Edge Costs Research Articles

Online graph exploration on a restricted graph class: Optimal solutions for tadpole graphs

We study the problem of online graph exploration on undirected graphs, where a searcher has to visit every vertex and return to the origin. Once a new vertex is visited, the searcher learns of all neighboring vertices and the connecting edge weights. The goal of such an exploration is to minimize its total cost, where each edge traversal incurs a cost of the corresponding edge weight.We investigate the problem on tadpole graphs (also known as dragons, kites), which consist of a cycle with an attached path. The construction by Miyazaki et al. (The online graph exploration problem on restricted graphs, IEICE Transactions 92-D (9), 2009) can be extended to show that every online algorithm on these graphs must have a competitive ratio of 2−ε, but the authors did not investigate non-unit edge weights. We show via amortized analysis that a greedy approach yields a matching competitive ratio of 2 on tadpole graphs, for arbitrary non-negative edge weights. Moreover, we also briefly discuss the topic of advice complexity on cycle and tadpole graphs.

Approximating Spanners and Directed Steiner Forest

It was recently found that there are very close connections between the existence of additive spanners (subgraphs where all distances are preserved up to an additive stretch), distance preservers (subgraphs in which demand pairs have their distance preserved exactly), and pairwise spanners (subgraphs in which demand pairs have their distance preserved up to a multiplicative or additive stretch) [Abboud-Bodwin SODA’16 8 J.ACM’17, Bodwin-Williams SODA’16]. We study these problems from an optimization point of view, where rather than studying the existence of extremal instances, we are given an instance and are asked to find the sparsest possible spanner/preserver. We give an O ( n 3/5 + ε )-approximation for distance preservers and pairwise spanners (for arbitrary constant ε > 0). This is the first nontrivial upper bound for either problem, both of which are known to be as hard to approximate as Label Cover. We also prove Label Cover hardness for approximating additive spanners, even for the cases of additive 1 stretch (where one might expect a polylogarithmic approximation, since the related multiplicative 2-spanner problem admits an O (log n )-approximation) and additive polylogarithmic stretch (where the related multiplicative spanner problem has an O (1)-approximation). Interestingly, the techniques we use in our approximation algorithm extend beyond distance-based problem to pure connectivity network design problems. In particular, our techniques allow us to give an O ( n 3/5 + ε )-approximation for the Directed Steiner Forest problem (for arbitrary constant ε > 0) when all edges have uniform costs, improving the previous best O ( n 2/3 + ε )-approximation due to Berman et al. [ICALP’11] (which holds for general edge costs).

General restricted inverse assignment problems under $$l_1$$ and $$l_\infty $$ norms

In this paper, we study the general restricted inverse assignment problems, in which we can only change the costs of some specific edges of an assignment problem as less as possible, so that a given assignment becomes the optimal one. Under $$l_1$$ norm, we formulate this problem as a linear programming. Then we mainly consider two cases. For the case when the specific edges are only belong to the given assignment, we show that this problem can be reduced to some variations of the minimum cost flow problems. For the case when every specific edge does not belong to the given assignment, we show that this problem can be solved by a minimum cost circulation problem. In both cases, we present some combinatorial algorithms which are strongly polynomial. We also study this problem under the $$l_\infty $$ norm. We propose a binary search algorithm and prove that the optimal solution can be obtained in polynomial time.

Combinatorial approximation algorithms for the submodular multicut problem in trees with submodular penalties

In this paper, we introduce the submodular multicut problem in trees with submodular penalties, which generalizes the prize-collecting multicut problem in trees and the submodular vertex cover with submodular penalties. We present a combinatorial approximation algorithm, based on the primal-dual algorithm for the submodular set cover problem. In addition, we present a combinatorial 3-approximation algorithm for a special case where the edge cost is a modular function, based on the primal-dual scheme for the multicut problem in trees.

Linear edge costs and labeling algorithms: The case of the time‐dependent vehicle routing problem with time windows

AbstractIn this paper we implement a branch‐price and cut algorithm for a time dependent vehicle routing problem with time windows in which the goal is to minimize the total route duration. The travel time between two customers is given by a piecewise linear function on the departure time and, thus, it need not remain fixed along the planning horizon. We discuss different alternatives for the implementation of these linear functions within the labeling algorithm applied to solve the pricing problem. We also provide a tailored implementation for one of these alternatives, relying on efficient data structures for storing the labels, and show several strategies to accelerate the algorithm. Computational results show that the proposed techniques are effective and improve the column generation step, solving all instances with 25 customers, 49 of 56 with 50 customers, and many instances with 100 customers. Furthermore, heuristic adaptations are able to find good quality solutions in reasonable computation times.

Smoothed Analysis of the Minimum-Mean Cycle Canceling Algorithm and the Network Simplex Algorithm

The minimum-cost flow (MCF) problem is a fundamental optimization problem with many applications and seems to be well understood. Over the last half century many algorithms have been developed to solve the MCF problem, and these algorithms have varying worst-case bounds on their running time. However, these worst-case bounds are not always a good indication of the algorithms' performance in practice. The Network Simplex (NS) algorithm needs an exponential number of iterations for some instances, but it is among the best algorithms in practice and performs very well in experimental studies. On the other hand, the Minimum-Mean Cycle Canceling (MMCC) algorithm is strongly polynomial, but performs badly in experimental studies. To explain these differences in performance in practice we apply the framework of smoothed analysis to NS and MMCC. We show an upper bound of $O(mn^2\log(n)\log(\phi))$ for the number of iterations of the MMCC algorithm. Here, $n$ is the number of nodes, $m$ is the number of edges, and $\phi$ is a parameter limiting the degree to which the edge costs are perturbed. We also show a lower bound of $\Omega(m\log(\phi))$ for the number of iterations of the MMCC algorithm, which can be strengthened to $\Omega(mn)$ when $\phi=\Theta(n^2)$. For the number of iterations of the NS algorithm we show a smoothed lower bound of $\Omega(m \cdot \min \{ n, \phi \} \cdot \phi)$.

Modeling the capacitated p-cable trench problem with facility costs

In this work, two network design concepts which have been proven to be relevant with respect to telecommunication applications, are joined within a unified approach. First, the cable trench problem searches for cost-minimizing network structures that take into account two types of edge costs appearing in the installation of wire-based networks, namely trenching costs and cable costs. Second, the facility location problem considers the placement of shared telecommunication equipment, like switches or concentrators, together with an assignment of entities to demand nodes. Following practical needs, we join these concepts within the new capacitated cable trench problem with facility costs and service capacity in terms of number of customers that can be served by facilities. Within this setting, facility location decisions in wire-based networks can be taken under a more realistic cost scenario. A mixed-integer linear program and valid inequalities are proposed. Experiments indicate a positive impact of the valid inequalities on computational time and integrality gap.

Dolha - an efficient and exact data structure for streaming graphs

A streaming graph is a graph formed by a sequence of incoming edges with time stamps. Unlike static graphs, the streaming graph is highly dynamic and time related. In the real world, the high volume and velocity streaming graphs such as internet traffic data, social network communication data and financial transfer data are bringing challenges to the classic graph data structures. We present a new data structure: double orthogonal list in hash table (Dolha) which is a high speed and high memory efficiency graph structure applicable to streaming graph. Dolha has constant time cost for single edge and near linear space cost that we can contain billions of edges information in memory size and process an incoming edge in nanoseconds. Dolha also has linear time cost for neighborhood queries, which allow it to support most algorithms in graphs without extra cost. We also present a persistent structure based on Dolha that has the ability to handle the sliding window update and time related queries.

An Eco-Routing Algorithm for HEVs Under Traffic Conditions

In a previous work, a bi-level optimization approach was presented for the energy management of Hybrid Electric Vehicles (HEVs), using a statistical model for traffic conditions. The present work is an extension of this framework to the eco-routing problem. The optimal trajectory is computed as the shortest path on a weighted graph whose nodes are (position, state of charge) pairs for the vehicle. The edge costs are provided by cost maps from an offline optimization at the lower level of small road segments. The error due to the discretization of the state of charge is proven to be linear if the cost maps are Lipschitz. The classical A⁎ algorithm is used to solve the problem, with a heuristic based on a lower bound of the energy needed to complete the travel. The eco-routing method is compared to the fastest-path strategy by numerical simulations on a simple synthetic road network.

Hunting multiple bumps in graphs

Bump hunting is an important approach to the extraction of insights from Euclidean datasets. Recently, it has been explored for graph datasets for the first time, and a single bump is hunted in an unweighted graph in this exploration. Here, we extend this exploration by hunting multiple bumps in a weighted graph. Given a weighted graph and a set of query nodes exhibiting a property of interest, our objective is to find k non-overlapping and connected subgraphs, i.e., bumps, in which the discrepancy between the numbers of query and non-query nodes is maximized and the sum of edge costs is minimized simultaneously. We prove that our extended bump hunting problem can be transformed to a recently formulated Prize-Collecting Steiner Forest Problem (PCSFP). We further prove that PCSFP is NP-hard even in trees. Then, we propose a fast approximation algorithm for solving PCSFP in trees. Based on this algorithm, we improve the state-of-the-art approximation algorithm for solving PCSFP in graphs, and prove that the solutions of our improvement are always better than or equal to those of the state-of-the-art algorithm. Moreover, we adapt the existing bump hunting algorithms for solving our extended bump hunting problem. We evaluate our methodology via real datasets, and show that 1) our improvement scales well to large graphs, while producing solutions that dominate those of the state-of-the-art algorithm; and 2) our adaptation of an existing bump hunting algorithm can also produce solutions that are better than those of the state-of-the-art algorithm in some cases.

Multi-level Steiner Trees

In the classical Steiner tree problem, given an undirected, connected graph G =( V , E ) with non-negative edge costs and a set of terminals T ⊆ V , the objective is to find a minimum-cost tree E &prime ⊆ E that spans the terminals. The problem is APX-hard; the best-known approximation algorithm has a ratio of ρ = ln (4)+ε < 1.39. In this article, we study a natural generalization, the multi-level Steiner tree (MLST) problem: Given a nested sequence of terminals T ℓ ⊂ … ⊂ T 1 ⊆ V , compute nested trees E ℓ ⊆ … ⊆ E 1 ⊆ E that span the corresponding terminal sets with minimum total cost. The MLST problem and variants thereof have been studied under various names, including Multi-level Network Design, Quality-of-Service Multicast tree, Grade-of-Service Steiner tree, and Multi-tier tree. Several approximation results are known. We first present two simple O (ℓ)-approximation heuristics. Based on these, we introduce a rudimentary composite algorithm that generalizes the above heuristics, and determine its approximation ratio by solving a linear program. We then present a method that guarantees the same approximation ratio using at most 2ℓ Steiner tree computations. We compare these heuristics experimentally on various instances of up to 500 vertices using three different network generation models. We also present several integer linear programming formulations for the MLST problem and compare their running times on these instances. To our knowledge, the composite algorithm achieves the best approximation ratio for up to ℓ = 100 levels, which is sufficient for most applications, such as network visualization or designing multi-level infrastructure.

A Faster Algorithm for Minimum-cost Bipartite Perfect Matching in Planar Graphs

Given a weighted planar bipartite graph G ( A ∪ B , E ) where each edge has an integer edge cost, we give an Õ( n 4/3 log nC ) time algorithm to compute minimum-cost perfect matching; here C is the maximum edge cost in the graph. The previous best-known planarity exploiting algorithm has a running time of O ( n 3/2 log n ) and is achieved by using planar separators (Lipton and Tarjan ’80). Our algorithm is based on the bit-scaling paradigm (Gabow and Tarjan ’89). For each scale, our algorithm first executes O ( n 1/3 ) iterations of Gabow and Tarjan’s algorithm in O ( n 4/3 ) time leaving only O ( n 2/3 ) vertices unmatched. Next, it constructs a compressed residual graph H with O ( n 2/3 ) vertices and O ( n ) edges. This is achieved by using an r -division of the planar graph G with r = n 2/3 . For each partition of the r -division, there is an edge between two vertices of H if and only if they are connected by a directed path inside the partition. Using existing efficient shortest-path data structures, the remaining O ( n 2/3 ) vertices are matched by iteratively computing a minimum-cost augmenting path, each taking Õ( n 2/3 ) time. Augmentation changes the residual graph, so the algorithm updates the compressed representation for each partition affected by the change in Õ( n 2/3 ) time. We bound the total number of affected partitions over all the augmenting paths by O ( n 2/3 log n ). Therefore, the total time taken by the algorithm is Õ( n 4/3 ).

Network Reconfiguration Algorithm (NRA) for scheduling communication-intensive graphs in heterogeneous computing environment

Distributed environments are widely used for computing complex applications modeled as task graphs. The computer network becomes more complex when the compute nodes are heterogeneous, however by choosing the appropriate network communication links for communication between a pair of compute tasks can enhance the computing efficiency(called network reconfiguration). One of the steps in heterogeneous network reconfiguration problem is mapping the application task graph edges on the network links. High Performance Computing (HPC) systems are usually heterogeneous, therefore mapping task graph edges on the communication links should consider the two factors: communication cost of task graph edges and the communication capability of network links. The system performance enhances if tasks are mapped on the compute nodes based on the computational costs of the tasks and the processing capability of compute nodes in addition to the edge scheduling on network links. In our earlier algorithm, Heterogeneous Edge and Task Scheduling (HETS) both edge and task mapping simultaneously improve the execution performance of task graphs. The proposed Network Reconfiguration Algorithm (NRA) minimizes the communication overhead and optimizes the schedule length with contention-aware model. NRA reduces an attribute Kirchhoff Index (KI) for optimal network reconfiguration providing minimum execution time. Both synthesized and task graphs of real applications are used for evaluation. The simulation results prove the efficiency of NRA in terms of average schedule length, schedule length ratio, speedup and system throughput. Comparisons with the baseline algorithms show that NRA provides 36% improved results specially for communication-intensive applications.

An auxiliary-graph-based methodology for regenerator assignment problem optimization in translucent elastic optical networks

Abstract In this article, we propose an auxiliary-graph-based methodology to best assess regenerator assignment choices in translucent elastic optical networks. The proposed methodology considers that each possible transparent segment in a route is represented by an edge in an auxiliary graph. Proper edge cost functions are designed and the decision on where to use regeneration resources is performed by a standard routing algorithm in the built auxiliary graph. We propose four different edge cost functions and compare the performance, in terms of blocking probability of call requests, of some networks using our proposed methodology and other heuristics available in the literature that tackle with regenerator assignment problem in translucent elastic optical networks. The simulation scenarios involved three different network physical topologies, amplified spontaneous emission noise as physical impairment generated by boosters, in-line amplifiers and pre-amplifiers, besides losses and gains observed by the optical signals along network lightpaths. By properly defining the respective edge cost function, our proposed methodology can not only replicate some heuristics previously proposed in the literature, but also create strategies that present superior performance in all analyzed scenarios and conditions.

A 4-approximation algorithm for the TSP-Path satisfying a biased triangle inequality

In this paper, we study the path version of the Traveling Salesman Problem with the edge cost function satisfying a “relaxed” form of triangle inequality called the biased triangle inequality. We denote this problem as the Biased-TSP-Path. In this paper, we prove that the Biased-TSP-Path is approximable within a constant factor. Specifically, we design a 4-approximation algorithm by a suitable modification of the double-tree algorithm using effective shortcutting procedures.

A dual CNN–RNN for multiple people tracking

In this paper, we present a deep learning-based approach, namely a dual CNN–RNN for multiple people tracking. We follow tracking-by-detection paradigm by first training a CNN to measure the similarity of two detection boxes. To solve the data association (DA) problem, we build a graph with nodes as detections and edge costs that are the outputs of a CNN. The general minimum cost lifted multi-cut problem (LMP) and corresponding optimization algorithms are utilized to solve the DA problem. To tackle occlusion and ID-switch problems, an RNN network is proposed to predict the nonlinear motion of people. Moreover, we utilize target motion information to stitch tracklets and build long trajectories. The results of our experiments conducted on Multiple Object Tracking Benchmark 2016 (MOT2016) confirm the efficiency of the proposed algorithm.

A Flexible, Natural Formulation for the Network Design Problem with Vulnerability Constraints

Given a graph, a set of origin-destination (OD) pairs with communication requirements, and an integer k ≥ 2, the network design problem with vulnerability constraints (NDPVC) is to identify a subgraph with the minimum total edge costs such that, between each OD pair, there exist a hop-constrained primary path and a hop-constrained backup path after any k − 1 edges of the graph fail. Formulations exist for single-edge failures (i.e., k = 2). To solve the NDPVC for an arbitrary number of edge failures, we develop two natural formulations based on the notion of length-bounded cuts. We compare their strengths and flexibilities in solving the problem for k ≥ 3. We study different methods to separate infeasible solutions by computing length-bounded cuts of a given size. Experimental results show that, for single-edge failures, our formulation increases the number of solved benchmark instances from 61% (obtained within a two-hour limit by the best published algorithm) to more than 95%, thus increasing the number of solved instances by 1,065. Our formulation also accelerates the solution process for larger hop limits and efficiently solves the NDPVC for general k. We test our best algorithm for two to five simultaneous edge failures and investigate the impact of multiple failures on the network design.

Improved approximation algorithms for minimum power covering problems

Given an undirected graph with edge costs, the power of a node is the maximum cost of an edge incident to it, and the power of a graph is the sum of the powers of its nodes. Motivated by applications in wireless networks, we consider two network design problems under the power minimization criteria. In both problems we are given a graph G=(V,E) with edge costs and a set T⊆V of terminals. The goal is to find a minimum power edge subset F⊆E such that the graph H=(V,F) satisfies some prescribed requirements. In the Min-Power Edge-Cover problem, H should contain an edge incident to every terminal. Using the Iterative Randomized Rounding (IRR) method, we give an algorithm with expected approximation ratio 1.41; the ratio is reduced to 73/60<1.217 when T is an independent set in G. In the case of unit costs we also achieve ratio 73/60, and in addition give a simple efficient combinatorial algorithm with ratio 5/4. For all these NP-hard problems the previous best known ratio was 3/2. In the related Min-Power Terminal Backup problem, H should contain a path from every t∈T to some node in T∖{t}. We obtain ratio 3/2 for this NP-hard problem, improving the trivial ratio of 2.

Approximating constrained minimum cost input–output selection for generic arbitrary pole placement in structured systems

This paper deals with minimum cost constrained selection of inputs, outputs and feedback pattern in structured systems, referred to as optimal input–output and feedback co-design problem. The input–output set is constrained in the sense that the set of states that each input can influence and the set of states that each output can sense are pre-specified. Further, each input and each output are associated with a cost. The feedback pattern is unconstrained and the cost of a feedback edge is the sum of cost of the input and output associated with it. Our goal is to optimally select an input–output set and a feedback pattern such that the closed-loop system has no structurally fixed modes (SFMs). This problem is known to be NP-hard. In this paper, we show that the problem is inapproximable to factor (1−o(1))logn, where n denotes the number of states in the system. Then we present an approximation algorithm of time complexity O(n3) to approximate the problem. We prove that the proposed algorithm gives a (2logn)-approximate solution to the problem. Thus the algorithm given in this paper is an order optimal approximation algorithm to approximate the optimal input–output and feedback co-design problem.

Parameterized Analysis of the Online Priority and Node-Weighted Steiner Tree Problems

In this paper we study the online variant of two well-known Steiner tree problems. In the online setting, the input consists of a sequence of terminals; upon arrival of a terminal, the online algorithm must irrevocably buy a subset of edges and vertices of the graph so as to guarantee the connectivity of the currently revealed part of the input. More precisely, we first study the online node-weighted Steiner tree problem, in which both edges and vertices are weighted, and the objective is to minimize the total cost of edges and vertices in the solution. We then address the online priority Steiner tree problem, in which each edge and each request are associated with a priority value, which corresponds to their bandwidth support and requirement, respectively. Both problems have applications in the domain of multicast network communications and have been studied from the point of view of approximation algorithms. Motivated by the observation that competitive analysis gives very pessimistic and unsatisfactory results when the only relevant parameter is the number of terminals, we introduce an approach based on parameterized analysis of online algorithms. In particular, we base the analysis on additional parameters that help reveal the true complexity of the underlying problem, and allow a much finer classification of online algorithms based on their performance. More specifically, for the online node-weighted Steiner tree problem, we show a tight bound of Θ(max{min{α,k},log k}) on the competitive ratio, where α is the ratio of the maximum node weight to the minimum node weight and k is the number of terminals. For the online priority Steiner tree problem, we show corresponding tight bounds of ${\Theta }(b\log \frac {k}{b})$ , when k > b and Θ(k), when k ≤ b, where b is the number of priority levels and k is the number of terminals. Our main results apply to both deterministic and randomized algorithms, as well as to generalized versions of the problems (i.e., to Steiner forest variants).