We prove that it is NP-hard to decide whether two points in a polygonal domain with holes can be connected by a wire. This implies that finding any approximation to the shortest path for a long snake amidst polygonal obstacles is NP-hard. On the positive side, we show that snake's problem is "length-tractable": if the snake is "fat", i.e., its length width ratio is small, the shortest path can be computed in polynomial time.
In VLSI numerous extensions and generalizations of the basic problem were considered. These include routing multiple paths, on several levels, and with different constraints and objectives. It is impossible to survey all literature on the subject; we will only mention the books @cite_14 @cite_16 .
@cite_14 @cite_16
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During recent days, the large problem space of very large scale integrated (VLSI) circuits has led global routing problem to a NP Complete one. With the advent of three dimensional integrated circuits (3D IC) the problem has become more complex. In this paper, a multi-objective global routing technique is formulated using fuzzy logic to get rid of the limitations of deterministic approaches. During global routing the decision is taken from a fuzzy logic expert system depending upon some generated pre-routing guiding information. Proposed approach is mainly focused on the modern generation 3D ICs. This primary work is mainly incorporated in the standard cell design for 3D ICs. A two-pin global router tool is designed based on the proposed algorithm resulting 85-97 routability in negligible time for ISPD'98 benchmarks. Publisher Summary This chapter discusses some of the most important vehicle routing problem types. The vehicle routing problem lies at the heart of distribution management. It is faced each day by thousands of companies and organizations engaged in the delivery and collection of goods or people. Because conditions vary from one setting to the next, the objectives and constraints encountered in practice are highly variable. Most algorithmic research and software development in this area focus on a limited number of prototype problems. By building enough flexibility in optimization systems, these can be adapted to various practical contexts. The classical vehicle routing problem (VRP) is one of the most popular problems in combinatorial optimization, and its study has given rise to several exact and heuristic solution techniques of general applicability. The chapter presents a comprehensive overview of the available exact and heuristic algorithms for the VRP, most of which have been adapted to solve other variants. We present a polynomial-time algorithm that finds the maximum weighted independent set of a transitive graph. The studied problem finds applications in a variety of VLSI contexts, including path delay fault testing, scheduling in high-level synthesis, and channel routing in physical design automation. The algorithm has been implemented and incorporated in a CAD tool for path delay fault testing. We experimentally verify its impact in the latter context. This pioneering study of two-dimensional wiring patterns develops powerful algorithms for the physical design of VLSI circuits. Its homotopic approach to circuit layout advances the state of the art in wire routing and layout compaction, and will inspire future research. By viewing wires as flexible connections with fixed topology, the author obtains simple and efficient algorithms for CAD problems whose previous solutions employed, unreliable or inefficient heuristics."Single-Layer Wire Routing and Compaction" is the first rigorous treatment of homotopic layouts and the techniques for optimizing them. In a novel application of classical mathematics to computer science, Maley characterizes the ideal routing of a layout in terms of simple topological invariants. He derives practical algorithms from this theoretical insight. The algorithms and their underlying ideas are intuitive, widely applicable, and presented in a highly readable style.F. Miller Maley is a Research Associate in the Computer Science Department at Princeton University. "Single-Layer Wire Routing and Compaction" is included in the series Foundations of Computing, edited by Michael Garey and Albert Meyer.