2000/01/01 by Luca Trevisan, Gregory B. Sorkin, Madhu Sudan +1 · 207 citations
Computer Science · Mathematics · #Complexity and Algorithms in Graphs #Advanced Graph Theory Research #Optimization and Search Problems #Linear programming #Mathematics #Simple (philosophy) #Space (punctuation) #Mathematical proof #Combinatorics #Duality (order theory) #Maximum cut #Upper and lower bounds #Computer science #Discrete mathematics #Algorithm
paper · doi:10.1137/s0097539797328847
published in SIAM Journal on Computing 29(6), 2074-2097 (Society for Industrial and Applied Mathematics)
openalex publication_date 2000/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/06/11
We present a linear programming-based method for finding "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 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 [SIAM J. Comput., 27 (1998), pp. 804--915] of how to prove the optimality of gadgets: linear programming duality gives such proofs. The new gadgets, when combined with recent results of Håstad [ Proceedings of the 29th ACM Symposium on Theory of Computing, 1997, pp. 1--10], improve the known inapproximability results for MAX CUT and MAX DICUT, showing that approximating these problems to within factors of 16/17 + ε and 12/13+ ε, respectively, is NP-hard for every ε > 0. Prior to this work, the best-known inapproximability thresholds for both problems were 71/72 (M. Bellare, O. Goldreich, and M. Sudan [ SIAM J. Comput., 27 (1998), pp. 804--915]). Without using the gadgets from this paper, the best possible hardness that would follow from Bellare, Goldreich, and Sudan and Håstad is 18/19. We also use the gadgets to obtain an improved approximation algorithm for MAX3 SAT which guarantees an approximation ratio of .801. This improves upon the previous best bound (implicit from M. X. Goemans and D. P. Williamson [J. ACM, 42 (1995), pp. 1115--1145]; U. Feige and M. X. Goemans [Proceedings of the Third Israel Symposium on Theory of Computing and Systems, 1995, pp. 182--189]) of .7704.