Gadgets, Approximation, and Linear Programming

We present a linear programming-based method for nding \gadgets", i.e., combinatorial structures reducing constraints of one optimization problem to constraints of another. A key step in this method is a simple observation which limits the search space to a nite one. Using this new method we present a number of new, computer-constructed gadgets for several di erent reductions. This method also answers a question posed by Bellare, Goldreich and Sudan 2] of how to prove the optimality of gadgets: LP duality gives such proofs. The new gadgets, when combined with recent results of Hastad 9], improve the known inapproximability results for MAX CUT and MAX DICUT, showing that approximating these problems to within factors of 16=17 + and 12=13 + respectively is NP-hard, for every > 0. Prior to this work, the best known inapproximability thresholds for both problems was 71/72 2]. Without using the gadgets from this paper, the best possible hardness that would follow from 2, 9] is 18=1...

• Type: Journal
• Year: 2000
• Where: SIAMCOMP
• Authors: Luca Trevisan, Gregory B. Sorkin, Madhu Sudan, David P. Williamson
• Keywords: Computer-constructed Gadgets, Gadgets, Linear Programming-based Method, SIAMCOMP 2000