L. Trevisan, G.B. Sorkin, M. Sudan, D.P. Williamson.
Gadgets, Approximation, and Linear Programming.
In Proceedings of the 37th Symposium on Foundations of Computer Science (FOCS'96),
IEEE, 1996.
We present a linear-programming based method for finding ``gadgets'', i.e., combinatorial structures reducing constraints of one optimization problems to constraints of another. A key step in this method is a simple observation which limits the search space to a finite one. Using this new method we present a number of new, computer-constructed gadgets for several different reductions. This method also answers a question posed by Bellare, Goldreich and Sudan on how to prove the optimality of gadgets --- we show how LP duality gives such proofs. The new gadgets improve hardness results for MAX CUT and MAX DICUT, showing that approximating these problems to within factors of 60/61 and 44/45 respectively is NP-hard (improving upon the previous hardness of 71/72 for both problems). We also use the gadgets to obtain an improved approximation algorithm for Max 3SAT which guarantees an approximation ratio of .801. This improves upon the previous best bound of .7704.