Showing papers on "Longest path problem published in 2022"

Finding Hamiltonian and Longest (s,t)-Paths of C-Shaped Supergrid Graphs in Linear Time

Abstract: A graph is called Hamiltonian connected if it contains a Hamiltonian path between any two distinct vertices. In the past, we proved the Hamiltonian path and cycle problems for general supergrid graphs to be NP-complete. However, they are still open for solid supergrid graphs. In this paper, first we will verify the Hamiltonian cycle property of C-shaped supergrid graphs, which are a special case of solid supergrid graphs. Next, we show that C-shaped supergrid graphs are Hamiltonian connected except in a few conditions. For these excluding conditions of Hamiltonian connectivity, we compute their longest paths. Then, we design a linear-time algorithm to solve the longest path problem in these graphs. The Hamiltonian connectivity of C-shaped supergrid graphs can be applied to compute the optimal stitching trace of computer embroidery machines, and construct the minimum printing trace of 3D printers with a C-like component being printed.

Restless Temporal Path Parameterized Above Lower Bounds

Abstract: Reachability questions are one of the most fundamental algorithmic primitives in temporal graphs -- graphs whose edge set changes over discrete time steps. A core problem here is the NP-hard Short Restless Temporal Path: given a temporal graph $\mathcal G$, two distinct vertices $s$ and $z$, and two numbers $\delta$ and $k$, is there a $\delta$-restless temporal $s$-$z$ path of length at most $k$? A temporal path is a path whose edges appear in chronological order and a temporal path is $\delta$-restless if two consecutive path edges appear at most $\delta$ time steps apart from each other. Among others, this problem has applications in neuroscience and epidemiology. While Short Restless Temporal Path is known to be computationally hard, e.g., it is NP-hard for only three time steps and W[1]-hard when parameterized by the feedback vertex number of the underlying graph, it is fixed-parameter tractable when parameterized by the path length $k$. We improve on this by showing that Short Restless Temporal Path can be solved in (randomized) $4^{k-d}|\mathcal G|^{O(1)}$ time, where $d$ is the minimum length of a temporal $s$-$z$ path.

Graphs without a rainbow path of length 3

Abstract: In 1959 Erd\H os and Gallai proved the asymptotically optimal bound for the maximum number of edges in graphs not containing a path of a fixed length. We investigate a rainbow version of the theorem, in which one considers $k \geq 1$ graphs on a common set of vertices not creating a path having edges from different graphs and asks for the maximum number of edges in each graph. We prove the asymptotically optimal bound in the case of a path on three edges and any $k \geq 1$.

Analyzing the 3-path Vertex Cover Problem in Planar Bipartite Graphs

Abstract: Let $$G=(V,E)$$ be a simple graph. A set $$C \subseteq V$$ is called a k-path vertex cover of G, if each k-path in G contains at least one vertex from C. In the k-path vertex cover problem, we are given a graph G and asked to find a k-path vertex cover of minimum cardinality. For $$k=3$$ , the problem becomes the well-known 3-path vertex cover (3PVC) problem, which has been widely studied, as per the literature. In this paper, we focus on the 3PVC problem in planar bipartite (pipartite) graphs for the most part. We first show that the 3PVC problem is NP-hard, even in pipartite graphs in which the degree of all vertices is bounded by 4. We then show that the 3PVC problem on this class of graphs admits a linear time 1.5-approximation algorithm. Finally, we show that the 3PVC problem is APX-complete in bipartite graphs. The last result is particularly interesting, since the vertex cover problem in bipartite graphs is solvable in polynomial time.

Dynamic Discretization Discovery Algorithms for Time-Dependent Shortest Path Problems

Abstract: Finding a shortest path in a network is a fundamental optimization problem. We focus on settings in which the travel time on an arc in the network depends on the time at which traversal of the arc begins. In such settings, reaching the destination as early as possible is not the only objective of interest. Minimizing the duration of the path, that is, the difference between the arrival time at the destination and the departure from the origin, and minimizing the travel time along the path from origin to destination, are also of interest. We introduce dynamic discretization discovery algorithms to efficiently solve such time-dependent shortest path problems with piecewise linear arc travel time functions. The algorithms operate on partially time-expanded networks in which arc costs represent lower bounds on the arc travel time over the subsequent time interval. A shortest path in this partially time-expanded network yields a lower bound on the value of an optimal path. Upper bounds are easily obtained as by-products of the lower bound calculations. The algorithms iteratively refine the discretization by exploiting breakpoints of the arc travel time functions. In addition to time discretization refinement, the algorithms permit time intervals to be eliminated, improving lower and upper bounds, until, in a finite number of iterations, optimality is proved. Computational experiments show that only a small fraction of breakpoints must be explored and that the fraction decreases as the length of the time horizon and the size of the network increases, making the algorithms highly efficient and scalable. Summary of Contribution: New data collection techniques have increased the availability and fidelity of time-dependent travel time information, making the time-dependent variant of the classic shortest path problem an extremely relevant problem in the field of operations research. This paper provides novel algorithms for the time-dependent shortest path problem with both the minimum duration and minimum travel time objectives, which aims to address the computational challenges faced by existing algorithms. A computational study shows that our new algorithm is indeed significantly more efficient than existing approaches.

Comparison of Variants of Yen's Algorithm for Finding K-Simple Shortest Paths

Abstract: In directed and weighted graph, with n nodes and m edges, the K-shortest paths problem involve finding a set of k shortest paths between a defined source and destination pair where the first path is shortest, and the remaining k-1 paths are in increasing lengths. In K-shortest path problem there are two classes, k shortest simple path problem and k shortest non-simple path problem. The first algorithm to solve K shortest simple path problems is Yen's algorithm based on deviation path concept. Later many variants of Yen's algorithm are proposed with improved computational performance. In this paper some of the variants of Yen's algorithm for finding top k simple shortest path are studied and compared.

On the shortest path problem of uncertain random digraphs

Abstract: In the field of graph theory, the shortest path problem is one of the most significant problems. However, since varieties of indeterminated factors appear in complex networks, determining of the shortest path from one vertex to another in complex networks may be a lot more complicated than the cases in deterministic networks. To illustrate this problem, the model of uncertain random digraph will be proposed via chance theory, in which some arcs exist with degrees in probability measure and others exist with degrees in uncertain measure. The main focus of this paper is to investigate the main properties of the shortest path in uncertain random digraph. Methods and algorithms are designed to calculate the distribution of shortest path more efficiently. Besides, some numerical examples are presented to show the efficiency of these methods and algorithms.

Generalized Longest Path Problems

Abstract: The longest simple path and snake-in-a-box are combinatorial search problems of considerable research interest. We create a common framework of longest constrained path in a graph that contains these two problems, as well as other interesting maximum path problems, as special cases. We analyze properties of this general framework, produce bounds on the path length that can be used as admissible heuristics for all problem types therein. For the special cases of longest simple path and snakes, these heuristics are shown to reduce the number of expansions when searching for a maximal path, which in some cases leads to reduced search time despite the significant overhead of computing these heuristics.

A Bi-Criteria FPTAS for Scheduling with Memory Constraints on Graph with Bounded Tree-width

Abstract: In this paper we study a scheduling problem arising from executing numerical simulations on HPC architectures. With a constant number of parallel machines, the objective is to minimize the makespan under memory constraints for the machines. Those constraints come from a neighborhood graph G for the jobs. Motivated by a previous result on graphs G with bounded path-width, our focus is on the case when the neighborhood graph G has bounded tree-width. Our result is a bi-criteria fully polynomial time approximation algorithm based on a dynamic programming algorithm. It allows to find a solution within a factor of $$1+\epsilon $$ of the optimal makespan, where the memory capacity of the machines may be exceeded by a factor at most $$1+\epsilon $$ . This result relies on the use of a nice tree decomposition of G and its traversal in a specific way which may be useful on its own. The case of unrelated machines is also tractable with minor modifications.

Two New Characterizations of Path Graphs

Abstract: Path graphs are intersection graphs of paths in a tree. We start from the characterization of path graphs by Monma and Wei [C.L.~Monma,~and~V.K.~Wei, Intersection Graphs of Paths in a Tree, J. Combin. Theory Ser. B, 41:2 (1986) 141--181] and we reduce it to some 2-colorings subproblems, obtaining the first characterization that directly leads to a polynomial recognition algorithm. Then we introduce the collection of the attachedness graphs of a graph and we exhibit a list of minimal forbidden 2-edge colored subgraphs in each of the attachedness graph.

A review on the quickest flow problem

Abstract: Path problems were basically studied to find an alternate path that is, finding a second shortest route, if the route is blocked. The shortest path problem is mainly focused on finding the shortest paths between the vertices of a given network. A new variant of shortest path problem is quickest path problem, where the predetermined data is sent from the source to the sink. The quickest flow problem relaxes the limitations of single path to multiple paths. In this paper we reviewed the shortest path problem, quickest path problem and quickest flow problem. Later on, each problem is clarified with certain examples.

Reducing Computation Time with Priority Assignment in Distributed MPC

Abstract: <p>Networked control problems of multi-agent systems can be distributed to the agents to reduce computational effort. One distribution strategy is priority-based non-cooperative distributed model predictive control (P-DMPC), in which the computation time is mainly determined by the longest path in the coupling directed acyclic graph (DAG). The longest path is dependent on the undirected coupling graph, which is fixed, and the priority assignment, which is variable. This article presents an approach to assign priorities in P-DMPC to reduce the longest path length in the coupling DAG and therefore the computation time for the networked control system (NCS). We proof that this problem can be mapped to a graph-coloring problem, in which the number of needed colors corresponds to the longest path length in the coupling DAG. We present an efficient graph-coloring algorithm from which we determine priorities for the agents. We evaluate effect and effort of the approach before applying it to trajectory planning for networked vehicles at intersections.</p>

Constrained Shortest Path and Hierarchical Structures

Abstract: The Constrained Shortest Path (CSP) problem is as follows. An n-vertex graph is given, two weights are assigned to each edge: “cost” and “length”. It is required to find a min-cost bounded-length path between a given pair of vertices. The problem is NP-hard even when the lengths of all edges are the same. Therefore, various approximation algorithms have been proposed in the literature for it. The constraint on path length can be accounted for by considering one aggregated edge weight equals to the linear combination of the cost and length. By varying the value of the Lagrange multiplier in the linear combination, a feasible solution delivers a minimum to the objective function with new weights. At the same time, as usually, the Dijkstra’s algorithm or its modifications are used to construct a shortest path with the current weights of the edges. However, in the large graphs, this approach may turn out to be time-consuming. In this paper, we propose to search a solution, not in the original graph but in the specially constructed hierarchical structures (HS). We show that the shortest path in the HS is constructed with O(m)-time complexity, where m is the number of edges/arcs of the graph, and the approximate solution in the case of integer costs and lengths is found with $$O(m\log n)$$ -time complexity. In result of a priori analysis of the algorithm its accuracy estimation turned out to depend on the parameters of the problem and can be significant. Therefore, to evaluate the algorithm’s effectiveness, we conducted a numerical experiment on the graph of roads of megalopolis and randomly constructed metric unit-disk graphs (UDGs). The numerical experiment results show that in the HS, solution is built 10–100 times faster than in the methods which use Dijkstra’s like algorithm to build a min-weight path in the original graph.

A fast layered path planning algorithm for job shop scheduling problem

Abstract: Job shop scheduling problem (JSP) is a classical system resource optimisation problem and also an NP hard problem. The search algorithm based on Akers obstacle graph model is an effective algorithm to solve JSP, which first removes part of jobs from the original schedule, then constructs obstacle graph and finds the shortest path from the graph, and finally reinserts the jobs according to the shortest path decoding method to get the new schedule. Although the new scheduling can achieve good results, it is time-consuming to find the shortest path. Therefore, it is necessary to further study how to quickly plan the shortest path. This study presents a fast layered path search algorithm for solving the obstacle graph of job shop scheduling. The algorithm designs a node expansion method and a delay distance formula. The obstacles generated by different machines in the obstacle graph are layered. When the nodes expand, the extended nodes are compared with the parent layer nodes to quickly avoid closely arranged obstacles, and multiple child nodes are generated at one time through node expansion to improve the node expansion ability. At the same time, node expansion method and delay distance formula can be well integrated with A* algorithm. Finally, the test verifies that the algorithm can spend less time to find the shortest path.

On the Multistage Shortest Path Problem Under Distributional Uncertainty

Abstract: In this paper, we consider an ambiguity-averse multistage network game between a user and an attacker. The arc costs are assumed to be random variables that satisfy prescribed first-order moment constraints for some subsets of arcs and individual probability constraints for some particular arcs. The user aims at minimizing its cumulative expected loss by traversing between two fixed nodes in the network, while the attacker’s objective is to maximize the user’s expected loss by selecting a distribution of arc costs from the family of admissible distributions. In contrast to most of the related studies, both the user and the attacker can dynamically adjust their decisions at particular nodes of the user’s path. By observing the user’s decisions, the attacker may reveal some additional distributional information associated with the arcs emanated from the current user’s position. It is shown that the resulting multistage distributionally robust shortest path problem (DRSPP) admits a linear mixed-integer programming reformulation (MIP). In particular, we distinguish between acyclic and general graphs by introducing different forms of non-anticipativity constraints. Finally, we perform a numerical study, where the quality of adaptive decisions and computational tractability of the proposed MIP reformulation are explored with respect to several classes of synthetic network instances.

Finding shortest non-separating and non-disconnecting paths

Abstract: For a connected graph $G = (V, E)$ and $s, t \in V$, a non-separating $s$-$t$ path is a path $P$ between $s$ and $t$ such that the set of vertices of $P$ does not separate $G$, that is, $G - V(P)$ is connected. An $s$-$t$ path is non-disconnecting if $G - E(P)$ is connected. The problems of finding shortest non-separating and non-disconnecting paths are both known to be NP-hard. In this paper, we consider the problems from the viewpoint of parameterized complexity. We show that the problem of finding a non-separating $s$-$t$ path of length at most $k$ is W[1]-hard parameterized by $k$, while the non-disconnecting counterpart is fixed-parameter tractable parameterized by $k$. We also consider the shortest non-separating path problem on several classes of graphs and show that this problem is NP-hard even on bipartite graphs, split graphs, and planar graphs. As for positive results, the shortest non-separating path problem is fixed-parameter tractable parameterized by $k$ on planar graphs and polynomial-time solvable on chordal graphs if $k$ is the shortest path distance between $s$ and $t$.

Counting Dominating Sets in Directed Path Graphs

Abstract: A dominating set of a graph is a set of vertices such that every vertex not in the set has at least one neighbor in the set. The problem of counting dominating sets is #P-complete for chordal graphs but solvable in polynomial time for its subclass of interval graphs. The complexity status of the corresponding problem is still undetermined for directed path graphs, which are a well-known class of graphs that falls between chordal graphs and interval graphs. This paper reveals that the problem of counting dominating sets remains #P-complete for directed path graphs but a stricter constraint to rooted directed path graphs admits a polynomial-time solution.

Detours in Directed Graphs

Abstract: We study two "above guarantee" versions of the classical Longest Path problem on undirected and directed graphs and obtain the following results. In the first variant of Longest Path that we study, called Longest Detour, the task is to decide whether a graph has an (s,t)-path of length at least dist_G(s,t)+k (where dist_G(s,t) denotes the length of a shortest path from s to t). Bez\'akov\'a et al. proved that on undirected graphs the problem is fixed-parameter tractable (FPT) by providing an algorithm of running time 2^{O (k)} n. Further, they left the parameterized complexity of the problem on directed graphs open. Our first main result establishes a connection between Longest Detour on directed graphs and 3-Disjoint Paths on directed graphs. Using these new insights, we design a 2^{O(k)} n^{O(1)} time algorithm for the problem on directed planar graphs. Further, the new approach yields a significantly faster FPT algorithm on undirected graphs. In the second variant of Longest Path, namely Longest Path Above Diameter, the task is to decide whether the graph has a path of length at least diam(G)+k (diam(G) denotes the length of a longest shortest path in a graph G). We obtain dichotomy results about Longest Path Above Diameter on undirected and directed graphs. For (un)directed graphs, Longest Path Above Diameter is NP-complete even for k=1. However, if the input undirected graph is 2-connected, then the problem is FPT. On the other hand, for 2-connected directed graphs, we show that Longest Path Above Diameter is solvable in polynomial time for each k\in{1,\dots, 4} and is NP-complete for every k\geq 5. The parameterized complexity of Longest Path Above Diameter on general directed graphs remains an interesting open problem.

INT-react: An O(E) Path Planner for Resilient Network-Wide Telemetry Over Megascale Networks

Abstract: In-band network telemetry (INT) delivers high-precision network monitoring by collecting device-internal states entirely on the data plane. For rapid congestion awareness and network troubleshooting, it is necessary to conduct network-wide telemetry by generating multiple monitoring paths covering the entire network graph. Solving the optimal path planning problem used the eulerian trail initially at a time complexity of $O(k(3E+V-15k/2))$ . For mega-scale data center networks, such a high complexity is unacceptable because the algorithm cannot adapt well to occasional topology changes. In this work, we propose improved INT-path and INT-react, two refined path planning algorithms with a much reduced time complexity of only $O(E)$ . Furthermore, INT-react also considers balanced path generation to reduce the longest path length for synchronized collection of telemetry data from each monitoring path. The evaluation shows that on average it costs 2.10s for the improved INT-path to solve the optimal path planning for a network of 9500 switches, while the computation is completed within only 0.283s on average for INT-react. In addition, INT-react reduces the longest path length. INT-react's path planning is so fast that it promptly reacts to topology changes and is ready to be deployed in mega-scale production networks.

On the multi-stage shortest path problem under distributional uncertainty

Abstract: In this paper we consider an ambiguity-averse multi-stage network game between a user and an attacker. The arc costs are assumed to be random variables that satisfy prescribed first-order moment constraints for some subsets of arcs and individual probability constraints for some particular arcs. The user aims at minimizing its cumulative expected loss by traversing between two fixed nodes in the network, while the attacker's objective is to maximize the user's expected loss by selecting a distribution of arc costs from the family of admissible distributions. In contrast to most of the related studies, both the user and the attacker can dynamically adjust their decisions at particular nodes of the user's path. By observing the user's decisions, the attacker may reveal some additional distributional information associated with the arcs emanated from the current user's position. It is shown that the resulting multi-stage distributionally robust shortest path problem (DRSPP) admits a linear mixed-integer programming reformulation (MIP). In particular, we distinguish between acyclic and general graphs by introducing different forms of non-anticipativity constraints. Finally, we perform a numerical study, where the quality of adaptive decisions and computational tractability of the proposed MIP reformulation are explored with respect to several classes of synthetic network~instances.

Edge Intersection Graphs of Paths on a Triangular Grid

Abstract: We introduce a new class of intersection graphs, the edge intersection graphs of paths on a triangular grid, called EPGt graphs. We compare this new class with the well-known class of EPG graphs. A turn of a path at a grid point is called a bend. An EPGt representation in which every path has at most k bends is called a Bk-EPGt representation and the corresponding graphs are called Bk-EPGt graphs. We characterize the representation of cliques with three vertices and chordless 4-cycles in B1-EPGt representations.

Beyond the shortest path: the path length index as a distribution

Abstract: The traditional complex network approach considers only the shortest paths from one node to another, not taking into account several other possible paths. This limitation is significant, for example, in urban mobility studies. In this short report, as the first steps, we present an exhaustive approach to address that problem and show we can go beyond the shortest path, but we do not need to go so far: we present an interactive procedure and an early stop possibility. After presenting some fundamental concepts in graph theory, we presented an analytical solution for the problem of counting the number of possible paths between two nodes in complete graphs, and a depth-limited approach to get all possible paths between each pair of nodes in a general graph (an NP-hard problem). We do not collapse the distribution of path lengths between a pair of nodes into a scalar number, we look at the distribution itself - taking all paths up to a pre-defined path length (considering a truncated distribution), and show the impact of that approach on the most straightforward distance-based graph index: the walk/path length.

A Local-to-Global Theorem for Congested Shortest Paths

Abstract: Amiri and Wargalla (2020) proved the following local-to-global theorem in directed acyclic graphs (DAGs): if $G$ is a weighted DAG such that for each subset $S$ of 3 nodes there is a shortest path containing every node in $S$, then there exists a pair $(s,t)$ of nodes such that there is a shortest $st$-path containing every node in $G$. We extend this theorem to general graphs. For undirected graphs, we prove that the same theorem holds (up to a difference in the constant 3). For directed graphs, we provide a counterexample to the theorem (for any constant), and prove a roundtrip analogue of the theorem which shows there exists a pair $(s,t)$ of nodes such that every node in $G$ is contained in the union of a shortest $st$-path and a shortest $ts$-path. The original theorem for DAGs has an application to the $k$-Shortest Paths with Congestion $c$ (($k,c$)-SPC) problem. In this problem, we are given a weighted graph $G$, together with $k$ node pairs $(s_1,t_1),\dots,(s_k,t_k)$, and a positive integer $c\leq k$. We are tasked with finding paths $P_1,\dots, P_k$ such that each $P_i$ is a shortest path from $s_i$ to $t_i$, and every node in the graph is on at most $c$ paths $P_i$, or reporting that no such collection of paths exists. When $c=k$ the problem is easily solved by finding shortest paths for each pair $(s_i,t_i)$ independently. When $c=1$, the $(k,c)$-SPC problem recovers the $k$-Disjoint Shortest Paths ($k$-DSP) problem, where the collection of shortest paths must be node-disjoint. For fixed $k$, $k$-DSP can be solved in polynomial time on DAGs and undirected graphs. Previous work shows that the local-to-global theorem for DAGs implies that $(k,c)$-SPC on DAGs whenever $k-c$ is constant. In the same way, our work implies that $(k,c)$-SPC can be solved in polynomial time on undirected graphs whenever $k-c$ is constant.

A*Net: A Scalable Path-based Reasoning Approach for Knowledge Graphs

Abstract: Reasoning on large-scale knowledge graphs has been long dominated by embedding methods. While path-based methods possess the inductive capacity that embeddings lack, they suffer from the scalability issue due to the exponential number of paths. Here we present A*Net, a scalable path-based method for knowledge graph reasoning. Inspired by the A* algorithm for shortest path problems, our A*Net learns a priority function to select important nodes and edges at each iteration, to reduce time and memory footprint for both training and inference. The ratio of selected nodes and edges can be specified to trade off between performance and efficiency. Experiments on both transductive and inductive knowledge graph reasoning benchmarks show that A*Net achieves competitive performance with existing state-of-the-art path-based methods, while merely visiting 10% nodes and 10% edges at each iteration. On a million-scale dataset ogbl-wikikg2, A*Net achieves competitive performance with embedding methods and converges faster. To our best knowledge, A*Net is the first path-based method for knowledge graph reasoning at such a scale.