Parameterized Study of Steiner Tree on Unit Disk Graphs

We study the Steiner Tree problem on unit disk graphs. Given a n vertex unit disk graph G, a subset R? V(G) of t vertices and a positive integer k, the objective is to decide if there exists a tree T in G that spans over all vertices of R and uses at most k vertices from V? R. The vertices of R are referred to as terminals and the vertices of V(G)? R as Steiner vertices. First, we show that the problem is NP-hard. Next, we prove that the Steiner Tree problem on unit disk graphs can be solved in n^{O(?{t+k})} time. We also show that the Steiner Tree problem on unit disk graphs parameterized by k has an FPT algorithm with running time 2^{O(k)}n^{O(1)}. In fact, the algorithms are designed for a more general class of graphs, called clique-grid graphs [Fomin et al., 2019]. We mention that the algorithmic results can be made to work for Steiner Tree on disk graphs with bounded aspect ratio. Finally, we prove that Steiner Tree on disk graphs parameterized by k is W[1]-hard

Approximating Directed Steiner Problems via Tree Embedding

In the k-edge connected directed Steiner tree (k-DST) problem, we are given a directed graph G on n vertices with edge-costs, a root vertex r, a set of h terminals T and an integer k. The goal is to find a min-cost subgraph H of G that connects r to each terminal t by k edge-disjoint r,t-paths. This problem includes as special cases the well-known directed Steiner tree (DST) problem (the case k = 1) and the group Steiner tree (GST) problem. Despite having been studied and mentioned many times in literature, e.g., by Feldman et al. [SODA'09, JCSS'12], by Cheriyan et al. [SODA'12, TALG'14] and by Laekhanukit [SODA'14], there was no known non-trivial approximation algorithm for k-DST for k >= 2 even in the special case that an input graph is directed acyclic and has a constant number of layers. If an input graph is not acyclic, the complexity status of k-DST is not known even for a very strict special case that k= 2 and |T| = 2. In this paper, we make a progress toward developing a non-trivial approximation algorithm for k-DST. We present an O(D k^{D-1} log n)-approximation algorithm for k-DST on directed acyclic graphs (DAGs) with D layers, which can be extended to a special case of k-DST on "general graphs" when an instance has a D-shallow optimal solution, i.e., there exist k edge-disjoint r,t-paths, each of length at most D, for every terminal t. For the case k= 1 (DST), our algorithm yields an approximation ratio of O(D log h), thus implying an O(log^3 h)-approximation algorithm for DST that runs in quasi-polynomial-time (due to the height-reduction of Zelikovsky [Algorithmica'97]). Consequently, as our algorithm works for general graphs, we obtain an O(D k^{D-1} log n)-approximation algorithm for a D-shallow instance of the k-edge-connected directed Steiner subgraph problem, where we wish to connect every pair of terminals by k-edge-disjoint paths

Simplified Emanation Graphs: A Sparse Plane Spanner with Steiner Points

Emanation graphs of grade $k$, introduced by Hamedmohseni, Rahmati, and Mondal, are plane spanners made by shooting $2^{k+1}$ rays from each given point, where the shorter rays stop the longer ones upon collision. The collision points are the Steiner points of the spanner. We introduce a method of simplification for emanation graphs of grade $k=2$, which makes it a competent spanner for many possible use cases such as network visualization and geometric routing. In particular, the simplification reduces the number of Steiner points by half and also significantly decreases the total number of edges, without increasing the spanning ratio. Exact methods of simplification along with mathematical proofs on properties of the simplified graph is provided. We compare simplified emanation graphs against Shewchuk's constrained Delaunay triangulations on both synthetic and real-life datasets. Our experimental results reveal that the simplified emanation graphs outperform constrained Delaunay triangulations in common quality measures (e.g., edge count, angular resolution, average degree, total edge length) while maintain a comparable spanning ratio and Steiner point count.Comment: A preliminary and shorter version of this paper was accepted in SOFSEM 202

A Few Problems on the Steiner Distance and Crossing Number of Graphs

We provide a brief overview of the Steiner ratio problem in its original Euclidean context and briefly discuss the problem in other metric spaces. We then review literature in Steiner distance problems in general graphs as well as in trees. Given a connected graph G we examine the relationship between the Steiner k-diameter, sdiamk(G), and the Steiner k-radius, sradk(G). In 1990, Henning, Oellermann and Swart [Ars Combinatoria 12 13-19, (1990)] showed that for any connected graph G, sdiam3(G) ≤(8/5)srad3(G) and conjectured that for all k ≥ 2 and a connected graph G, sdiamk(G) ≤ (2(k+1))/(2k−1)sradk(G). The paper also included an incorrect proof that sdiam4(G) ≤ (10/7)srad4(G). We provide a correct proof that sdiam4(G) ≤ (10/7)srad4(G) and show that for k ≥ 5, sdiamk(G) ≤ (k+3)/(k+1)sradk(G). By construction, we also show that the latter of these bounds is tight for each k ≥ 5. We then examine the Steiner distance of large sets in hypercubes. In particular, we show that for k = O(2n/n), the Steiner k-diameter of the n-cube is k + Θ((2n)/√n) using a recent result of Griggs. This section is a joint work with Éva Czabarka and László Székely. Finally, we move to structural properties of graphs in the context of crossing numbers. For positive integers n and e, let κ(n, e) be the minimum number of crossings among all graphs with n vertices and at least e edges. Under the condition that n \u3c\u3c e \u3c\u3c n2, Pach, Spencer, and Tóth [Discrete and Computational Geometry 24 623-644, (2000)] showed that κ(n, e)(n2)/(e3) tends towards a positive constant (called the midrange crossing constant) as n → ∞. We extend their proof to show that the midrange crossing constant exists for graph classes that satisfy a certain set of graph properties. As a corollary, we show that the the midrange crossing constant exists for the family of bipartite graphs. This section is a joint work with Éva Czabarka, László Székely, and Zhiyu Wang

Multi-Level Steiner Trees

In the classical Steiner tree problem, one is given an undirected, connected graph G=(V,E) with non-negative edge costs and a set of terminals T subseteq V. The objective is to find a minimum-cost edge set E\u27 subseteq E that spans the terminals. The problem is APX-hard; the best known approximation algorithm has a ratio of rho = ln(4)+epsilon < 1.39. In this paper, we study a natural generalization, the multi-level Steiner tree (MLST) problem: given a nested sequence of terminals T_1 subset ... subset T_k subseteq V, compute nested edge sets E_1 subseteq ... subseteq E_k subseteq E that span the corresponding terminal sets with minimum total cost. The MLST problem and variants thereof have been studied under names such as Quality-of-Service Multicast tree, Grade-of-Service Steiner tree, and Multi-Tier tree. Several approximation results are known. We first present two natural heuristics with approximation factor O(k). Based on these, we introduce a composite algorithm that requires 2^k Steiner tree computations. We determine its approximation ratio by solving a linear program. We then present a method that guarantees the same approximation ratio and needs at most 2k Steiner tree computations. We compare five algorithms experimentally on several classes of graphs using four types of graph generators. We also implemented an integer linear program for MLST to provide ground truth. Our combined algorithm outperforms the others both in theory and in practice when the number of levels is small (k <= 22), which works well for applications such as designing multi-level infrastructure or network visualization

On the Integrality Gap of Directed Steiner Tree Problem

In the Directed Steiner Tree problem, we are given a directed graph G = (V,E) with edge costs, a root vertex r ∈ V, and a terminal set X ⊆ V . The goal is to find the cheapest subset of edges that contains an r-t path for every terminal t ∈ X. The only known polylogarithmic approximations for Directed Steiner Tree run in quasi-polynomial time and the best polynomial time approximations only achieve a guarantee of O(|X|^ε) for any constant ε > 0. Furthermore, the integrality gap of a natural LP relaxation can be as bad as Ω(√|X|).  We demonstrate that l rounds of the Sherali-Adams hierarchy suffice to reduce the integrality gap of a natural LP relaxation for Directed Steiner Tree in l-layered graphs from Ω( k) to O(l · log k) where k is the number of terminals. This is an improvement over Rothvoss’ result that 2l rounds of the considerably stronger Lasserre SDP hierarchy reduce the integrality gap of a similar formulation to O(l · log k). We also observe that Directed Steiner Tree instances with 3 layers of edges have only an O(logk) integrality gap bound in the standard LP relaxation, complementing the fact that the gap can be as large as Ω(√k) in graphs with 4 layers. Finally, we consider quasi-bipartite instances of Directed Steiner Tree meaning no edge in E connects two Steiner nodes V − (X ∪ {r}). By a simple reduction from Set Cover, it is still NP-hard to approximate quasi-bipartite instances within a ratio better than O(log|X|). We present a polynomial-time O(log |X|)-approximation for quasi-bipartite instances of Directed Steiner Tree. Our approach also bounds the integrality gap of the natural LP relaxation by the same quantity. A novel feature of our algorithm is that it is based on the primal-dual framework, which typically does not result in good approximations for network design problems in directed graphs

Approximability of the Minimum Steiner Cycle Problem

In this paper, we consider variants of a new problem that we call minimum Steiner cycle problem (SCP). The problem is defined as follows. Given is a weighted complete graph and a set of terminal vertices. In the SCP problem, we are looking for a minimum-cost cycle that passes through every terminal exactly once and through every other vertex of the graph at most once. We show that, if P&lt;&gt;NP, there is no approximation algorithm for SCP on directed graphs with an approximation ratio polynomial in the input size. Moreover, this result holds even in the case when the number of terminals is 4. On the contrary, we show that SCP on undirected graphs with constant number of terminals and edge costs satisfying the beta-relaxed triangle inequality is approximable with the ratio beta^2+beta. When the number of terminals k is a part of the input, the problem is obviously a generalization of TSP. For the metric case, we present a 3/2- and a 2/3 log_2 k-approximation algorithm for undirected and directed graphs G=(V,E), respectively. For the case with the beta-relaxed triangle inequality, we present a (beta^2+beta)-approximation algorithm