Showing papers on "Longest path problem published in 2020"

A Systematic Review of Hyper-Heuristics on Combinatorial Optimization Problems

Abstract: Hyper-heuristics aim at interchanging different solvers while solving a problem. The idea is to determine the best approach for solving a problem at its current state. This way, every time we make a move it gets us closer to a solution. The problem changes; so does its state. As a consequence, for the next move, a different solver may be invoked. Hyper-heuristics have been around for almost 20 years. However, combinatorial optimization problems date from way back. Thus, it is paramount to determine whether the efforts revolving around hyper-heuristic research have been targeted at the problems of the highest interest for the combinatorial optimization community. In this work, we tackle such an endeavor. We begin by determining the most relevant combinatorial optimization problems, and then we analyze them in the context of hyper-heuristics. The idea is to verify whether they remain as relevant when considering exclusively works related to hyper-heuristics. We find that some of the most relevant problem domains have also been popular for hyper-heuristics research. Alas, others have not and few efforts have been directed towards solving them. We identify the following problem domains, which may help in furthering the impact of hyper-heuristics: Shortest Path, Set Cover, Longest Path, and Minimum Spanning Tree. We believe that focusing research on ways for solving them may lead to an increase in the relevance and impact that hyper-heuristics have on combinatorial optimization problems.

Utilization-Tensity Bound for Real-Time DAG Tasks under Global EDF Scheduling

Abstract: Utilization bound is a well-known concept in real-time scheduling theory for sequential periodic tasks, which can be used both for quantifying the performance of scheduling algorithms and as efficient schedulability tests. However, the schedulability of parallel real time task graphs depends on not only utilization, but also another parameter tensity , the ratio between the longest path length and period. In this paper, we use utilization-tensity bounds to better characterize the schedulability of parallel real-time tasks. In particular, we derive utilization-tensity bounds for parallel DAG tasks under global EDF scheduling, which facilitate significantly more precise schedulability analysis than the state-of-the-art analysis techniques based on capacity augmentation bound and response time analysis. Moreover, we apply the above results to the federated scheduling paradigm to improve the system schedulability by choosing proper scheduling strategies for tasks with different workload and structure features.

A path recorder algorithm for Multiple Longest Common Subsequences (MLCS) problems.

Abstract: MOTIVATION Searching the Longest Common Subsequences of many sequences is called a Multiple Longest Common Subsequence (MLCS) problem which is a very fundamental and challenging problem in many fields of data mining. The existing algorithms cannot be applicable to problems with long and large-scale sequences due to their huge time and space consumption. To efficiently handle large-scale MLCS problems, a Path Recorder Directed Acyclic Graph (PRDAG) model and a novel Path Recorder Algorithm (PRA) are proposed. RESULTS In PRDAG, we transform the MLCS problem into searching the longest path from the Directed Acyclic Graph (DAG), where each longest path in DAG corresponds to an MLCS. To tackle the problem efficiently, we eliminate all redundant and repeated nodes during the construction of DAG, and for each node, we only maintain the longest paths from the source node to it but ignore all non-longest paths. As a result, the size of the DAG becomes very small, and the memory space and search time will be greatly saved. Empirical experiments have been performed on a standard benchmark set of both DNA sequences and protein sequences. The experimental results demonstrate that our model and algorithm outperform the related leading algorithms, especially for large-scale MLCS problems. AVAILABILITY AND IMPLEMENTATION This program code is written by the first author and can be available at https://www.ncbi.nlm.nih.gov/nuccore and https://blog.csdn.net/wswguilin. SUPPLEMENTARY INFORMATION Supplementary data are available at Bioinformatics online.

Contracting to a longest path in H-free graphs

Abstract: The Path Contraction problem has as input a graph G and an integer k and is to decide if G can be modified to the k-vertex path P_k by a sequence of edge contractions. A graph G is H-free for some graph H if G does not contain H as an induced subgraph. The Path Contraction problem restricted to H-free graphs is known to be NP-complete if H = claw or H = P₆ and polynomial-time solvable if H = P₅. We first settle the complexity of Path Contraction on H-free graphs for every H by developing a common technique. We then compare our classification with a (new) classification of the complexity of the problem Long Induced Path, which is to decide for a given integer k, if a given graph can be modified to P_k by a sequence of vertex deletions. Finally, we prove that the complexity classifications of Path Contraction and Cycle Contraction for H-free graphs do not coincide. The latter problem, which has not been fully classified for H-free graphs yet, is to decide if for some given integer k, a given graph contains the k-vertex cycle C_k as a contraction.

The Distribution of Path Lengths On Directed Weighted Graphs

Abstract: We consider directed weighted graphs and define various families of path counting functions. Our main results are explicit formulas for the main term of the asymptotic growth rate of these counting functions, under some irrationality assumptions on the lengths of all closed orbits on the graph. In addition we assign transition probabilities to such graphs and compute statistics of the corresponding random walks. Some examples and applications are reviewed.

The longest path in the Price model

Abstract: The Price model, the directed version of the Barabasi-Albert model, produces a growing directed acyclic graph. We look at variants of the model in which directed edges are added to the new vertex in one of two ways: using cumulative advantage (preferential attachment) choosing vertices in proportion to their degree, or with random attachment in which vertices are chosen uniformly at random. In such networks, the longest path is well defined and in some cases is known to be a better approximation to geodesics than the shortest path. We define a reverse greedy path and show both analytically and numerically that this scales with the logarithm of the size of the network with a coefficient given by the number of edges added using random attachment. This is a lower bound on the length of the longest path to any given vertex and we show numerically that the longest path also scales with the logarithm of the size of the network but with a larger coefficient that has some weak dependence on the parameters of the model.

On path partitions of the divisor graph

Abstract: It is known that the longest simple path in the divisor graph that uses integers $\leq N$ is of length $\asymp N/\log N$. We study the partitions of $\{1,2,\dots, N\}$ into a minimal number of paths of the divisor graph, and we show that in such a partition, the longest path can have length asymptotically $N^{1-o(1)}$.

2D Qubit Placement of Quantum Circuits Using LONGPATH.

Abstract: In order to achieve speedup over conventional classical computing for finding solution of computationally hard problems, quantum computing was introduced. Quantum algorithms can be simulated in a pseudo quantum environment, but implementation involves realization of quantum circuits through physical synthesis of quantum gates. This requires decomposition of complex quantum gates into a cascade of simple one-qubit and two-qubit gates. The methodological framework for physical synthesis imposes a constraint regarding placement of operands (qubits) and operators. If physical qubits can be placed on a grid, where each node of the grid represents a qubit, then quantum gates can only be operated on adjacent qubits, otherwise SWAP gates must be inserted to convert nonlinear nearest neighbour architecture to linear nearest neighbour architecture. Insertion of SWAP gates should be made optimal to reduce cumulative cost of physical implementation. A schedule layout generation is required for placement and routing a priori to actual implementation. In this paper, two algorithms are proposed to optimize the number of SWAP gates in any arbitrary quantum circuit. The first algorithm is intended to start with generation of an interaction graph followed by finding the longest path starting from the node with maximum degree. The second algorithm optimizes the number of SWAP gates between any pair of non-neighbouring qubits. Our proposed approach has a significant reduction in number of SWAP gates in 1D and 2D NTC architecture.

On the Analysis of Parallel Real-Time Tasks with Spin Locks

Abstract: Locking protocol is an essential component in resource management of real-time systems, which coordinates mutually exclusive accesses to shared resources from different tasks. Although the design and analysis of locking protocols have been intensively studied for sequential real-time tasks, there has been little work on this topic for parallel real-time tasks. In this paper, we study the analysis of parallel real-time tasks using spin locks to protect accesses to shared resources in three commonly used request serving orders (unordered, FIFO-order and priority-order). A remarkable feature making our analysis method more accurate is to systematically analyze the blocking time which may delay a task's finishing time, where the impact to the total workload and the longest path length is jointly considered, rather than analyzing them separately and counting all blocking time as the workload that delays a task's finishing time, as commonly assumed in the state-of-the-art.

The Maximum Binary Tree Problem.

Abstract: We introduce and investigate the approximability of the maximum binary tree problem (MBT) in directed and undirected graphs. The goal in MBT is to find a maximum-sized binary tree in a given graph. MBT is a natural variant of the well-studied longest path problem, since both can be viewed as finding a maximum-sized tree of bounded degree in a given graph. The connection to longest path motivates the study of MBT in directed acyclic graphs (DAGs), since the longest path problem is solvable efficiently in DAGs. In contrast, we show that MBT in DAGs is in fact hard: it has no efficient exp(-O(log n/ log log n))-approximation algorithm under the exponential time hypothesis, where n is the number of vertices in the input graph. In undirected graphs, we show that MBT has no efficient exp(-O(log^0.63 n))-approximation under the exponential time hypothesis. Our inapproximability results rely on self-improving reductions and structural properties of binary trees. We also show constant-factor inapproximability assuming P ≠ NP. In addition to inapproximability results, we present algorithmic results along two different flavors: (1) We design a randomized algorithm to verify if a given directed graph on n vertices contains a binary tree of size k in 2^k poly(n) time. (2) Motivated by the longest heapable subsequence problem, introduced by Byers, Heeringa, Mitzenmacher, and Zervas, ANALCO 2011, which is equivalent to MBT in permutation DAGs, we design efficient algorithms for MBT in bipartite permutation graphs.

Intersecting longest paths in chordal graphs

Abstract: We consider the size of the smallest set of vertices required to intersect every longest path in a chordal graph. Such sets are known as longest path transversals. We show that if $\omega(G)$ is the clique number of a chordal graph $G$, then there is a transversal of order at most $4\lceil\frac{\omega(G)}{5}\rceil$. We also consider the analogous question for longest cycles, and show that if $G$ is a 2-connected chordal graph then there is a transversal intersecting all longest cycles of order at most $2\lceil\frac{\omega(G)}{3}\rceil$.

2D Qubit Placement of Quantum Circuits using LONGPATH

Abstract: In order to achieve speedup over conventional classical computing for finding solution of computationally hard problems, quantum computing was introduced. Quantum algorithms can be simulated in a pseudo quantum environment, but implementation involves realization of quantum circuits through physical synthesis of quantum gates. This requires decomposition of complex quantum gates into a cascade of simple one qubit and two qubit gates. The methodological framework for physical synthesis imposes a constraint regarding placement of operands (qubits) and operators. If physical qubits can be placed on a grid, where each node of the grid represents a qubit then quantum gates can only be operated on adjacent qubits, otherwise SWAP gates must be inserted to convert non-Linear Nearest Neighbor architecture to Linear Nearest Neighbor architecture. Insertion of SWAP gates should be made optimal to reduce cumulative cost of physical implementation. A schedule layout generation is required for placement and routing apriori to actual implementation. In this paper, two algorithms are proposed to optimize the number of SWAP gates in any arbitrary quantum circuit. The first algorithm is intended to start with generation of an interaction graph followed by finding the longest path starting from the node with maximum degree. The second algorithm optimizes the number of SWAP gates between any pair of non-neighbouring qubits. Our proposed approach has a significant reduction in number of SWAP gates in 1D and 2D NTC architecture.

Some exact values for p(t , d)

Abstract: Let G = (V, E) be a graph. When a graph is used to model the linkage structure of communication networks, the diameter of a graph gives the length of longest path among all the shortest paths betwe...

The conjugacy class graphs of non-abelian 3-groups

Abstract: A graph is formed by a pair of vertices and edges. It can be related to groups by using the groups’ properties for its vertices and edges. The set of vertices of the graph comprises the elements or sets from the group while the set of edges of the graph is the properties and condition for the graph. A conjugacy class of an element is the set of elements that are conjugated with . Any element of a group , labelled as , is conjugated to if it satisfies for some elements in with its inverse . A conjugacy class graph of a group is defined when its vertex set is the set of non-central conjugacy classes of . Two distinct vertices and are connected by an edge if and only if their cardinalities are not co-prime, which means that the order of the conjugacy classes of and have common factors. Meanwhile, a simple graph is the graph that contains no loop and no multiple edges. A complete graph is a simple graph in which every pair of distinct vertices is adjacent. Moreover, a -group is the group with prime power order. In this paper, the conjugacy class graphs for some non-abelian 3-groups are determined by using the group’s presentations and the definition of conjugacy class graph. There are two classifications of the non-abelian 3-groups which are used in this research. In addition, some properties of the conjugacy class graph such as the chromatic number, the dominating number, and the diameter are computed. A chromatic number is the minimum number of vertices that have the same colours where the adjacent vertices have distinct colours. Besides, a dominating number is the minimum number of vertices that is required to connect all the vertices while a diameter is the longest path between any two vertices. As a result of this research, the conjugacy class graphs of these groups are found to be complete graphs with chromatic number, dominating number and diameter that are equal to eight, one and one, respectively.

Algorithms and Experiments Comparing Two Hierarchical Drawing Frameworks.

Abstract: We present algorithms that extend the path-based hierarchical drawing framework and give experimental results. Our algorithms run in $O(km)$ time, where $k$ is the number of paths and $m$ is the number of edges of the graph, and provide better upper bounds than the original path based framework: e.g., the height of the resulting drawings is equal to the length of the longest path of $G$, instead of $n-1$, where $n$ is the number of nodes. Additionally, we extend this framework, by bundling and drawing all the edges of the DAG in $O(m + n \log n)$ time, using minimum extra width per path. We also provide some comparison to a well known hierarchical drawing framework, widely known as the Sugiyama framework, as a proof of concept. The experimental results show that our algorithms produce drawings that are better in area and number of bends, but worse for crossings in sparse graphs. Hence, our technique offers an interesting alternative for drawing hierarchical graphs. Finally, we present an $O(m + k \log k)$ time algorithm that computes a specific order of the paths in order to reduce the total edge length and number of crossings and bends.

Enhanced Algorithm for Logical Topology-Based Fault Link Recovery in Crossbar Networks

Abstract: Conventionally, virtual topology nodes are mapped with actual physical topology nodes and links mapped with lightpaths (LPs). But this is computationally unwieldy becoming huge and more complex with increasing network size. This work proposes a scheme for optimal utilization of the network, minimizing network congestion by allocating efficient LPs in existing physical systems. When network size increases, the degree of the sub-trees also increases but without any increase of carrier wavelengths; the number of hops (and carrier wavelengths) decreases as it generates direct paths between the longest path nodes. Structural metrics of the physical and logical topologies of this method are more efficient. It supports multicast connections and minimizes data replication. The outer iteration of the proposed algorithm is executed twice; the first iteration yields the network’s logical topology, while the second gives “complete bipartite graph.” Using the logical topology, the path length of the longest distance node is reduced, and direct connections between nodes are created. The reconfiguration is used only for the nearest node of logical topology, and this reduces the number of hops between the nearest nodes to three. The maximum path length is 6 for any n × n crossbar network. This will cause reduction in the number of protection cycle.

Generalized Path Optimization Problem for a Weighted Digraph over an Additively Idempotent Semiring

Abstract: In this paper, a generalized path optimization problem for a weighted digraph (i.e., directed graph) over an additively idempotent semiring was considered. First, the conditions for power convergence of a matrix over an additively idempotent semiring were investigated. Then we proved that the path optimization problem is associated with powers of the adjacency matrix of the weighted digraph. The classical matrix power method for the shortest path problem on the min-plus algebra was generalized to the generalized path optimization problem. The proposed generalized path optimization model encompasses different path optimization problems, including the longest path problem, the shortest path problem, the maximum reliability path problem, and the maximum capacity path problem. Finally, for the four special cases, we illustrate the pictorial representations of the graphs with example data and the proposed method.

Methods, systems, and media for resolving graph database queries

Abstract: Method, systems, and media for resolving a database query are provided, comprising: identifying a connected component in a query graph corresponding to the database query; determining a longest path length for the connected component; selecting a path having the longest path length; building an algebraic expression for the path; solving the algebraic expression using matrix-matrix multiplication to provide a solution; and responding to the query based on the solution.

Sensing the insensible using optical schemes: converting the maze problem into a quantum search problem

Abstract: Many large image processing and data processing scenarios can soon develop into maze problems requiring, say, finding the longest possible path, which are unresponsive, intractable, and challenging to analyze! Such maze problems, of which the traveling salesman decision problem is a special case, are of the Non-determinate Polynomial (NPcomplete) class of problems that are often impossible to solve with finite time and storage. We propose a novel methodology to approach this class of NP-complete problems. We convert a suitably formulated maze problem into a Quantum Search Problem (QSP), and the desired solutions are then sought using the iterative Grover’s Search Algorithm. Thus, we reformulate the entire class of such NP-problems into QSPs. Our current solution deals with two-dimensional perfect mazes with no closed loops. We encode all possible individual paths from the starting point of the maze into a quantum register. A quantum fitness operator applied to the register encodes each qubit with its fitness value. We propose the design of an optical oracle that marks all entities above a certain fitness value and uses the Grover search algorithm to find the optimal marked state in an iterative manner.

Methods, systems, and media for resolving database queries using algebraic expressions using matrix-matrix multiplication

Abstract: Mechanisms are provided for resolving database queries. These mechanisms identify a connected component in a query graph corresponding to a database query. They then determine a longest path length for the connected component. Next, the mechanisms select a path having the longest path length and build an algebraic expression for the path. Finally, the mechanisms solve the algebraic expression using matrix-matrix multiplication to provide a solution, and then respond to the query based on the solution.

Sublinear Longest Path Transversals and Gallai Families.

Abstract: We show that connected graphs admit sublinear longest path transversals. This improves an earlier result of Rautenbach and Sereni and is related to the fifty-year-old question of whether connected graphs admit constant-size longest path transversals. The same technique allows us to show that $2$-connected graphs admit sublinear longest cycle transversals. We also make progress toward a characterization of the graphs $H$ such that every connected $H$-free graph has a longest path transversal of size $1$. In particular, we show that the graphs $H$ on at most $4$ vertices satisfying this property are exactly the linear forests. Finally, we show that if the order of a connected graph $G$ is large relative to its connectivity $\kappa(G)$ and $\alpha(G) \le \kappa(G) + 2$, then each vertex of maximum degree forms a longest path transversal of size $1$.

Sensitivity Analysis of ARC Criticalities in Stochastic Activity Networks

Abstract: Using Monte Carlo simulation, this paper proposes a new algorithm for estimating the arc criticalities of stochastic activity networks. The algorithm is based on the following result: given the length of all arcs in a network except for the one arc of interest, which is on the critical path (longest path) if and only if its length is greater than a threshold. Therefore, the new algorithm is named Threshold Arc Criticality (TAC). By applying Infinitesimal Perturbation Analysis (IPA) to TAC, an unbiased estimator of the stochastic derivative of the arc criticalities with respect to parameters of arc length distributions can be derived. With a valid estimator of stochastic derivative of arc criticalities, sensitivity analysis of arc criticalities is carried out via simulation of a small test network.

Approximate Risk Function for Project Durations

Abstract: In this chapter we discuss the application of the second moment method in scheduling networks. The issue we focus here is on finding the probability distribution for the project completion time when there are multiple network paths in the project and therefore the critical path is itself uncertain. We present an approximate method to this problem and discuss its validity in a larger managerial context.

Some Classical Problems on K 1,3 -free Split Graphs.

Abstract: A set D ⊆ V of a graph G is called a total outerconnected dominating set of G if D is a total dominating set of G and G[V \ D] is connected. The Minimum Total Outerconnected Domination (MTOCD) problem is NP-complete in general graphs, chordal graphs and split graphs. Hence we look into MTOCD problem and found that in K1,3-free split graphs the MTOCD problem is polynomial-time solvable. We present an polynomial-time algorithm which computes minimum total outer-connected dominating set in K1,3-free split graphs since the problem is NP-complete in K1,5-free split graphs as we can observe that from the NP-completeness reduction of split graphs has K1,5 as its induced subgraph. The Longest path problem is the problem of finding a simple maximum length path in a graph. Longest path is NP-complete for all graph classes in which Hamiltonian path problem is NPComplete. Since Hamiltonian path problem is known to be NPComplete in K1,5-free split graphs, it is obvious that in K1,5-free split graphs longest path problem is also NP-complete. We present a polynomial-time algorithm to find a longest path in K1,3-free split graphs. The decision version of Maximum-Cut problem is known to be NP-complete in 2-split graphs. We propose a polynomial-time algorithm to solve the Maximum-Cut problem in K1,3-free split graphs with d(v) ≤ 2, Vv ∈ I, which are a subclass of 2split graphs. We also present polynomial-time algorithms for the Steiner Path problem, the dominating set problem the Steiner cycle problem and cutwidth problem restricted to K1,3-free split graphs.

Hamiltonian Cycle Parameterized by Treedepth in Single Exponential Time and Polynomial Space

Abstract: For many algorithmic problems on graphs of treewidth $t$, a standard dynamic programming approach gives an algorithm with time and space complexity $2^{\mathcal{O}(t)}\cdot n^{\mathcal{O}(1)}$. It turns out that when one considers the more restrictive parameter treedepth, it is often the case that a variation of this technique can be used to reduce the space complexity to polynomial, while retaining time complexity of the form $2^{\mathcal{O}(d)}\cdot n^{\mathcal{O}(1)}$, where $d$ is the treedepth. This transfer of methodology is, however, far from automatic. For instance, for problems with connectivity constraints, standard dynamic programming techniques give algorithms with time and space complexity $2^{\mathcal{O}(t\log t)}\cdot n^{\mathcal{O}(1)}$ on graphs of treewidth $t$, but it is not clear how to convert them into time-efficient polynomial space algorithms for graphs of low treedepth. Cygan et al. (FOCS'11) introduced the Cut&Count technique and showed that a certain class of problems with connectivity constraints can be solved in time and space complexity $2^{\mathcal{O}(t)}\cdot n^{\mathcal{O}(1)}$. Recently, Hegerfeld and Kratsch (STACS'20) showed that, for some of those problems, the Cut&Count technique can be also applied in the setting of treedepth, and it gives algorithms with running time $2^{\mathcal{O}(d)}\cdot n^{\mathcal{O}(1)}$ and polynomial space usage. However, a number of important problems eluded such a treatment, with the most prominent examples being Hamiltonian Cycle and Longest Path. In this paper we clarify the situation by showing that Hamiltonian Cycle, Hamiltonian Path, Long Cycle, Long Path, and Min Cycle Cover all admit $5^d\cdot n^{\mathcal{O}(1)}$-time and polynomial space algorithms on graphs of treedepth $d$. The algorithms are randomized Monte Carlo with only false negatives.