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

• Authors: David R. Karger;Jacob Scott
• Venue: APPROX '08 / RANDOM '08 Proceedings of the 11th international workshop, APPROX 2008, and 12th international workshop, RANDOM 2008 on Approximation, Randomization and Combinatorial Optimization: Algorithms and Techniques
• Year: 2008

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 iwith $i = a_i2^{t_i}$ has mantissa aiand exponent ti. This natural representation is the norm in real-world optimization. A set of integers Ihas L-bit precisionif $\max_{i \in I} a_i. 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/茂戮驴), respectively, when Lis a fixed constant. Our algorithm for the later problem is exact when distances are given by the L1norm. 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 classof fixed precision polynomial time solvable problems. Taken together with algorithms for the Knapsack and Pm||Cmaxproblems 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.