Inapproximability of Combinatorial Optimization Problems

We survey results on the hardness of approximating combinatorial optimization problems

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

New Tools and Connections for Exponential-Time Approximation

In this paper, we develop new tools and connections for exponential time approximation. In this setting, we are given a problem instance and an integer r>1, and the goal is to design an approximation algorithm with the fastest possible running time. We give randomized algorithms that establish an approximation ratio of 1. r for maximum independent set in O∗(exp(O~(n/rlog2r+rlog2r))) time, 2. r for chromatic number in O∗(exp(O~(n/rlogr+rlog2r))) time, 3. (2−1/r) for minimum vertex cover in O∗(exp(n/rΩ(r))) time, and 4. (k−1/r) for minimum k-hypergraph vertex cover in O∗(exp(n/(kr)Ω(kr))) time. (Throughout, O~ and O∗ omit polyloglog(r) and factors polynomial in the input size, respectively.) The best known time bounds for all problems were O∗(2n/r) (Bourgeois et al. i

New Tools and Connections for Exponential-Time Approximation

In this paper, we develop new tools and connections for exponential time approximation. In this setting, we are given a problem instance and an integer r>1, and the goal is to design an approximation algorithm with the fastest possible running time. We give randomized algorithms that establish an approximation ratio of 1. r for maximum independent set in O∗(exp(O~(n/rlog2r+rlog2r))) time, 2. r for chromatic number in O∗(exp(O~(n/rlogr+rlog2r))) time, 3. (2−1/r) for minimum vertex cover in O∗(exp(n/rΩ(r))) time, and 4. (k−1/r) for minimum k-hypergraph vertex cover in O∗(exp(n/(kr)Ω(kr))) time. (Throughout, O~ and O∗ omit polyloglog(r) and factors polynomial in the input size, respectively.) The best known time bounds for all problems were O∗(2n/r) (Bourgeois et al. in Discret Appl Math 159(17):1954–1970, 2011; Cygan et al. in Exponential-time approximation of hard problems, 2008). For maximum independent set and chromatic number, these bounds were complemented by exp(n1−o(1)/r1+o(1)) lower bounds (under the Exponential Time Hypothesis (ETH)) (Chalermsook et al. in Foundations of computer science, FOCS, pp. 370–379, 2013; Laekhanukit in Inapproximability of combinatorial problems in subexponential-time. Ph.D. thesis, 2014). Our results show that the naturally-looking O∗(2n/r) bounds are not tight for all these problems. The key to these results is a sparsification procedure that reduces a problem to a bounded-degree variant, allowing the use of approximation algorithms for bounded-degree graphs. To obtain the first two results, we introduce a new randomized branching rule. Finally, we show a connection between PCP parameters and exponential-time approximation algorithms. This connection together with our independent set algorithm refute the possibility to overly reduce the size of Chan’s PCP (Chan in J. ACM 63(3):27:1–27:32, 2016). It also implies that a (significant) improvement over our result will refute the gap-ETH conjecture (Dinur in Electron Colloq Comput Complex (ECCC) 23:128, 2016; Manurangsi and Raghavendra in A birthday repetition theorem and complexity of approximating dense CSPs, 2016)

Some Applications of Coding Theory in Computational Complexity

Error-correcting codes and related combinatorial constructs play an important role in several recent (and old) results in computational complexity theory. In this paper we survey results on locally-testable and locally-decodable error-correcting codes, and their applications to complexity theory and to cryptography. Locally decodable codes are error-correcting codes with sub-linear time error-correcting algorithms. They are related to private information retrieval (a type of cryptographic protocol), and they are used in average-case complexity and to construct ``hard-core predicates'' for one-way permutations. Locally testable codes are error-correcting codes with sub-linear time error-detection algorithms, and they are the combinatorial core of probabilistically checkable proofs

Approximation Algorithms for NP-Hard Problems

The workshop was concerned with the most important recent developments in the area of efficient approximation algorithms for NP-hard optimization problems as well as with new techniques for proving intrinsic lower bounds for efficient approximation