Paths $P_1,\ldots,P_k$ in a graph $G=(V,E)$ are mutually induced if any two distinct $P_i$ and $P_j$ have neither common vertices nor adjacent vertices (except perhaps their end-vertices). The Induced Disjoint Paths problem is to decide if a graph $G$ with $k$ pairs of specified vertices $(s_i,t_i)$ contains $k$ mutually induced paths $P_i$ such that each $P_i$ connects $s_i$ and $t_i$. This is a classical graph problem that is NP-complete even for $k=2$. We study it for AT-free graphs.
Unlike its subclasses of permutation graphs and cocomparability graphs, the class of AT-free graphs has no geometric intersection model. However, by a new, structural analysis of the behaviour of Induced Disjoint Paths for AT-free graphs, we prove that it can be solved in polynomial time for AT-free graphs even when $k$ is part of the input. This is in contrast to the situation for other well-known graph classes, such as planar graphs, claw-free graphs, or more recently, (theta,wheel)-free graphs, for which such a result only holds if $k$ is fixed.
As a consequence of our main result, the problem of deciding if a given AT-free graph contains a fixed graph $H$ as an induced topological minor admits a polynomial-time algorithm. In addition, we show that such an algorithm is essentially optimal by proving that the problem is W[1]-hard with parameter $|V_H|$, even on a subclass of AT-free graph, namely cobipartite graphs. We also show that the problems $k$-in-a-Path and $k$-in-a-Tree are polynomial-time solvable on AT-free graphs even if $k$ is part of the input. These problems are to test if a graph has an induced path or induced tree, respectively, spanning $k$ given vertices.
The well-known Disjoint Paths problem is to decide if a graph contains k pairwise disjoint paths, each connecting a different terminal pair from a set of k distinct pairs. We determine, with an exception of two cases, the complexity of the Disjoint Paths problem for $H$-free graphs. If $k$ is fixed, we obtain the $k$-Disjoint Paths problem, which is known to be polynomial-time solvable on the class of all graphs for every $k \geq 1$. The latter does no longer hold if we need to connect vertices from terminal sets instead of terminal pairs. We completely classify the complexity of $k$-Disjoint …
Paths $P^1,\ldots,P^k$ in a graph $G=(V,E)$ are mutually induced if any two distinct $P^i$ and $P^j$ have neither common vertices nor adjacent vertices. For a fixed integer $k$, the $k$-Induced Disjoint Paths problem is to decide if a graph $G$ with $k$ pairs of specified vertices $(s_i,t_i)$ contains $k$ mutually induced paths $P^i$ such that each $P^i$ starts from $s_i$ and ends at $t_i$. Whereas the non-induced version is well-known to be polynomial-time solvable for every fixed integer $k$, a classical result from the literature states that even $2$-Induced Disjoint Paths is NP-complete. W…
Given a graph G and k pairs of vertices (s_1,t_1), ..., (s_k,t_k), the k-Vertex-Disjoint Paths problem asks for pairwise vertex-disjoint paths P_1, ..., P_k such that P_i goes from s_i to t_i. Schrijver [SICOMP'94] proved that the k-Vertex-Disjoint Paths problem on planar directed graphs can be solved in time n^{O(k)}. We give an algorithm with running time 2^{2^{O(k^2)}} n^{O(1)} for the problem, that is, we show the fixed-parameter tractability of the problem.
Paths $P_1,\ldots, P_k$ in a graph $G=(V,E)$ are mutually induced if any two distinct $P_i$ and $P_j$ have neither common vertices nor adjacent vertices. The Induced Disjoint Paths problem is to decide if a graph $G$ with $k$ pairs of specified vertices $(s_i,t_i)$ contains $k$ mutually induced paths $P_i$ such that each $P_i$ starts from $s_i$ and ends at $t_i$. This is a classical graph problem that is NP-complete even for $k=2$. We introduce a natural generalization, Induced Disjoint Connected Subgraphs: instead of connecting pairs of terminals, we must connect sets of terminals. We give al…
We devise a constant-factor approximation algorithm for the maximization version of the edge-disjoint paths problem if the supply graph together with the demand edges form a planar graph. By planar duality this is equivalent to packing cuts in a planar graph such that each cut contains exactly one demand edge. We also show that the natural linear programming relaxations have constant integrality gap, yielding an approximate max-multiflow min-multicut theorem.
In this paper, we consider a transformation of $k$ disjoint paths in a graph. For a graph and a pair of $k$ disjoint paths $\mathcal{P}$ and $\mathcal{Q}$ connecting the same set of terminal pairs, we aim to determine whether $\mathcal{P}$ can be transformed to $\mathcal{Q}$ by repeatedly replacing one path with another path so that the intermediates are also $k$ disjoint paths. The problem is called Disjoint Paths Reconfiguration. We first show that Disjoint Paths Reconfiguration is PSPACE-complete even when $k=2$. On the other hand, we prove that, when the graph is embedded on a plane and al…
introduce {\sc Planar Disjoint Paths Completion}, a completion counterpart of the Disjoint Paths problem, and study its parameterized complexity. The problem can be stated as follows: given a, not necessarily connected, plane graph $G,$ $k$ pairs of terminals, and a face $F$ of $G,$ find a minimum-size set of edges, if one exists, to be added inside $F$ so that the embedding remains planar and the pairs become connected by $k$ disjoint paths in the augmented network. Our results are twofold: first, we give an upper bound on the number of necessary additional edges when a solution exists. This …
We investigate the computational complexity of finding temporally disjoint paths or walks in temporal graphs. There, the edge set changes over discrete time steps and a temporal path (resp. walk) uses edges that appear at monotonically increasing time steps. Two paths (or walks) are temporally disjoint if they never use the same vertex at the same time; otherwise, they interfere. This reflects applications in robotics, traffic routing, or finding safe pathways in dynamically changing networks. On the one extreme, we show that on general graphs the problem is computationally hard. The "walk ver…
Let G be a graph with minimum degree $δ$. It is well-known that maximal adjacency orderings (MAOs) compute a vertex set S such that every pair of S is connected by at least $δ$ internally vertex-disjoint paths in G. We present an algorithm that, given any pair of S, computes these $δ$ paths in linear time O(n+m). This improves the previously best solutions for these special vertex pairs, which were flow-based. Our algorithm simplifies a proof about pendant pairs of Mader and makes a purely existential proof of Nagamochi algorithmic.
We study the following general disjoint paths problem: given a supply graph $G$, a set $T\subseteq V(G)$ of terminals, a demand graph $H$ on the vertices $T$, and an integer $k$, the task is to find a set of $k$ pairwise vertex-disjoint valid paths, where we say that a path of the supply graph $G$ is valid if its endpoints are in $T$ and adjacent in the demand graph $H$. For a class $\mathcal{H}$ of graphs, we denote by $\mathcal{H}$-Maximum Disjoint Paths the restriction of this problem when the demand graph $H$ is assumed to be a member of $\mathcal{H}$. We study the fixed-parameter tractabi…
Let $G=(V,E)$ be a multigraph with a set $T\subseteq V$ of terminals. A path in $G$ is called a $T$-path if its ends are distinct vertices in $T$ and no internal vertices belong to $T$. In 1978, Mader showed a characterization of the maximum number of edge-disjoint $T$-paths. In this paper, we provide a combinatorial, deterministic algorithm for finding the maximum number of edge-disjoint $T$-paths. The algorithm adopts an augmenting path approach. More specifically, we utilize a new concept of short augmenting walks in auxiliary labeled graphs to capture a possible augmentation of the number …
(see paper for full abstract) We show that the Edge-Disjoint Paths problem is W[1]-hard parameterized by the number $k$ of terminal pairs, even when the input graph is a planar directed acyclic graph (DAG). This answers a question of Slivkins (ESA '03, SIDMA '10). Moreover, under the Exponential Time Hypothesis (ETH), we show that there is no $f(k)\cdot n^{o(k)}$ algorithm for Edge-Disjoint Paths on planar DAGs, where $k$ is the number of terminal pairs, $n$ is the number of vertices and $f$ is any computable function. Our hardness holds even if both the maximum in-degree and maximum out-degre…
Let $G$ be a directed planar graph of complexity $n$, each arc having a nonnegative length. Let $s$ and $t$ be two distinct faces of $G$; let $s_1,...,s_k$ be vertices incident with $s$; let $t_1,...,t_k$ be vertices incident with $t$. We give an algorithm to compute $k$ pairwise vertex-disjoint paths connecting the pairs $(s_i,t_i)$ in $G$, with minimal total length, in $O(kn\log n)$ time.
The Induced Disjoint Paths problem is to test whether a graph G with k distinct pairs of vertices (s_i,t_i) contains paths P_1,...,P_k such that P_i connects s_i and t_i for i=1,...,k, and P_i and P_j have neither common vertices nor adjacent vertices (except perhaps their ends) for 1<=i < j<=k. We present a linear-time algorithm for Induced Disjoint Paths on circular-arc graphs. For interval graphs, we exhibit a linear-time algorithm for the generalization of Induced Disjoint Paths where the pairs (s_i,t_i) are not necessarily distinct.
The disjoint paths problem is a fundamental problem in algorithmic graph theory and combinatorial optimization. For a given graph $G$ and a set of $k$ pairs of terminals in $G$, it asks for the existence of $k$ vertex-disjoint paths connecting each pair of terminals. The proof of Robertson and Seymour [JCTB 1995] of the existence of an $n^3$ algorithm for any fixed $k$ is one of the highlights of their Graph Minors project. In this paper, we focus on the version of the problem where all the paths are required to be shortest paths. This problem, called the disjoint shortest paths problem, was i…
In the Directed Disjoint Paths problem, we are given a digraph $D$ and a set of requests $\{(s_1, t_1), \ldots, (s_k, t_k)\}$, and the task is to find a collection of pairwise vertex-disjoint paths $\{P_1, \ldots, P_k\}$ such that each $P_i$ is a path from $s_i$ to $t_i$ in $D$. This problem is NP-complete for fixed $k=2$ and W[1]-hard with parameter $k$ in DAGs. A few positive results are known under restrictions on the input digraph, such as being planar or having bounded directed tree-width, or under relaxations of the problem, such as allowing for vertex congestion. Positive results are sc…
In the Disjoint Paths problem, the input consists of an $n$-vertex graph $G$ and a collection of $k$ vertex pairs, $\{(s_i,t_i)\}_{i=1}^k$, and the objective is to determine whether there exists a collection $\{P_i\}_{i=1}^k$ of $k$ pairwise vertex-disjoint paths in $G$ where the end-vertices of $P_i$ are $s_i$ and $t_i$. This problem was shown to admit an $f(k)n^3$-time algorithm by Robertson and Seymour (Graph Minors XIII, The Disjoint Paths Problem, JCTB). In modern terminology, this means that Disjoint Paths is fixed parameter tractable (FPT) with respect to $k$. Remarkably, the above algo…
In the Disjoint Paths problem, the input is an undirected graph $G$ on $n$ vertices and a set of $k$ vertex pairs, $\{s_i,t_i\}_{i=1}^k$, and the task is to find $k$ pairwise vertex-disjoint paths connecting $s_i$ to $t_i$. The problem was shown to have an $f(k)n^3$ algorithm by Robertson and Seymour. In modern terminology, this means that Disjoint Paths is fixed parameter tractable (FPT), parameterized by the number of vertex pairs. This algorithm is the cornerstone of the entire graph minor theory, and a vital ingredient in the $g(k)n^3$ algorithm for Minor Testing (given two undirected grap…
In this paper, we study the \textsf{Planar Disjoint Paths} problem: Given an undirected planar graph $G$ with $n$ vertices and a set $T$ of $k$ pairs $(s_i,t_i)_{i=1}^k$ of vertices, the goal is to find a set $\mathcal P$ of $k$ pairwise vertex-disjoint paths connecting $s_i$ and $t_i$ for all indices $i\in\{1,\ldots,k\}$. We present a $2^{O(k^2)}n$-time algorithm for the \textsf{Planar Disjoint Paths} problem. This improves the two previously best-known algorithms: $2^{2^{O(k)}}n$-time algorithm [Discrete Applied Mathematics 1995] and $2^{O(k^2)}n^6$-time algorithm [STOC 2020].
Enumerating chemical graphs satisfying given constraints is a fundamental problem in mathematical and computational chemistry, and plays an essential part in a recently proposed framework for the inverse QSAR/QSPR. In this paper, constraints are given by feature vectors each of which consists of the frequencies of paths in a given set of paths. We consider the problem of enumerating chemical graphs that satisfy the path frequency constraints, which are given by a pair of feature vectors specifying upper and lower bounds of the frequency of each path. We design a branch-and-bound algorithm for …
The problem of finding the maximum number of vertex-disjoint uni-color paths in an edge-colored graph (called MaxCDP) has been recently introduced in literature, motivated by applications in social network analysis. In this paper we investigate how the complexity of the problem depends on graph parameters (namely the number of vertices to remove to make the graph a collection of disjoint paths and the size of the vertex cover of the graph), which makes sense since graphs in social networks are not random and have structure. The problem was known to be hard to approximate in polynomial time and…
A temporal graph has an edge set that may change over discrete time steps, and a temporal path (or walk) must traverse edges that appear at increasing time steps. Accordingly, two temporal paths (or walks) are temporally disjoint if they do not visit any vertex at the same time. The study of the computational complexity of finding temporally disjoint paths or walks in temporal graphs has recently been initiated by Klobas et al. [IJCAI '21]. This problem is motivated by applications in multi-agent path finding (MAPF), which include robotics, warehouse management, aircraft management, and traffi…
We study an "above guarantee" version of the {\sc Longest Path} problem in directed graphs: We are given a graph $G$, two vertices $s$ and $t$ of $G$, and a non-negative integer $k$, and the objective is to determine whether $G$ contains a path of length at least $dist_G(s,t) +k$ where $dist_G(s,t)$ is the length of a shortest path from $s$ to $t$ in $G$ (assuming that one exists). We show that the problem is fixed parameter tractable (FPT) parameterized by $k$ in the class of graphs where {\sc $2$-Disjoint Paths} problem is polynomial time solvable.
Given an undirected $n$-vertex graph and $k$ pairs of terminal vertices $(s_1,t_1), \ldots, (s_k,t_k)$, the $k$-Disjoint Shortest Paths ($k$-DSP)-problem asks whether there are $k$ pairwise vertex-disjoint paths $P_1,\ldots, P_k$ such that $P_i$ is a shortest $s_i$-$t_i$-path for each $i \in [k]$. Recently, Lochet [SODA 2021] provided an algorithm that solves $k$-DSP in $n^{O(k^{5^k})}$ time, answering a 20-year old question about the computational complexity of $k$-DSP for constant $k$. On the one hand, we present an improved $n^{O(k!k)}$-time algorithm based on a novel geometric view on this…
The Disjoint Paths Problem asks, given a graph $G$ and a set of pairs of terminals $(s_{1},t_{1}),\ldots,(s_{k},t_{k})$, whether there is a collection of $k$ pairwise vertex-disjoint paths linking $s_{i}$ and $t_{i}$, for $i=1,\ldots,k.$ In their $f(k)\cdot n^{3}$ algorithm for this problem, Robertson and Seymour introduced the irrelevant vertex technique according to which in every instance of treewidth greater than $g(k)$ there is an "irrelevant" vertex whose removal creates an equivalent instance of the problem. This fact is based on the celebrated Unique Linkage Theorem, whose - very techn…
We introduce and study the complexity of Path Packing. Given a graph $G$ and a list of paths, the task is to embed the paths edge-disjoint in $G$. This generalizes the well known Hamiltonian-Path problem. Since Hamiltonian Path is efficiently solvable for graphs of small treewidth, we study how this result translates to the much more general Path Packing. On the positive side, we give an FPT-algorithm on trees for the number of paths as parameter. Further, we give an XP-algorithm with the combined parameters maximal degree, number of connected components and number of nodes of degree at least …
We give the first polynomial-time algorithms on graphs of bounded maximum induced matching width (mim-width) for problems that are not locally checkable. In particular, we give $n^{\mathcal{O}(w)}$-time algorithms on graphs of mim-width at most $w$, when given a decomposition, for the following problems: Longest Induced Path, Induced Disjoint Paths and $H$-Induced Topological Minor for fixed $H$. Our results imply that the following graph classes have polynomial-time algorithms for these three problems: Interval and Bi-Interval graphs, Circular Arc, Permutation and Circular Permutation graphs,…
In the k-Disjoint Shortest Paths problem, a set of terminal pairs of vertices $\{(s_i,t_i)\mid 1\le i\le k\}$ is given and we are asked to find paths $P_1,\ldots,P_k$ such that each path $P_i$ is a shortest path from $s_i$ to $t_i$ and every vertex of the graph routes at most one of them. We introduce a generalization of the problem, namely, $k$-Disjoint Shortest Paths with Congestion-$c$ where every vertex is allowed to route up to $c$ paths. We provide a simple algorithm to solve the problem in time $f(k) n^{O(k-c)}$ on DAGs. Using the techniques for DAGs, we show the problem is solvable in …
Given an undirected graph and two disjoint vertex pairs $s_1,t_1$ and $s_2,t_2$, the Shortest two disjoint paths problem (S2DP) asks for the minimum total length of two vertex disjoint paths connecting $s_1$ with $t_1$, and $s_2$ with $t_2$, respectively. We show that for cubic planar graphs there are NC algorithms, uniform circuits of polynomial size and polylogarithmic depth, that compute the S2DP and moreover also output the number of such minimum length path pairs. Previously, to the best of our knowledge, no deterministic polynomial time algorithm was known for S2DP in cubic planar graphs…
Paths P1,...,Pk in a graph G=(V,E) are said to be mutually induced if for any 1 <= i < j <= k, Pi and Pj have neither common vertices nor adjacent vertices (except perhaps their end-vertices). The Induced Disjoint Paths problem is to test whether a graph G with k pairs of specified vertices (si,ti) contains k mutually induced paths Pi such that Pi connects si and ti for i=1,...,k. We show that this problem is fixed-parameter tractable for claw-free graphs when parameterized by k. Several related problems, such as the k-in-a-Path problem, are proven to be fixed-parameter tractable for claw-free…
Björklund and Husfeldt developed a randomized polynomial time algorithm to solve the shortest two disjoint paths problem. Their algorithm is based on computation of permanents modulo 4 and the isolation lemma. In this paper, we consider the following generalization of the shortest two disjoint paths problem, and develop a similar algebraic algorithm. The shortest perfect $(A+B)$-path packing problem is: given an undirected graph $G$ and two disjoint node subsets $A,B$ with even cardinalities, find a shortest $|A|/2+|B|/2$ disjoint paths whose ends are both in $A$ or both in $B$. Besides its NP…
On the Parameterized Tractability of Packing Vertex-Disjoint A-Paths with Length Constraints
Tata Institute of Fundamental Research（塔塔基础研究所）；National Institute of Science, Education and Research（国家科学教育与研究学院）；Indraprastha Institute of Information Technology Delhi（德里因德拉普拉斯塔信息技术学院）
For two positive integers $k$ and $\ell$, a $(k \times \ell)$-spindle is the union of $k$ pairwise internally vertex-disjoint directed paths with $\ell$ arcs between two vertices $u$ and $v$. We are interested in the (parameterized) complexity of several problems consisting in deciding whether a given digraph contains a subdivision of a spindle, which generalize both the Maximum Flow and Longest Path problems. We obtain the following complexity dichotomy: for a fixed $\ell \geq 1$, finding the largest $k$ such that an input digraph $G$ contains a subdivision of a $(k \times \ell)$-spindle is p…
We introduce a new subclass of chordal graphs that generalizes split graphs, which we call well-partitioned chordal graphs. Split graphs are graphs that admit a partition of the vertex set into cliques that can be arranged in a star structure, the leaves of which are of size one. Well-partitioned chordal graphs are a generalization of this concept in the following two ways. First, the cliques in the partition can be arranged in a tree structure, and second, each clique is of arbitrary size. We provide a characterization of well-partitioned chordal graphs by forbidden induced subgraphs, and giv…
A localized method to distribute paths on random graphs is devised, aimed at finding the shortest paths between given source/destination pairs while avoiding path overlaps at nodes. We propose a method based on message-passing techniques to process global information and distribute paths optimally. Statistical properties such as scaling with system size and number of paths, average path-length and the transition to the frustrated regime are analysed. The performance of the suggested algorithm is evaluated through a comparison against a greedy algorithm.
We study the problem of routing on disjoint paths in bounded treewidth graphs with both edge and node capacities. The input consists of a capacitated graph $G$ and a collection of $k$ source-destination pairs $\mathcal{M} = \{(s_1, t_1), \dots, (s_k, t_k)\}$. The goal is to maximize the number of pairs that can be routed subject to the capacities in the graph. A routing of a subset $\mathcal{M}'$ of the pairs is a collection $\mathcal{P}$ of paths such that, for each pair $(s_i, t_i) \in \mathcal{M}'$, there is a path in $\mathcal{P}$ connecting $s_i$ to $t_i$. In the Maximum Edge Disjoint Pat…
The shortest secure path (routing) problem in communication networks has to deal with multiple attack layers e.g., man-in-the-middle, eavesdropping, packet injection, packet insertion, etc. Consider different probabilities for each such attack over an edge, probabilities that can differ across edges. Furthermore, usage of a single shortest path (for routing) implies possible traffic bottleneck, which should be avoided if possible, which we term pathneck security avoidance. Finding all Pareto-optimal solutions for the multi-criteria single-source single-destination shortest secure path problem …
The overwhelming majority of survivable (fault-tolerant) network design models assume a uniform fault model. Such a model assumes that every subset of the network resources (edges or vertices) of a given cardinality $k$ may fail. While this approach yields problems with clean combinatorial structure and good algorithms, it often fails to capture the true nature of the scenario set coming from applications. One natural refinement of the uniform model is obtained by partitioning the set of resources into vulnerable and safe resources. The scenario set contains every subset of at most $k$ faulty …
In this paper, we continue our development of algorithms used for topological network discovery. We present native P system versions of two fundamental problems in graph theory: finding the maximum number of edge- and node-disjoint paths between a source node and target node. We start from the standard depth-first-search maximum flow algorithms, but our approach is totally distributed, when initially no structural information is available and each P system cell has to even learn its immediate neighbors. For the node-disjoint version, our P system rules are designed to enforce node weight capac…
Technische Universität Berlin（柏林工业大学）；Discrete Mathematics Group, Institute for Basic Science (IBS)（基础科学研究所离散数学组）；LIRMM, Univ Montpellier, CNRS（蒙彼利埃大学CNRS LIRMM实验室）；School of Computing, KAIST（韩国科学技术院计算机学院）
A solution of the $k$ shortest paths problem may output paths that are identical up to a single edge. On the other hand, a solution of the $k$ independent shortest paths problem consists of paths that share neither an edge nor an intermediate node. We investigate the case in which the number of edges that are not shared among any two paths in the output $k$-set is a parameter. We study two main directions: exploring \emph{near-shortest} paths and exploring \emph{exactly shortest paths}. We assume that the weighted graph $G=(V,E,w)$ has no parallel edges and that the edge lengths (weights) are …
In activation network design problems we are given an undirected graph $G=(V,E)$ and a pair of activation costs $\{c_e^u,c_e^v\}$ for each $e=uv \in E$. The goal is to find an edge set $F \subseteq E$ that satisfies a prescribed property of minimum activation cost $τ(F)=\sum_{v \in V} \max \{c_e^v: e \in F \mbox{ is incident to } v\}$. In the Activation $k$ Disjoint Paths problem we are given $s,t \in V$ and an integer $k$, and seek an edge set $F \subseteq E$ of $k$ internally disjoint $st$-paths of minimum activation cost. The problem admits an easy $2$-approximation algorithm. However, it w…
Motivated by applications to social network analysis (SNA), we study the problem of finding the maximum number of disjoint uni-color paths in an edge-colored graph. We show the NP-hardness and the approximability of the problem, and both approximation and exact algorithms are proposed. Since short paths are much more significant in SNA, we also study the length-bounded version of the problem, in which the lengths of paths are required to be upper bounded by a fixed integer $l$. It is shown that the problem can be solved in polynomial time for $l=3$ and is NP-hard for $l\geq 4$. We also show th…
Let $G$ be a directed graph with $n$ vertices, $m$ edges, and non-negative edge costs. Given $G$, a fixed source vertex $s$, and a positive integer $p$, we consider the problem of computing, for each vertex $t\neq s$, $p$ edge-disjoint paths of minimum total cost from $s$ to $t$ in $G$. Suurballe and Tarjan~[Networks, 1984] solved the above problem for $p=2$ by designing a $O(m+n\log n)$ time algorithm which also computes a sparse \emph{single-source $2$-multipath preserver}, i.e., a subgraph containing $2$ edge-disjoint paths of minimum total cost from $s$ to every other vertex of $G$. The ca…
We consider the problem of finding, for two pairs $(s_1,t_1)$ and $(s_2,t_2)$ of vertices in an undirected graphs, an $(s_1,t_1)$-path $P_1$ and an $(s_2,t_2)$-path $P_2$ such that $P_1$ and $P_2$ share no edges and the length of each $P_i$ satisfies $L_i$, where $L_i \in \{ \le k_i, \; = k_i, \; \ge k_i, \; \le \infty\}$. We regard $k_1$ and $k_2$ as parameters and investigate the parameterized complexity of the above problem when at least one of $P_1$ and $P_2$ has a length constraint (note that $L_i = "\le \infty"$ indicates that $P_i$ has no length constraint). For the nine different cases…
We analyze the Disjoint Path Allocation problem (DPA) in the priority framework. Motivated by the problem of traffic regulation in communication networks, DPA consists of allocating edge-disjoint paths in a graph. While online algorithms for DPA have been thoroughly studied in the past, we extend the analysis of this optimization problem by considering the more powerful class of priority algorithms. Like an online algorithm, a priority algorithm receives its input only sequentially and must output irrevocable decisions for individual input items before having seen the input in its entirety. Ho…
"Minimum Falling Path Sum" (MFPS) is classic question in programming - "Given a grid of size $N{\times}N$ with integers in cells, return the minimum sum of a falling path through grid. A falling path starts at any cell in the first row and ends in last row, with the rule of motion - the next element after the cell $(i,j)$ is one of the cells $(i+1,j-1),(i+1,j)$ and $(i+1,j+1)$". This problem has linear solution (LS) (i.e. O($N^2$)) using dynamic programming method (DPM). There is an Multi-Agent version of MFPS called "Cherry Pickup II" (CP2). CP2 is a search for the maximum sum of 2 falling pa…
Consider an undirected graph $G = (VG, EG)$ and a set of six \emph{terminals} $T = \set{s_1, s_2, s_3, t_1, t_2, t_3} \subseteq VG$. The goal is to find a collection $\calP$ of three edge-disjoint paths $P_1$, $P_2$, and $P_3$, where $P_i$ connects nodes $s_i$ and $t_i$ ($i = 1, 2, 3$). Results obtained by Robertson and Seymour by graph minor techniques imply a polynomial time solvability of this problem. The time bound of their algorithm is $O(m^3)$ (hereinafter we assume $n := \abs{VG}$, $m := \abs{EG}$, $n = O(m)$). In this paper we consider a special, \emph{Eulerian} case of $G$ and $T$. N…
This paper revisits the classical Edge Disjoint Paths (EDP) problem, where one is given an undirected graph $G$ and a set of terminal pairs $P$ and asks whether $G$ contains a set of pairwise edge-disjoint paths connecting every terminal pair in $P$. Our aim is to identify structural properties (parameters) of graphs which allow the efficient solution of EDP without restricting the placement of terminals in $P$ in any way. In this setting, EDP is known to remain NP-hard even on extremely restricted graph classes, such as graphs with a vertex cover of size $3$. We present three results which us…
