Improved NP-Inapproximability for 2-Variable Linear Equations

An instance of the 2-Lin(2) problem is a system of equations of the form "x_i + x_j = b (mod 2)". Given such a system in which it\u27s possible to satisfy all but an epsilon fraction of the equations, we show it is NP-hard to satisfy all but a C*epsilon fraction of the equations, for any C < 11/8 = 1.375 (and any 0 < epsilon <= 1/8). The previous best result, standing for over 15 years, had 5/4 in place of 11/8. Our result provides the best known NP-hardness even for the Unique Games problem, and it also holds for the special case of Max-Cut. The precise factor 11/8 is unlikely to be best possible; we also give a conjecture concerning analysis of Boolean functions which, if true, would yield a larger hardness factor of 3/2. Our proof is by a modified gadget reduction from a pairwise-independent predicate. We also show an inherent limitation to this type of gadget reduction. In particular, any such reduction can never establish a hardness factor C greater than 2.54. Previously, no such limitation on gadget reductions was known

Inapproximability of Combinatorial Optimization Problems

We survey results on the hardness of approximating combinatorial optimization problems

Approximation Hardness of Graphic TSP on Cubic Graphs

We prove explicit approximation hardness results for the Graphic TSP on cubic and subcubic graphs as well as the new inapproximability bounds for the corresponding instances of the (1,2)-TSP. The proof technique uses new modular constructions of simulating gadgets for the restricted cubic and subcubic instances. The modular constructions used in the paper could be also of independent interest

New Inapproximability Bounds for TSP

In this paper, we study the approximability of the metric Traveling Salesman Problem (TSP) and prove new explicit inapproximability bounds for that problem. The best up to now known hardness of approximation bounds were 185/184 for the symmetric case (due to Lampis) and 117/116 for the asymmetric case (due to Papadimitriou and Vempala). We construct here two new bounded occurrence CSP reductions which improve these bounds to 123/122 and 75/74, respectively. The latter bound is the first improvement in more than a decade for the case of the asymmetric TSP. One of our main tools, which may be of independent interest, is a new construction of a bounded degree wheel amplifier used in the proof of our results

On the complexity of nonlinear mixed-integer optimization

This is a survey on the computational complexity of nonlinear mixed-integer optimization. It highlights a selection of important topics, ranging from incomputability results that arise from number theory and logic, to recently obtained fully polynomial time approximation schemes in fixed dimension, and to strongly polynomial-time algorithms for special cases.Comment: 26 pages, 5 figures; to appear in: Mixed-Integer Nonlinear Optimization, IMA Volumes, Springer-Verla

Independent Set, Induced Matching, and Pricing: Connections and Tight (Subexponential Time) Approximation Hardnesses

We present a series of almost settled inapproximability results for three fundamental problems. The first in our series is the subexponential-time inapproximability of the maximum independent set problem, a question studied in the area of parameterized complexity. The second is the hardness of approximating the maximum induced matching problem on bounded-degree bipartite graphs. The last in our series is the tight hardness of approximating the k-hypergraph pricing problem, a fundamental problem arising from the area of algorithmic game theory. In particular, assuming the Exponential Time Hypothesis, our two main results are: - For any r larger than some constant, any r-approximation algorithm for the maximum independent set problem must run in at least 2^{n^{1-\epsilon}/r^{1+\epsilon}} time. This nearly matches the upper bound of 2^{n/r} (Cygan et al., 2008). It also improves some hardness results in the domain of parameterized complexity (e.g., Escoffier et al., 2012 and Chitnis et al., 2013) - For any k larger than some constant, there is no polynomial time min (k^{1-\epsilon}, n^{1/2-\epsilon})-approximation algorithm for the k-hypergraph pricing problem, where n is the number of vertices in an input graph. This almost matches the upper bound of min (O(k), \tilde O(\sqrt{n})) (by Balcan and Blum, 2007 and an algorithm in this paper). We note an interesting fact that, in contrast to n^{1/2-\epsilon} hardness for polynomial-time algorithms, the k-hypergraph pricing problem admits n^{\delta} approximation for any \delta >0 in quasi-polynomial time. This puts this problem in a rare approximability class in which approximability thresholds can be improved significantly by allowing algorithms to run in quasi-polynomial time.Comment: The full version of FOCS 201