Nearly ETH-Tight Algorithms for Planar Steiner Tree with Terminals on Few Faces

The Planar Steiner Tree problem is one of the most fundamental NP-complete problems as it models many network design problems. Recall that an instance of this problem consists of a graph with edge weights, and a subset of vertices (often called terminals); the goal is to find a subtree of the graph of minimum total weight that connects all terminals. A seminal paper by Erickson et al. [Math. Oper. Res., 1987] considers instances where the underlying graph is planar and all terminals can be covered by the boundary of $k$ faces. Erickson et al. show that the problem can be solved by an algorithm using $n^{O(k)}$ time and $n^{O(k)}$ space, where $n$ denotes the number of vertices of the input graph. In the past 30 years there has been no significant improvement of this algorithm, despite several efforts. In this work, we give an algorithm for Planar Steiner Tree with running time $2^{O(k)} n^{O(\sqrt{k})}$ using only polynomial space. Furthermore, we show that the running time of our algorithm is almost tight: we prove that there is no $f(k)n^{o(\sqrt{k})}$ algorithm for Planar Steiner Tree for any computable function $f$, unless the Exponential Time Hypothesis fails.Comment: 32 pages, 8 figures, accepted at SODA 201

Nearly ETH-tight algorithms for Planar Steiner Tree with Terminals on Few Faces

The Steiner Tree problem is one of the most fundamental NP-complete problems as it models many network design problems. Recall that an instance of this problem consists of a graph with edge weights, and a subset of vertices (often called terminals); the goal is to find a subtree of the graph of minimum total weight that connects all terminals. A seminal paper by Erickson et al. [Math. Oper. Res., 1987] considers instances where the underlying graph is planar and all terminals can be covered by the boundary of k faces. Erickson et al. show that the problem can be solved by an algorithm using nO(k) time and nO(k) space, where n denotes the number of vertices of the input graph. In the past 30 years there has been no significant improvement of this algorithm, despite several efforts. In this work, we give an algorithm for Planar Steiner Tree with running time using only polynomial space. Furthermore, we show the running time of our algorithm is almost tight: we prove that there is no algorithm for Planar Steiner Tree for any computable function f, unless the Exponential Time Hypothesis fails. Read More: https://epubs.siam.org/doi/10.1137/1.9781611975482.6

Nearly ETH-tight algorithms for planar Steiner Tree with terminals on few faces

\u3cp\u3eThe Steiner Tree problem is one of the most fundamental NP-complete problems as it models many network design problems. Recall that an instance of this problem consists of a graph with edge weights, and a subset of vertices (often called terminals); the goal is to find a subtree of the graph of minimum total weight that connects all terminals. A seminal paper by Erickson et al. [Math. Oper. Res., 1987] considers instances where the underlying graph is planar and all terminals can be covered by the boundary of k faces. Erickson et al. show that the problem can be solved by an algorithm using n\u3csup\u3eO\u3c/sup\u3e(k\u3csup\u3e)\u3c/sup\u3e time and n\u3csup\u3eO\u3c/sup\u3e(k\u3csup\u3e)\u3c/sup\u3e space, where n denotes the number of vertices of the input graph. In the past 30 years there has been no significant improvement of this algorithm, despite several efforts. In this work, we give an algorithm for Planar Steiner Tree with running time 2\u3csup\u3eO\u3c/sup\u3e(k\u3csup\u3e)\u3c/sup\u3en\u3csup\u3eO\u3c/sup\u3e(k\u3csup\u3e)\u3c/sup\u3e using only polynomial space. Furthermore, we show the running time of our algo-rithm is almost tight: we prove that there is no f(k)n\u3csup\u3eo\u3c/sup\u3e(k) algorithm for Planar Steiner Tree for any computable function f, unless the Exponential Time Hypothesis fails.\u3c/p\u3

Nearly ETH-tight algorithms for Planar Steiner Tree with Terminals on Few Faces

The Steiner Tree problem is one of the most fundamental NP-complete problems as it models many network design problems. Recall that an instance of this problem consists of a graph with edge weights, and a subset of vertices (often called terminals); the goal is to find a subtree of the graph of minimum total weight that connects all terminals. A seminal paper by Erickson et al. [Math. Oper. Res., 1987] considers instances where the underlying graph is planar and all terminals can be covered by the boundary of k faces. Erickson et al. show that the problem can be solved by an algorithm using nO(k) time and nO(k) space, where n denotes the number of vertices of the input graph. In the past 30 years there has been no significant improvement of this algorithm, despite several efforts. In this work, we give an algorithm for Planar Steiner Tree with running time using only polynomial space. Furthermore, we show the running time of our algorithm is almost tight: we prove that there is no algorithm for Planar Steiner Tree for any computable function f, unless the Exponential Time Hypothesis fails. Read More: https://epubs.siam.org/doi/10.1137/1.9781611975482.6

Nearly ETH-tight algorithms for planar Steiner Tree with terminals on few faces

The Steiner Tree problem is one of the most fundamental NP-complete problems as it models many network design problems. Recall that an instance of this problem consists of a graph with edge weights, and a subset of vertices (often called terminals); the goal is to find a subtree of the graph of minimum total weight that connects all terminals. A seminal paper by Erickson et al. [Math. Oper. Res., 1987] considers instances where the underlying graph is planar and all terminals can be covered by the boundary of k faces. Erickson et al. show that the problem can be solved by an algorithm using nO(k) time and nO(k) space, where n denotes the number of vertices of the input graph. In the past 30 years there has been no significant improvement of this algorithm, despite several efforts. In this work, we give an algorithm for Planar Steiner Tree with running time 2O(k)nO(k) using only polynomial space. Furthermore, we show the running time of our algo-rithm is almost tight: we prove that there is no f(k)no(k) algorithm for Planar Steiner Tree for any computable function f, unless the Exponential Time Hypothesis fails