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1. A Constant Factor Approximation for Navigating Through Connected Obstacles in the Plane

   Kumar, Neeraj; Lokshtanov, Daniel; Saurabh, Saket; Suri, Subhash (January 2021, Proceedings of the annual ACM-SIAM Symposium on Discrete Algorithms)

   null (Ed.)

   Given two points s and t in the plane and a set of obstacles defined by closed curves, what is the minimum number of obstacles touched by a path connecting s and t? This is a fundamental and well-studied problem arising naturally in computational geometry, graph theory (under the names Min-Color Path and Minimum Label Path), wireless sensor networks (Barrier Resilience) and motion planning (Minimum Constraint Removal). It remains NP-hard even for very simple-shaped obstacles such as unit-length line segments. In this paper we give the first constant factor approximation algorithm for this problem, resolving an open problem of [Chan and Kirkpatrick, TCS, 2014] and [Bandyapadhyay et al., CGTA, 2020]. We also obtain a constant factor approximation for the Minimum Color Prize Collecting Steiner Forest where the goal is to connect multiple request pairs (s1, t1), . . . , (sk, tk) while minimizing the number of obstacles touched by any (si, ti) path plus a fixed cost of wi for each pair (si, ti) left disconnected. This generalizes the classic Steiner Forest and Prize-Collecting Steiner Forest problems on planar graphs, for which intricate PTASes are known. In contrast, no PTAS is possible for Min-Color Path even on planar graphs since the problem is known to be APX- hard [Eiben and Kanj, TALG, 2020]. Additionally, we show that generalizations of the problem to disconnected obstacles in the plane or connected obstacles in higher dimensions are strongly inapproximable assuming some well-known hardness conjectures. 

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2. Multi-Priority Graph Sparsification

   https://doi.org/10.1007/978-3-031-34347-6_1  

   Ahmed, R.; Hamm, K.; Kobourov, S.; Jebelli, M.; Sahneh, F.; Spence, R (April 2023, 34th Intl. Workshop on Combinatorial Algorithms (IWOCA))

   A sparsification of a given graph G is a sparser graph (typically a subgraph) which aims to approximate or preserve some property of G. Examples of sparsifications include but are not limited to spanning trees, Steiner trees, spanners, emulators, and distance preservers. Each vertex has the same priority in all of these problems. However, real-world graphs typically assign different “priorities” or “levels” to different vertices, in which higher-priority vertices require higher-quality connectivity between them. Multi-priority variants of the Steiner tree problem have been studied in prior literature but this generalization is much less studied for other sparsification problems. In this paper, we define a generalized multi-priority problem and present a rounding-up approach that can be used for a variety of graph sparsifications. Our analysis provides a systematic way to compute approximate solutions to multi-priority variants of a wide range of graph sparsification problems given access to a single-priority subroutine. 

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3. Quasi-Polynomial Algorithms for Submodular Tree Orienteering and Other Directed Network Design Problems

   https://doi.org/10.1137/1.9781611975994.63  

   Ghuge, Rohan; Nagarajan, Viswanath (January 2020, Proceedings of the Annual ACMSIAM Symposium on Discrete Algorithms)

   We consider the following general network design problem on directed graphs. The input is an asymmetric metric (V, c), root r in V, monotone submodular function f and budget B. The goal is to find an r-rooted arborescence T of cost at most B that maximizes f(T). Our main result is a very simple quasi-polynomial time -approximation algorithm for this problem, where k ≤ |V| is the number of vertices in an optimal solution. To the best of our knowledge, this is the first non-trivial approximation ratio for this problem. As a consequence we obtain an O(log^2 k / loglog k) approximation algorithm for directed (polymatroid) Steiner tree in quasi-polynomial time. We also extend our main result to a setting with additional length bounds at vertices, which leads to improved approximation algorithms for the single-source buy-at-bulk and priority Steiner tree problems. For the usual directed Steiner tree problem, our result matches the best previous approximation ratio, but improves significantly on the running time. For polymatroid Steiner tree and single-source buy-at-bulk, our result improves prior approximation ratios by a logarithmic factor. For directed priority Steiner tree, our result seems to be the first non-trivial approximation ratio. Under certain complexity assumptions, our approximation ratios are best possible (up to constant factors). 

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4. 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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5. On Steiner Trees of the regular simplex

   https://doi.org/10.20382/jocg.v16i1a1  

   Fleischmann, Henry; Gamboa_Quintero, Guillermo; Karthik_C, S; Matějka, Josef; Petr, Jakub (January 2025, Journal of computational geometry)

   In the Euclidean Steiner Tree problem, we are given as input a set of points (called terminals) in the $$\ell_2$$-metric space and the goal is to find the minimum-cost tree connecting them. Additional points (called Steiner points) from the space can be introduced as nodes in the solution.  The seminal works of Arora [JACM'98] and Mitchell [SICOMP'99] provide a Polynomial Time Approximation Scheme (PTAS) for solving the Euclidean Steiner Tree problem in fixed dimensions. However, the problem remains poorly understood in higher dimensions (such as when the dimension is logarithmic in the number of terminals) and ruling out a PTAS for the problem in high dimensions is a notoriously long standing open problem (for example, see Trevisan [SICOMP'00]). Moreover, the explicit construction of optimal Steiner trees remains unknown for almost all well-studied high-dimensional point configurations. Furthermore, a vast majority the state-of-the-art structural results on (high-dimensional) Euclidean Steiner trees were established in the 1960s, with no noteworthy update in over half a century. In this paper, we revisit high-dimensional Euclidean Steiner trees, proving new structural results. We also establish a link between the computational hardness of the Euclidean Steiner Tree problem and understanding the optimal Steiner trees of regular simplices (and simplicial complexes), proposing several conjectures and showing that some of them suffice to resolve the status of the inapproximability of the Euclidean Steiner Tree problem. Motivated by this connection, we investigate optimal Steiner trees of regular simplices, proving new structural properties of their optimal Steiner trees, revisiting an old conjecture of Smith [Algorithmica'92] about their optimal topology, and providing the first explicit, general construction of candidate optimal Steiner trees for that topology. 

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