We examine the problem of reporting all intersecting pairs in a set of c-oriented polygons, each having at most k edges (for some constant k). Polygons are called c-oriented if the edges of all polygons have a constant number of orientations. The problem arises in many applications such as VLSI design rule checking and architecture databases. We present an asymptotically optimal algorithm that reports all pairs in O(n log nt) time and O(n) space, where n is the number of polygons and t the number of intersecting pairs. Since the optimal algorithm may report an intersecting pair more than once (but at most a constant number of times), we also give a further algorithm which reports each pair exactly once but is not space-optimal, namely, requires O(n log n) space.

In this paper, we consider the net assignment problem in the logic emulation system. This problem is also known as the board-level-routing problem. There are field programmable logic arrays (FPGAs) and crossbars on an emulator board. Each FPGA is connected to each crossbar. Connection requests between FPGAs are called nets, and FPGAs are interconnected through crossbars. We are required to assign each net to the suitable crossbar. This problem is known to be NP-complete in general. A polynomial time algorithm is known for a certain restricted case, in which we treat only 2-terminal nets. In this paper we propose a new polynomial time algorithm for this case.

A nearly equitable edge-coloring of a multigraph is a coloring such that edges incident to each vertex are colored equitably in number. This problem was solved in O(kn2) time, where n and k are the numbers of the edges and the colors, respectively. The running time was improved to be O(n2/k + n|V|) later. We present a more efficient algorithm for this problem that runs in O(n2/k) time.

For a property π on graphs, the edge-contraction problem with respect to π is defined as a problem of finding a set of edges of minimum cardinality whose contraction results in a graph satisfying the property π. This paper gives a lower bound for the approximation ratio for the problem for any property π that is hereditary on contractions and determined by biconnected components.

Persistent data structures, introduced by Sarnak and Tarjan, have been found especially useful in designing geometric algorithms. In this paper, we present a persistent form of binary-binary search tree, and then apply this data structure to solve various geometric searching problems, such as, three dimensional ray-shooting, hidden surface removal, polygonal point enclosure searching and so on. In all applications, we are able to either improve existing bounds or establish new bounds.

For a graph G, a biclique edge partition SBP(G) is a collection of bicliques (complete bipartite subgraphs) Bi such that each edge of G is contained in exactly one Bi. The Minimum Biclique Edge Partition Problem (MBEPP) asks for SBP(G) with the minimum size. In this paper, we show that for arbitrary small ε>0, (6053/6052-ε)-approximation of MBEPP is NP-hard.

The minimum biclique cover problem is known to be NP-hard for general bipartite graphs. It can be solved in polynomial time for C4-free bipartite graphs, bipartite distance hereditary graphs and bipartite domino-free graphs. In this paper, we define the modified Galois lattice Gm(B) for a bipartite graph B and introduce the redundant parameter R(B). We show that R(B)=0 if and only if B is domino-free. Furthermore, for an input graph such that R(B)=1, we show that the minimum biclique cover problem can be solved in polynomial time.

This paper focuses on algorithms for an efficient scalar multiplication. It proposes two algorithms for computing points of the form 2kP in affine coordinates. One works for k=2, and the other works for an arbitrary natural number k. The efficiency of these algorithms is based on a trade-off between a field inversion and several field multiplications. Montgomery trick is used to implement this trade-off. Since a field inversion is usually more expensive than 10 field multiplications, the proposed algorithms are efficient in comparison with existing ones.

Recently much attention has been devoted to the problem of translating a set of geometrical objects in a given direction, one at a time, without allowing collisions between the objects. This paper studies the translation problem in three dimensions on a set of c-oriented faces", that is, the faces whose bounding edges have a constant number c of orientations. We solve the problem in O(N log2 NK) time and O(N log N) space, where N is the total number of edges of the faces and K is the number of edge intersections in the projection plane. As an intermediate step, we also solve a problem related to ray-shooting. The algorithm for translating c-oriented faces finds uses in computer graphic systems.

In VLSI and printed wiring board design, routing process usually consists of two stages: the global routing and the detailed routing. The routability checking is to decide whether the global wires can be transformed into the detailed ones or not. In this paper, we propose two graphs, the capacity checking graph and the initial flow graph, for efficient routability checking in planar layouts.

Karger, Motwani and Sudan presented a graph coloring algorithm based on semidefinite programming, which colors any k-colorable graph with maximum degree Δ using (Δ1-2/k) colors. This algorithm leads to an algorithm for k-colorable graph using (n 1-3/(k+1)) colors. This improved Wigderson's algorithm, which uses O(n1-1/(k-1)) colors, containing as a subroutine an algorithm using (Δ+1) colors for graphs with maximum degree Δ. It is easy to imagine that an algorithm which uses less colors in terms of Δ leads to an algorithm which uses less colors in terms of n. In this paper, we consider this influence assuming that we have an algorithm which uses (Δ 1-x/k) colors for 2

Maximum Satisfiability Problem (MAX SAT) is one of the most natural optimization problems. Since it is known to be NP-hard, approximation algorithms have been considered. The aim of this survey is to show recent developments of approximation algorithms for MAX SAT.