ABSTRACT We show that the k-Vertex Cover problem in degree-3 graphs can be solved in O *(1.1616 k... more ABSTRACT We show that the k-Vertex Cover problem in degree-3 graphs can be solved in O *(1.1616 k ) time, which improves previous results of O *(1.1940 k ) by Chen, Kanj and Xia and O *(1.1864 k ) by Razgon. In this paper, we will present a new way to analyze algorithms for the problem. We use r=k-\frac25nr=k-\frac{2}{5}n to measure the size of the search tree, and then get a simple O(1.6651k-\frac25n0)O(1.6651^{k-\frac{2}{5}n_0})-time algorithm, where n 0 is the number of vertices with degree ≥ 2 in the graph. Combining this result with fast algorithms for the Maximum Independent Set problem in degree-3 graphs, we improve the upper bound for the k-Vertex Cover problem in degree-3 graphs.
ABSTRACT Given a tree with nonnegative edge cost and nonnegative vertex weight, and a number k ≥ ... more ABSTRACT Given a tree with nonnegative edge cost and nonnegative vertex weight, and a number k ≥ 0, we consider the following four cut problems: cutting vertices of weight at most or at least k from the tree by deleting some edges such that the remaining part of the graph is still a tree and the total cost of the edges being deleted is minimized or maximized. The MinMstCut problem (cut vertices of weight at most k and minimize the total cost of the edges being deleted) can be solved in linear time and space and the other three problems are NP-hard. In this paper, we design an O(ln/ε)-time O(l 2/ε + n)-space algorithm for MaxMstCut, and O(ln(1/ε + logn))-time O(l 2/ε + n)-space algorithms for MinLstCut and MaxLstCut, where n is the number of vertices in the tree, l the number of leaves, and ε> 0 the prescribed error bound.
ABSTRACT The paper presents an O * (1·716 n )-time polynomial-space algorithm for the traveling s... more ABSTRACT The paper presents an O * (1·716 n )-time polynomial-space algorithm for the traveling salesman problem in an n-vertex edge-weighted graph with maximum degree 4, which improves the previous results of the O * (1·890 n )-time polynomial-space algorithm by Eppstein and the O * (1·733 n )-time exponential-space algorithm by Gebauer.
ABSTRACT Given a graph G = (V,E), the edge dominating set problem is to find a minimum set M ⊆ E ... more ABSTRACT Given a graph G = (V,E), the edge dominating set problem is to find a minimum set M ⊆ E such that each edge in E – M has at least one common endpoint with an edge in M. The edge dominating set problem is an important graph problem and has been extensively studied. It is well known that the problem is NP-hard, even when the graph is restricted to a planar or bipartite graph with maximum degree 3. In this paper, we show that the edge dominating set problem in graphs with maximum degree 3 can be solved in O *(1.2721 n ) time and polynomial space, where n is the number of vertices in the graph. We also show that there is an O *(2.2306 k )-time polynomial-space algorithm to decide whether a graph with maximum degree 3 has an edge dominating set of size k or not. Above two results improve previously known results on exact and parameterized algorithms for this problem.
ABSTRACT We show that the k-Vertex Cover problem in degree-3 graphs can be solved in O *(1.1616 k... more ABSTRACT We show that the k-Vertex Cover problem in degree-3 graphs can be solved in O *(1.1616 k ) time, which improves previous results of O *(1.1940 k ) by Chen, Kanj and Xia and O *(1.1864 k ) by Razgon. In this paper, we will present a new way to analyze algorithms for the problem. We use r=k-\frac25nr=k-\frac{2}{5}n to measure the size of the search tree, and then get a simple O(1.6651k-\frac25n0)O(1.6651^{k-\frac{2}{5}n_0})-time algorithm, where n 0 is the number of vertices with degree ≥ 2 in the graph. Combining this result with fast algorithms for the Maximum Independent Set problem in degree-3 graphs, we improve the upper bound for the k-Vertex Cover problem in degree-3 graphs.
ABSTRACT Given a tree with nonnegative edge cost and nonnegative vertex weight, and a number k ≥ ... more ABSTRACT Given a tree with nonnegative edge cost and nonnegative vertex weight, and a number k ≥ 0, we consider the following four cut problems: cutting vertices of weight at most or at least k from the tree by deleting some edges such that the remaining part of the graph is still a tree and the total cost of the edges being deleted is minimized or maximized. The MinMstCut problem (cut vertices of weight at most k and minimize the total cost of the edges being deleted) can be solved in linear time and space and the other three problems are NP-hard. In this paper, we design an O(ln/ε)-time O(l 2/ε + n)-space algorithm for MaxMstCut, and O(ln(1/ε + logn))-time O(l 2/ε + n)-space algorithms for MinLstCut and MaxLstCut, where n is the number of vertices in the tree, l the number of leaves, and ε> 0 the prescribed error bound.
ABSTRACT The paper presents an O * (1·716 n )-time polynomial-space algorithm for the traveling s... more ABSTRACT The paper presents an O * (1·716 n )-time polynomial-space algorithm for the traveling salesman problem in an n-vertex edge-weighted graph with maximum degree 4, which improves the previous results of the O * (1·890 n )-time polynomial-space algorithm by Eppstein and the O * (1·733 n )-time exponential-space algorithm by Gebauer.
ABSTRACT Given a graph G = (V,E), the edge dominating set problem is to find a minimum set M ⊆ E ... more ABSTRACT Given a graph G = (V,E), the edge dominating set problem is to find a minimum set M ⊆ E such that each edge in E – M has at least one common endpoint with an edge in M. The edge dominating set problem is an important graph problem and has been extensively studied. It is well known that the problem is NP-hard, even when the graph is restricted to a planar or bipartite graph with maximum degree 3. In this paper, we show that the edge dominating set problem in graphs with maximum degree 3 can be solved in O *(1.2721 n ) time and polynomial space, where n is the number of vertices in the graph. We also show that there is an O *(2.2306 k )-time polynomial-space algorithm to decide whether a graph with maximum degree 3 has an edge dominating set of size k or not. Above two results improve previously known results on exact and parameterized algorithms for this problem.
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