The Directed Disjoint Shortest Paths Problem

In the k disjoint shortest paths problem (k-DSPP), we are given a graph and its vertex pairs (s_1, t_1), ...(s_k, t_k), and the objective is to find k pairwise disjoint paths P_1, ...P_k such that each path P_i is a shortest path from s_i to t_i, if they exist. If the length of each edge is equal to zero, then this problem amounts to the disjoint paths problem, which is one of the well-studied problems in algorithmic graph theory and combinatorial optimization. Eilam-Tzoreff (1998) focused on the case when the length of each edge is positive, and showed that the undirected version of 2-DSPP can be solved in polynomial time. Polynomial solvability of the directed version was posed as an open problem by Eilam-Tzoreff (1998). In this paper, we solve this problem affirmatively, that is, we give a first polynomial time algorithm for the directed version of 2-DSPP when the length of each edge is positive. Note that the 2 disjoint paths problem in digraphs is NP-hard, which implies that the directed 2-DSPP is NP-hard if the length of each edge can be zero. We extend our result to the case when the instance has two terminal pairs and the number of paths is a fixed constant greater than two. We also show that the undirected k-DSPP and the vertex-disjoint version of the directed k-DSPP can be solved in polynomial time if the input graph is planar and k is a fixed constant

A Local-To-Global Theorem for Congested Shortest Paths

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, there are no congestion constraints, and the problem can be solved easily by running a shortest path algorithm for each pair (s_i,t_i) independently. At the other extreme, when c = 1, the (k,c)-SPC problem is equivalent to the k-Disjoint Shortest Paths (k-DSP) problem, where the collection of shortest paths must be node-disjoint. For fixed k, k-DSP is polynomial-time solvable on DAGs and undirected graphs. Amiri and Wargalla interpolated between these two extreme values of c, to obtain an algorithm for (k,c)-SPC on DAGs that runs in polynomial time when k-c is constant. In the same way, we prove that (k,c)-SPC can be solved in polynomial time on undirected graphs whenever k-c is constant. For directed graphs, because of our counterexample to the original theorem statement, our roundtrip local-to-global result does not imply such an algorithm (k,c)-SPC. Even without an algorithmic application, our proof for directed graphs may be of broader interest because it characterizes intriguing structural properties of shortest paths in directed graphs

Tight Approximation and Kernelization Bounds for Vertex-Disjoint Shortest Paths

We examine the possibility of approximating Maximum Vertex-Disjoint Shortest Paths. In this problem, the input is an edge-weighted (directed or undirected) $n$-vertex graph $G$ along with $k$ terminal pairs $(s_1,t_1),(s_2,t_2),\ldots,(s_k,t_k)$. The task is to connect as many terminal pairs as possible by pairwise vertex-disjoint paths such that each path is a shortest path between the respective terminals. Our work is anchored in the recent breakthrough by Lochet [SODA '21], which demonstrates the polynomial-time solvability of the problem for a fixed value of $k$. Lochet's result implies the existence of a polynomial-time $ck$-approximation for Maximum Vertex-Disjoint Shortest Paths, where $c \leq 1$ is a constant. Our first result suggests that this approximation algorithm is, in a sense, the best we can hope for. More precisely, assuming the gap-ETH, we exclude the existence of an $o(k)$-approximations within $f(k) \cdot$poly($n$) time for any function $f$ that only depends on $k$. Our second result demonstrates the infeasibility of achieving an approximation ratio of $n^{\frac{1}{2}-\varepsilon}$ in polynomial time, unless P = NP. It is not difficult to show that a greedy algorithm selecting a path with the minimum number of arcs results in a $\lceil\sqrt{\ell}\rceil$-approximation, where $\ell$ is the number of edges in all the paths of an optimal solution. Since $\ell \leq n$, this underscores the tightness of the $n^{\frac{1}{2}-\varepsilon}$-inapproximability bound. Additionally, we establish that Maximum Vertex-Disjoint Shortest Paths is fixed-parameter tractable when parameterized by $\ell$ but does not admit a polynomial kernel. Our hardness results hold for undirected graphs with unit weights, while our positive results extend to scenarios where the input graph is directed and features arbitrary (non-negative) edge weights

A Local-to-Global Theorem for Congested Shortest Paths

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.Comment: Updated to reflect reviewer comment

Permutation Routing in the Hypercube and Grid Topologies

The problem of edge disjoint path routing arises from applications in distributed memory parallel computing. We examine this problem in both the directed hypercube and two-dimensional grid topologies. Complexity results are obtained for these problems where the routing must consist entirely of shortest length paths. Additionally, approximation algorithms are presented for the case when the routing request is of a special form known as a permutation. Permutations simply require that no vertex in the graph may be used more than once as either a source or target for a routing request. Szymanski conjectured that permutations are always routable in the directed hypercube, and this remains an open problem

Evaluation and Enumeration Problems for Regular Path Queries

Regular path queries (RPQs) are a central component of graph databases. We investigate decision- and enumeration problems concerning the evaluation of RPQs under several semantics that have recently been considered: arbitrary paths, shortest paths, and simple paths. Whereas arbitrary and shortest paths can be enumerated in polynomial delay, the situation is much more intricate for simple paths. For instance, already the question if a given graph contains a simple path of a certain length has cases with highly non-trivial solutions and cases that are long-standing open problems. We study RPQ evaluation for simple paths from a parameterized complexity perspective and define a class of simple transitive expressions that is prominent in practice and for which we can prove a dichotomy for the evaluation problem. We observe that, even though simple path semantics is intractable for RPQs in general, it is feasible for the vast majority of RPQs that are used in practice. At the heart of our study on simple paths is a result of independent interest: the two disjoint paths problem in directed graphs is W[1]-hard if parameterized by the length of one of the two paths

Finding a Shortest Non-zero Path in Group-Labeled Graphs via Permanent Computation

A group-labeled graph is a directed graph with each arc labeled by a group element, and the label of a path is defined as the sum of the labels of the traversed arcs. In this paper, we propose a polynomial time randomized algorithm for the problem of finding a shortest s-t path with a non-zero label in a given group-labeled graph (which we call the Shortest Non-Zero Path Problem). This problem generalizes the problem of finding a shortest path with an odd number of edges, which is known to be solvable in polynomial time by using matching algorithms. Our algorithm for the Shortest Non-Zero Path Problem is based on the ideas of Björklund and Husfeldt (Proceedings of the 41st international colloquium on automata, languages and programming, part I. LNCS 8572, pp 211–222, 2014). We reduce the problem to the computation of the permanent of a polynomial matrix modulo two. Furthermore, by devising an algorithm for computing the permanent of a polynomial matrix modulo 2r for any fixed integer r, we extend our result to the problem of packing internally-disjoint s-t paths

Polynomial Time Algorithm for Determining Max-Min Paths in Networks and Solving Zero Value Cyclic Games

We study the max-min paths problem, which represents a game version of the shortest and the longest paths problem in a weighted directed graph. In this problem the vertex set V of the weighted directed graph G=(V,E) is divided into two disjoint subsets VA and VB which are regarded as positional sets of two players. The players are seeking for a directed path from the given starting position ν 0 to the final position ν f , where the first player intends to maximize the integral cost of the path while the second one has aim to minimize it. Polynomial-time algorithm for determining max-min path in networks is proposed and its application for solving zero value cyclic games is developed. Mathematics Subject Classification 2000: 90B10, 90C35, 90C27

Parameterizing Path Partitions

We study the algorithmic complexity of partitioning the vertex set of a given (di)graph into a small number of paths. The Path Partition problem (PP) has been studied extensively, as it includes Hamiltonian Path as a special case. The natural variants where the paths are required to be either \emph{induced} (Induced Path Partition, IPP) or \emph{shortest} (Shortest Path Partition, SPP), have received much less attention. Both problems are known to be NP-complete on undirected graphs; we strengthen this by showing that they remain so even on planar bipartite directed acyclic graphs (DAGs), and that SPP remains \NP-hard on undirected bipartite graphs. When parameterized by the natural parameter ``number of paths'', both SPP and IPP are shown to be W{1}-hard on DAGs. We also show that SPP is in \XP both for DAGs and undirected graphs for the same parameter, as well as for other special subclasses of directed graphs (IPP is known to be NP-hard on undirected graphs, even for two paths). On the positive side, we show that for undirected graphs, both problems are in FPT, parameterized by neighborhood diversity. We also give an explicit algorithm for the vertex cover parameterization of PP. When considering the dual parameterization (graph order minus number of paths), all three variants, IPP, SPP and PP, are shown to be in FPT for undirected graphs. We also lift the mentioned neighborhood diversity and dual parameterization results to directed graphs; here, we need to define a proper novel notion of directed neighborhood diversity. As we also show, most of our results also transfer to the case of covering by edge-disjoint paths, and purely covering.Comment: 27 pages, 8 figures. A short version appeared in the proceedings of the CIAC 2023 conferenc