Constructing disjoint Steiner trees in Sierpiński graphs

This dissertation examines a number of geometric interconnection, partitioning, and placement problems arising in the field of VLSI physical design automation. In particular, many of the results concern the geometric Steiner tree problem: given a set of terminals in the plane, find a minimum-length interconnection of those terminals according to some geometric distance metric. Two new algorithms are introduced that compute optimal rectilinear Steiner trees. The first algorithm is a dynamic programming algorithm based on decomposing a rectilinear Steiner tree into full trees. A full tree is a Steiner tree in which every terminal is a leaf. Its time complexity is $O(n3\sp{n}$), where n is the number of terminals. The second algorithm modifies the first by the use of full-set screening, which is a process by which some candidate full trees are eliminated from consideration. Its time complexity is approximately 0($n\sp22.62\sp{n}$). We demonstrate that for instances small enough to reasonably solve in practice, these algorithms are faster than has been proven for any previous algorithm. Empirical evidence is also given indicating that both algorithms are faster in practice than several other popular algorithms for computing optimal rectilinear Steiner trees. A new theorem is proven that allows efficient computation of rectilinear Steiner trees in the presence of obstacles. This theorem leads to the first algorithms that compute optimal obstacle-avoiding rectilinear Steiner trees in time corresponding to the size of the instance (the number of terminals and obstacle border segments) rather than the size of the routing area. Two types of algorithms are presented. The first computes optimal rectilinear Steiner trees in the presence of obstacles for small instances. The second computes heuristic solutions to arbitrary instances. The power-p Steiner tree problem is introduced, in which the weight of an edge is its geometric length raised to the p power. A special case of the power-p Steiner tree problem is also considered: the bottleneck Steiner tree problem, which is to find a geometric Steiner tree with the length of the longest edge minimized. Algorithms are described for computing optimal optimal Euclidean power-2 Steiner trees, rectilinear bottleneck Steiner trees, and rectilinear Steiner trees that minimize a combination of bottleneck weight and total length. The power-p Steiner tree problem for p $>$ 2 is also considered, and evidence given that the problem is essentially unsolvable for large p. In addition, bounds are given on the power-p Steiner ratio, which measures the quality of a minimum spanning tree as an approximation of a power-p Steiner tree. The exact value of the bottleneck Steiner ratio is also derived, and a fully polynomial-time approximation scheme is given for computing a bottleneck Steiner tree with a given topology in any distance metric. Finally, a system called M scONDRIAN is presented that performs simultaneous placement and global routing in field-programmable gate arrays (FPGAs). M scONDRIAN is a recursive geometric partitioning strategy using optimal rectilinear Steiner trees of terminals that lie in a small grid. Experimental results show that M scONDRIAN produces results significantly superior to those of previous FPGA layout algorithms. In addition, we present a modified version of M scONDRIAN that computes performance-driven placement and routing solutions, in which more accurate estimates of electrical delay are optimized. (Abstract shortened by UMI.)

We investigate the complexity of finding a minimum Steiner tree in new subclasses of split graphs namely tree-convex split graphs and circular-convex split graphs. It is known that the Steiner tree problem (STREE) is NP-complete on split graphs [1]. To strengthen this result, we introduce convex ordering on one of the partitions (clique or independent set), and prove that STREE is polynomial-time solvable for tree-convex split graphs with convexity on clique (K), whereas STREE is NP-complete on tree-convex split graphs with convexity on independent set (I). Further, we show that STREE is polynomial-time solvable for path (triad)-convex split graphs with convexity on I, and circular-convex split graphs. Finally, we show that STREE can be used as a framework for the dominating set problem in split graphs, and hence the complexity of STREE and the dominating set problem is the same for all these graph classes. KeywordsSteiner treeTree-convexPath-convexTriad-convexCircular-convexDomination

We study the Steiner Tree problem on unit disk graphs. Given a n vertex unit disk graph G, a subset \(R\subseteq V(G)\) 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 \(V\setminus R\). The vertices of R are referred to as terminals and the vertices of \(V(G)\setminus R\) 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 \(n^{O(\sqrt{t+k})}\) 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 (Discret. Comput. Geometry 62(4):879–911, 2019). We mention that the algorithmic results can be made to work for Steiner Tree problem on disk graphs with bounded aspect ratio. Finally, we prove that Steiner Tree problem on disk graphs parameterized by k, is W[1]-hard.

Computing a tree having a small vertex cover

The Steiner tree problem (STP) aims to determine some Steiner nodes such that the minimum spanning tree over these Steiner nodes and a given set of special nodes has the minimum weight, which is NP-hard. STP includes several important cases. The Steiner tree problem in graphs (GSTP) is one of them. Many heuristics have been proposed for STP, and some of them have proved to be performance guarantee approximation algorithms for this problem. Since evolutionary algorithms (EAs) are general and popular randomized heuristics, it is significant to investigate the performance of EAs for STP. Several empirical investigations have shown that EAs are efficient for STP. However, up to now, there is no theoretical work on the performance of EAs for STP. In this article, we reveal that the (1+1) EA achieves 3/2-approximation ratio for STP in a special class of quasi-bipartite graphs in expected runtime [Formula: see text], where [Formula: see text], [Formula: see text], and [Formula: see text] are, respectively, the number of Steiner nodes, the number of special nodes, and the largest weight among all edges in the input graph. We also show that the (1+1) EA is better than two other heuristics on two GSTP instances, and the (1+1) EA may be inefficient on a constructed GSTP instance.

Determining the integrality gap of the bidirected cut relaxation for the metric Steiner tree problem, and exploiting it algorithmically, is a long-standing open problem. We use geometry to define an LP whose dual is equivalent to this relaxation. This opens up the possibility of using the primal-dual schema in a geometric setting for designing an algorithm for this problem. Using this approach, we obtain a 4/3 factor algorithm and integrality gap bound for the case of quasi-bipartite graphs; the previous best integrality gap upper bound being 3/2 (Rajagopalan and Vazirani in On the bidirected cut relaxation for the metric Steiner tree problem, 1999). We also obtain a factor $${\sqrt{2}}$$ strongly polynomial algorithm for this class of graphs. A key difficulty experienced by researchers in working with the bidirected cut relaxation was that any reasonable dual growth procedure produces extremely unwieldy dual solutions. A new algorithmic idea helps finesse this difficulty—that of reducing the cost of certain edges and constructing the dual in this altered instance—and this idea can be extracted into a new technique for running the primal-dual schema in the setting of approximation algorithms.

Steiner trees for hereditary graph classes: A treewidth perspective

We consider the classical problems (Edge) Steiner Tree and Vertex Steiner Tree after restricting the input to some class of graphs characterized by a small set of forbidden induced subgraphs. We show a dichotomy for the former problem restricted to \((H_1,H_2)\)-free graphs and a dichotomy for the latter problem restricted to H-free graphs. We find that there exists an infinite family of graphs H such that Vertex Steiner Tree is polynomial-time solvable for H-free graphs, whereas there exist only two graphs H for which this holds for Edge Steiner Tree. We also find that Edge Steiner Tree is polynomial-time solvable for \((H_1,H_2)\)-free graphs if and only if the treewidth of the class of \((H_1,H_2)\)-free graphs is bounded (subject to \(\mathsf{P}\ne \mathsf{NP}\)). To obtain the latter result, we determine all pairs \((H_1,H_2)\) for which the class of \((H_1,H_2)\)-free graphs has bounded treewidth.

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' 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.

The Steiner tree problem has been extensively studied in the literature because of its various applications (e.g., network design, circuit layout and database query answering). The main result in the paper states that an instance of this problem on a graph G is polynomially reducible to an instance of the same problem on an induced subgraph of G whenever G contains a homogeneous set of nodes (i.e., a set of two or more nodes such that every reamining node in G is adjacent either to all or to none of its nodes). This result allows, in particular, to solve in polynomial time the Steiner tree problem on a new class of graphs, called homogeneous graphs, defined in terms of homogeneous sets, that properly contains some classes of graphs on which the problem is polynomially solvable. A polynomial algorithm to recognize homogeneous graphs is also given.

The belief propagation approximation, or cavity method, has been recently applied to several combinatorial optimization problems in its zero-temperature implementation, the max-sum algorithm. In particular, recent developments to solve the edge-disjoint paths problem and the prize-collecting Steiner tree problem on graphs have shown remarkable results for several classes of graphs and for benchmark instances. Here we propose a generalization of these techniques for two variants of the Steiner trees packing problem where multiple ‘interacting’ trees have to be sought within a given graph. Depending on the interaction among trees we distinguish the vertex-disjoint Steiner trees problem, where trees cannot share nodes, from the edge-disjoint Steiner trees problem, where edges cannot be shared by trees but nodes can be members of multiple trees. Several practical problems of huge interest in network design can be mapped into these two variants, for instance, the physical design of very large scale integration (VLSI) chips.The formalism described here relies on two components edge-variables that allows us to formulate a massage-passing algorithm for the V-DStP and two algorithms for the E-DStP differing in the scaling of the computational time with respect to some relevant parameters. We will show that through one of the two formalisms used for the edge-disjoint variant it is possible to map the max-sum update equations into a weighted maximum matching problem over proper bipartite graphs. We developed a heuristic procedure based on the max-sum equations that shows excellent performance in synthetic networks (in particular outperforming standard multi-step greedy procedures by large margins) and on large benchmark instances of VLSI for which the optimal solution is known, on which the algorithm found the optimum in two cases and the gap to optimality was never larger than 4%.

We present some fundamental structural properties for minimum length networks (known as Steiner minimum trees) interconnecting a given set of points in an environment in which edge segments are restricted to λ uniformly oriented directions. We show that the edge segments of any full component of such a tree contain a total of at most four directions if λ is not a multiple of 3, or six directions if λ is a multiple of 3. This result allows us to develop useful canonical forms for these full components. The structural properties of these Steiner minimum trees are then used to resolve an important open problem in the area: does there exist a polynomial time algorithm for constructing a Steiner minimum tree if the topology of the tree is known? We obtain a simple linear time algorithm for constructing a Steiner minimum tree for any given set of points and a given Steiner topology.

Steiner's problem in double trees

Chordal graphs are the intersection graphs of subtrees of a tree, while interval graphs of subpaths of a path. Undirected path graphs, directed path graphs and rooted directed path graphs are intermediate graph classes, defined, respectively, as the intersection graphs of paths of a tree, of directed paths of an oriented tree, and of directed paths of an out branching. All of these path graphs have vertex leafage 2. Dominating Set, Connected Dominating Set, and Steiner tree problems are ‐hard parameterized by the size of the solution on chordal graphs, ‐complete on undirected path graphs, and polynomial‐time solvable on rooted directed path graphs, and hence also on interval graphs. We further investigate the (parameterized) complexity of all these problems when constrained to chordal graphs, taking the vertex leafage and the aforementioned classes into consideration. We prove that Dominating Set, Connected Dominating Set, and Steiner tree are on chordal graphs when parameterized by the size of the solution plus the vertex leafage, and that Weighted Connected Dominating Set is polynomial‐time solvable on strongly chordal graphs. We also introduce a new subclass of undirected path graphs, which we call in–out rooted directed path graphs, as the intersection graphs of directed paths of an in–out branching. We prove that Dominating Set, Connected Dominating Set, and Steiner tree are solvable in polynomial time on this class, generalizing the polynomiality for rooted directed path graphs proved by Booth and Johnson (SIAM J. Comput. 11 (1982), 191‐199.) and by White et al. (Networks 15 (1985), 109‐124.).

The Geometric Steiner Minimum Tree problem (GSMT) is to connect at minimum cost n given points (called terminals) in d‐dimensional Euclidean space. We generalize a GSMT approximation partitioning algorithm by Komlós and Shing (KS) and analyze its performance under more relaxed conditions. These generalizations have practical applications for multilayer VLSI routing. Whereas KS assumed d = 2, our algorithm works for any dimension d ⩾ 2. Moreover, whereas their analysis assumed the rectilinear norm and a uniform distribution of input points, our analysis holds for any norm on Rd and whenever the terminals are any independent identically distributed random variables taking on values in any bounded subset of Rd. Both algorithms depend on a parameter t through which the user can trade off time for solution quality. We evaluate our algorithm in terms of its performance ratio–the ratio of the cost of the Steiner tree computed by the algorithm divided by the cost of a Steiner minimum tree. Applying a probability theorem on subadditive Euclidean functionals by Steele, we prove the following: Under the aforementioned distribution of inputs, the limit as n → ∞ of the supremum of the performance ratio of our algorithm is 1 + Ot−1/d(d‐1), almost surely. This result generalizes the corresponding 1 + O(t−1/2) bound proven by KS. Along the way, we prove a useful combinatorial lemma about d‐dimensional rectangle slicings. We prove that the worst‐case time and space complexity of our algorithm is θ(n lg (n/t) + TMST(v, v) + nTSMT(t)/t) and θ(SMST(v, v) + SSMT(t)), respectively, where v ⩽ n + 2d+1 (n/t) σ(t) is the number of vertices in the resulting Steiner tree. Here, TSMT(t) and SSMT(t) are the time and space required to solve exactly any GSMT problem of size less than t; TMST(n, m) and SMST(n, m) are the time and space required to find a minimum spanning tree of a graph with n nodes and m edges; and σ(t) is the maximum number of Steiner points for any Steiner minimum tree with t terminals. For example, for R2 and the rectilinear norm, the time is O(n lg(n/t) + n lg*n + nTSMT(t)/t) and the space is O(n lg *n + SSMT(t)). © 1994 by John Wiley &amp; Sons, Inc.