Efficient Algorithms for Fixed-Precision Instances of Bin Packing and Euclidean TSP

Abstract

This paper presents new, polynomial time algorithms for Bin Packing and Euclidean TSP under fixed precision. In this model, integers are encoded as floating point numbers, each with a mantissa and an exponent. Thus, an integer i with $i = a_i2^{t_i}$ has mantissa a _i and exponent t _i . This natural representation is the norm in real-world optimization. A set of integers I has L-bit precision if $\max_{i \in I} a_i< 2^L$. In this framework, we show an exact algorithm for Bin Packing and an FPTAS for Euclidean TSP which run in time poly(n) and poly(n + log1/ε), respectively, when L is a fixed constant. Our algorithm for the later problem is exact when distances are given by the L ₁ norm. In contrast, both problems are strongly NP-Hard (and yield PTASs) when precision is unbounded. These algorithms serve as evidence of the significance of the class of fixed precision polynomial time solvable problems. Taken together with algorithms for the Knapsack and Pm||C _max problems introduced by Orlin et al., [10] we see that fixed precision defines a class incomparable to polynomial time approximation schemes, covering at least four distinct natural NP-hard problems.