Minimum-size Set Research Articles

Perfect edge domination and efficient edge domination in graphs

Let G=( V, E) be a finite and undirected graph without loops and multiple edges. An edge is said to dominate itself and any edge adjacent to it. A subset D of E is called a perfect edge dominating set if every edge of E⧹ D is dominated by exactly one edge in D and an efficient edge dominating set if every edge of E is dominated by exactly one edge in D. The perfect (efficient) edge domination problem is to find a perfect (efficient) edge dominating set of minimum size in G. Suppose that each edge e is associated with a real number w( e) as its weight. Then, the weighted perfect (efficient) edge domination problem is to calculate a perfect (efficient) edge dominating set D such that the weight w( D) of D is minimum, where w( D)=∑ e∈ D w( e). In this paper, we show that the perfect (efficient) edge domination problem is NP-complete on bipartite (planar bipartite) graphs. Moreover, we present linear-time algorithms to solve the weighted perfect (efficient) edge domination problem on generalized series–parallel graphs and chordal graphs.

KINETIC COLLISION DETECTION FOR SIMPLE POLYGONS

We design a simple and elegant kinetic data structure for detecting collisions between polygonal (but not necessarily convex) objects in motion in the plane. Our structure is compact, maintaining an active set of certificates whose number is proportional to a minimum-size set of separating polygons for the objects. It is also responsive; on the failure of a certificate invariants can be restored in time logarithmic in the total number of object vertices. It is difficult to characterize the efficiency of our structure for lack of a canonical definition of external events. Nevertheless we give an easy upper bound on the worst case number of certificate failures.

Wavelength Conversion in Optical Networks

In many models of optical routing, we are given a set of communication paths in a network, and we must assign a wavelength to each path so that paths sharing an edge receive different wavelengths. The goal is to assign as few wavelengths as possible in order to make as efficient use as possible of the optical bandwidth. A lot of work in the area of optical networks has considered the use of wavelength converters: if a node of a network contains a wavelength converter, any path that passes through this node may change its wavelength. Having converters at some of the nodes can reduce the number of wavelengths required for routing, down to the following natural congestion bound: even with converters, we will always need at least as many wavelengths as the maximum number of paths sharing a single edge. Thus Wilfong and Winkler defined a set S of nodes in a network to be sufficient if, placing converters at the nodes in S, every set of paths can be routed with a number of wavelengths equal to its congestion bound. They showed that finding a sufficient set of minimum size is NP-complete, even in planar graphs.In this paper, we provide a polynomial-time algorithm to find a sufficient set for an arbitrary directed network whose size is within a factor of 2 of minimum. We also observe that improving on the factor of 2 would lead to a corresponding improvement for the vertex cover problem. For the case of planar graphs, we provide a polynomial-time approximation scheme. The algorithms are based on connections between the minimum sufficient set problem and the undirected feedback vertex set problem. In particular, as a component of the algorithm on planar graphs, we develop the first polynomial-time approximation scheme for the undirected feedback vertex set problem in planar graphs, a result that we feel to be of interest in its own right.

Solving the weighted efficient edge domination problem on bipartite permutation graphs

Given a simple graph G = ( V, E), an edge ( u, v) ϵ E is said to dominate itself and any edge ( u, x) or ( v, x), where xϵ V. A subset D⊂- E is called an efficient edge dominating set of G if all edges in E are dominated by exactly one edge of D. The efficient edge domination problem is to find an efficient edge dominating set of minimum size in G. Suppose that each edge eϵ E is associated with a real number w( e), called the weight of e. The weighted efficient edge domination problem is to calculate an efficient edge dominating set D of G such that the weight w( D) of D is minimum, where w(D) = ∑{w(e)¦eϵD}. In this paper, we show that the problem of determining whether G has an efficient edge dominating set is NP-complete when G is restricted to a bipartite graph. Consequently, the decision problem of efficient (vertex) domination remains NP-complete for the line graphs of bipartite graphs. Moreover, we present a linear time algorithm to solve the weighted efficient edge domination problem on bipartite permutation graphs, which form a subclass of bipartite graphs, using the technique of dynamic programming.

Optimal Testing and Design of Adders

On-the-fly calculations of area and performance are a typical part of the computer-aided iterative design process in VLSI, which aims at a satisfactory tradeoff of various conflicting objectives, among which are test-generation time and test-set size. However, determining test sets on-the-fly as one circuit is transformed into another is extremely difficult. Our goal is to add a test dimension to the design optimization process that complements methods concerned with area and performance optimization. We define a set of logic transformations that result in easily computed changes to test sets. Test-set preserving (TSP) transformations preserve a combinational circuit’s test sets, while test-set altering (TSA) transformations introduce a minimum number of tests needed to maintain completeness. We illustrate our approach with a family of adders that share area-efficient tree structures and differ in the amount of carry-lookahead used to accelerate carry computation. Members include the ripple-carry adder, which has no lookahead, and the standard carry-lookahead adder, which exploits lookahead across all inputs. It is straightforward to derive area and performance measures for this class of adders. Given an n-bit adder with lookahead degree k, we determine a sequence of circuit transformations that produce the adder of degree k2 and test sets of minimum size. Optimal test sets of size k(logkn + 1) + 2 result for arbitrary n and k, which improve significantly upon previously reported tests.

The Boolean Basis Problem and How to Cover Some Polygons by Rectangles

For $S \subseteq \{ 0,1\} ^n $ the Boolean basis (or set basis) problem is to find a minimum size set $B \subseteq \{ 0,1\} ^n $ such that each $s \in S$ is a Boolean sum of vectors from B. This paper examines tractable special cases of this NP-complete problem. In particular a construction preserving tractability is given. The rectangle cover problem—expressing a rectilinear polygon as the union of a minimum number of rectangles—is a main application.

A unified framework for race analysis of asynchronous networks

A unified framework is developed for the study of asynchronous circuits of both gate and MOS type. A basic network model consisting of a directed graph and a set of vertex excitation functions is introduced. A race analysis model, using three values (0, 1, and x), is developed for studying state transitions in the network. It is shown that the results obtained using this model are equivalent to those using ternary simulation. It is also proved that the set of state variables can be reduced to a minimum size set of feedback variables, and the analysis still yields both the correct state transitions and output hazard information. Finally, it is shown how the general results above are applicable to both gate and MOS circuits.

Dimensional analysis, using computer symbolic mathematics

This paper describes a computer-algebra program that performs automatic dimensional analysis. The user merely provides a list of the types of physical quantities involved in a problem, and the program produces a minimum-size set of nondimensional groupings of these quantities. These groupings reveal qualitative information about the solution, and they may reduce the necessary number of variables that must be varied independently in an experimental or numerical solution of the problem.