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1. Approximation algorithms for the vertex-weighted grade-of-service Steiner tree problem

   Ahmed, R. ; Efrat, A. ; Kobourov, S. ; Krieger, S. ; Spence, R. ( January 2019 , ArXiv.org)

   Given a graph G = (V, E) and a subset T ⊆ V of terminals, a Steiner tree of G is a tree that spans T. In the vertex-weighted Steiner tree (VST) problem, each vertex is assigned a non-negative weight, and the goal is to compute a minimum weight Steiner tree of G. Vertex-weighted problems have applications in network design and routing, where there are different costs for installing or maintaining facilities at different vertices. We study a natural generalization of the VST problem motivated by multi-level graph construction, the vertex-weighted grade-of-service Steiner tree problem (V-GSST), which can be stated as follows: given a graph G and terminals T, where each terminal v ∈ T requires a facility of a minimum grade of service R(v) ∈ {1, 2, . . . `}, compute a Steiner tree G0 by installing facilities on a subset of vertices, such that any two vertices requiring a certain grade of service are connected by a path in G 0 with the minimum grade of service or better. Facilities of higher grade are more costly than facilities of lower grade. Multi-level variants such as this one can be useful in network design problems where vertices may require facilities of varying priority. While similar problems have been studied in the edge-weighted case, they have not been studied as well in the more general vertex-weighted case. We first describe a simple heuristic for the V-GSST problem whose approximation ratio depends on `, the number of grades of service. We then generalize the greedy algorithm of [Klein & Ravi, 1995] to show that the V-GSST problem admits a (2 ln |T|)-approximation, where T is the set of terminals requiring some facility. This result is surprising, as it shows that the (seemingly harder) multi-grade problem can be approximated as well as the VST problem, and that the approximation ratio does not depend on the number of grades of service. Finally, we show that this problem is a special case of the directed Steiner tree problem and provide an integer linear programming (ILP) formulation for the V-GSST problem. 

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2. Slice-Aware Service Restoration with RecoveryTrucks for Optical Metro-Access Networks

   Sifat Ferdousi, Massimo Tornatore ( December 2019 , IEEE Globecom 2019)

   Next-generation optical metro-access networks are expected to support end-to-end virtual network slices for critical 5G services. However, disasters affecting physical infrastructures upon which network slices are mapped can cause significant disruption in these services. Operators can deploy recovery units or trucks to restore services based on slice requirements. In this study, we investigate the problem of slice-aware service restoration in metro-access networks with specialized recovery trucks to restore services after a disaster failure. We model the problem based on classical vehicle-routing problem to find optimal routes for recovery trucks to failure sites to provide temporary backup service until the network components are repaired. Our proposed slice-aware service-restoration approach is formulated as a mixed integer linear program with the objective to minimize penalty of service disruption across different network slices.We compare our slice-aware approach with a slice-unaware approach and show that our proposed approach can achieve significant reduction in service-disruption penalty 

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3. Multi-level Steiner trees

   Ahmed, R. ; Angelini, P. ; Sahneh, F. ; Efrat, A. ; Glickenstein, D. ; Gronemann, M. ; Heinsohn, N. ; Kobourov, S. ; Spence, R. ; Watkins, J. ; et al ( January 2018 , Symposium on Experimental Algorithms)

   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′⊆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 paper, we study a natural generalization, the multi-level Steiner tree (MLST) problem: given a nested sequence of terminals Tℓ⊂⋯⊂T1⊆V, compute nested trees Eℓ⊆⋯⊆E1⊆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 various integer linear programming (ILP) 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. 

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4. Multi-level Steiner Trees

   https://doi.org/10.1145/3368621  

   Ahmed, Reyan ; Angelini, Patrizio ; Sahneh, Faryad Darabi ; Efrat, Alon ; Glickenstein, David ; Gronemann, Martin ; Heinsohn, Niklas ; Kobourov, Stephen G. ; Spence, Richard ; Watkins, Joseph ; et al ( December 2019 , ACM Journal of Experimental Algorithmics)

   null (Ed.)

   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. 

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5. Online proximal gradient for learning graphs from streaming signals

   https://doi.org/10.23919/Eusipco47968.2020.9287550  

   Shafipour, Rasoul ; Mateos, Gonzalo ( January 2021 , 2020 28th European Signal Processing Conference (EUSIPCO))

   null (Ed.)

   We leverage proximal gradient iterations to develop an online graph learning algorithm from streaming network data. Our goal is to track the (possibly) time-varying network topology, and effect memory and computational savings by processing the data on-the-fly as they are acquired. The setup entails observations modeled as stationary graph signals generated by local diffusion dynamics on the unknown network. Moreover, we may have a priori information on the presence or absence of a few edges as in the link prediction problem. The stationarity assumption implies that the observations' covariance matrix and the so-called graph shift operator (GSO - a matrix encoding the graph topology) commute under mild requirements. This motivates formulating the topology inference task as an inverse problem, whereby one searches for a (e.g., sparse) GSO that is structurally admissible and approximately commutes with the observations' empirical covariance matrix. For streaming data said covariance can be updated recursively, and we show online proximal gradient iterations can be brought to bear to efficiently track the time-varying solution of the inverse problem with quantifiable guarantees. Specifically, we derive conditions under which the GSO recovery cost is strongly convex and use this property to prove that the online algorithm converges to within a neighborhood of the optimal time-varying batch solution. Preliminary numerical tests illustrate the effectiveness of the proposed graph learning approach in adapting to streaming information and tracking changes in the sought dynamic network. 

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