Given a vertex-weighted undirected graph, to find the vertex cover of minimum weight is called minimum weight vertex cover problem (MWVCP). It is known as an NP-hard problem. In this paper, a fast heuristic for MWVCP is proposed. Our algorithm is based on a simple algorithm called “list-heuristic.” Experimenal results show that our algorithm calculates better solutions in shorter time than approximation algorithms for MWVCP.

Abstract Let S be a family of arcs on a circle. A graph G = (V,E) is called a circular‐arc overlap graph for S if (i) there is a one‐to‐one correspondence between V and S and (ii) two vertices in V are adjacent if and only if the corresponding arcs in S intersect but neither one of them contains the other. In this article, we present two polynomial time algorithms on circular‐arc overlap graphs. Given a circular‐arc overlap graph in the form of a family of n arcs, the first algorithm obtains a maximum independent set in O(n2) time and the second one finds a maximum clique in O(n5) time.

Abstract Let F = {I1, I2,…,In} be a finite family of closed intervals on the real line. Two intervals Ij and Ik in F are said to overlap each other if they intersect but neither one of them contains the other. A graph G = (V, E) is called an overlap graph for F if there is a one‐to‐one correspondence between V and F such that two vertices in V are adjacent to each other if and only if the corresponding intervals in F overlap each other. In this paper, we present two efficient algorithms for finding maximum cliques of an overlap graph when it is given in the form of a family of n intervals. The first algorithm finds a maximum clique in O (n. log n + Min {m, n‐ ω}) time, where m is the number of edges and ω is the size of a maximum clique, respectively, of the graph. The second algorithm generates all maximum cliques in O (n ‐ log n + m + γ) time, where γ is the total sum of their sizes.

The problem of embedding a graph in the plane with the minimum number of edge crossings arises in some circuit layout problems. It has been known to be NP-hard in general. Recently, in the area of book embedding, this problem was shown to be NP-hard even when the vertices are placed on a straight line l. The authors show that the problem remains NP-hard even if, in addition to these constraints, the positions of the vertices on l are predetermined.>

The authors analyze the computational complexity of the constrained via minimization problem. Given an already routed circuit, the problem is to find a minimum cardinality set of vias for which a valid layer assignment exists using only two layers. The authors first prove that the corresponding decision problem is NP-complete. They then show that it remains NP-complete even if one or more of the following restrictions are imposed: (1) the input layout must conform to a rectangular grid; (2) vias are restricted to lie at junctions which existed in the input layout; and (3) the maximum junction degree is limited to six or more.>

A new algorithm is presented for finding a maximum independent set of a circular-arc graph. When the graph is given in the form of a family of n arcs, our algorithm requires only $O(n \cdot \log n)$ time and $O(n)$ space. Furthermore, if the endpoints of the arcs are already sorted, it runs in $O(n)$ time. This algorithm is time- and space-optimal to within a constant factor.

This paper describes a graph theoretic algorithm which, given a particular layout, finds a layer assignment that requires the minimum number of vias. The time complexity of the algorithm is Ο(n3) where n is the number of routing segments in the given layout. Unlike previous algorithms, this algorithm does not require the layout to be grid based and places no constraints on the location of vias or the number of wires that may be joined at a single junction. The algorithm yields globally optimum results when the maximum junction degree is limited to three and has been fully implemented.