcase is always easier than the former one. In particular, we show that the short paths node-wise interdiction problem can be efficiently solved by an extension of Dijkstra’s algorithm. In contrast, the short paths total interdiction problem is known to be NP-hard. We strengthen this hardness result by deriving the following inapproximability bounds: Given k, it is NP-hard to approximate within a factor c<2 the maximum s–t distance d(s,t) obtainable by removing (at most) k arcs from G. Furthermore, given d, it is NP-hard to approximate within a factor the minimum number of arcs which has to be removed to guarantee d(s,t)≥d. Finally, we also show that the same inapproximability bounds hold for undirected graphs and/or node elimination. This research was supported in part by NSF grant IIS-0118635 and by DIMACS, the NSF Center for Discrete Mathematics & Theoretical Computer Science. Preprints DTR-2005-04 and DTR-2006-13 are available at and . Our co-author Leonid Khachiyan passed away with tragic suddenness on April 29th, 2005.     

Parsimonious flooding in dynamic graphs

An edge-Markovian process with birth-rate p and death-rate q generates infinite sequences of graphs (G ₀, G ₁, G ₂,…) with the same node set [n] such that G _t is obtained from G _t-1 as follows: if e ? E(G_t-1){e\notin E(G_{t-1})} then e ? E(G_t){e\in E(G_{t})} with probability p, and if e ? E(G_t-1){e\in E(G_{t-1})} then e ? E(G_t){e\notin E(G_{t})} with probability q. In this paper, we establish tight bounds on the complexity of flooding in edge-Markovian graphs, where flooding is the basic mechanism in which every node becoming aware of an information at step t forwards this information to all its neighbors at all forthcoming steps t′ > t. These bounds complete previous results obtained by Clementi et al. Moreover, we also show that flooding in dynamic graphs can be implemented in a parsimonious manner, so that to save bandwidth, yet preserving efficiency in term of simplicity and completion time. For a positive integer k, we say that the flooding protocol is k-active if each node forwards an information only during the k time steps immediately following the step at which the node receives that information for the first time. We define the reachability threshold for the flooding protocol as the smallest integer k such that, for any source s ? [n]{s\in [n]} , the k-active flooding protocol from s completes (i.e., reaches all nodes), and we establish tight bounds for this parameter. We show that, for a large spectrum of parameters p and q, the reachability threshold is by several orders of magnitude smaller than the flooding time. In particular, we show that it is even constant whenever the ratio p/(p + q) exceeds log n/n. Moreover, we also show that being active for a number of steps equal to the reachability threshold (up to a multiplicative constant) allows the flooding protocol to complete in optimal time, i.e., in asymptotically the same number of steps as when being perpetually active. These results demonstrate that flooding can be implemented in a practical and efficient manner in dynamic graphs. The main ingredient in the proofs of our results is a reduction lemma enabling to overcome the time dependencies in edge-Markovian dynamic graphs.     

The super spanning connectivity and super spanning laceability of the enhanced hypercubes

A k -container C(u,v) of a graph G is a set of k disjoint paths between u and v. A k-container C(u,v) of G is a k ^* -container if it contains all vertices of G. A graph G is k ^* -connected if there exists a k ^*-container between any two distinct vertices of G. Therefore, a graph is 1^*-connected (respectively, 2^*-connected) if and only if it is Hamiltonian connected (respectively, Hamiltonian). A graph G is super spanning connected if there exists a k ^*-container between any two distinct vertices of G for every k with 1≤k≤κ(G) where κ(G) is the connectivity of G. A bipartite graph G is k ^* -laceable if there exists a k ^*-container between any two vertices from different partite set of G. A bipartite graph G is super spanning laceable if there exists a k ^*-container between any two vertices from different partite set of G for every k with 1≤k≤κ(G). In this paper, we prove that the enhanced hypercube Q _n,m is super spanning laceable if m is an odd integer and super spanning connected if otherwise.