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

Parinya Chalermsook, Bundit Laekhanukit+1 more|arXiv (Cornell University)|Aug 12, 2013

Complexity and Algorithms in Graphs46 references8 citations

TL;DR

This paper establishes nearly tight subexponential-time inapproximability results for three fundamental NP-hard problems—maximum independent set, induced matching, and k-hypergraph pricing—using a novel connection between them. Under the Exponential Time Hypothesis (ETH), it proves that no r-approximation algorithm for maximum independent set can run in time below 2^{n^{1−ε}/r^{1+ε}}, nearly matching the best-known upper bound, and similarly establishes near-optimal hardness for k-hypergraph pricing and induced matching on bounded-degree bipartite graphs.

ABSTRACT

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-ε}/r^{1+ε}} 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-ε}, n^{1/2-ε})-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), ilde 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-ε} hardness for polynomial-time algorithms, the k-hypergraph pricing problem admits n^δ approximation for any δ>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.

Motivation & Objective

• To establish nearly tight inapproximability bounds for the maximum independent set problem in subexponential time under the Exponential Time Hypothesis (ETH).
• To prove strong hardness of approximation for the induced matching problem on bounded-degree bipartite graphs.
• To resolve the approximability threshold of the k-hypergraph pricing problem, showing near-optimal inapproximability in polynomial time.
• To unify and strengthen hardness results across three distinct problems via novel reductions and connections.
• To explore the role of quasi-polynomial time algorithms in improving approximation ratios for problems resistant to polynomial-time approximation.

Proposed method

• Constructs a nearly-linear size, sparse PCP with large degree and small free-bit complexity, using a careful assembly of existing PCP components.
• Establishes a tight connection between the independent set and induced matching problems via a new graph-theoretic property of dispersers related to induced matching number.
• Reduces the independent set problem to the induced matching problem using a specialized graph transformation that preserves approximation ratios.
• Transfers subexponential-time inapproximability from independent set to the k-hypergraph pricing problem via a reduction from semi-induced matching.
• Designs a quasi-polynomial time O(n^δ)-approximation algorithm for k-hypergraph pricing, demonstrating that approximation thresholds improve significantly with more time.
• Uses price ladder constraints and rounding techniques to design a constant-factor approximation algorithm with time complexity O((log nm)^n poly(n,m)).

Experimental results

Research questions

• RQ1 Can the subexponential-time inapproximability of the maximum independent set problem be tightened under the Exponential Time Hypothesis?
• RQ2 What is the best possible approximation ratio achievable in polynomial time for the k-hypergraph pricing problem?
• RQ3 How does the approximability of induced matching on bounded-degree bipartite graphs relate to that of independent set and pricing problems?
• RQ4 Can quasi-polynomial time algorithms significantly improve approximation ratios for problems that resist polynomial-time approximation?
• RQ5 Is the ETH necessary to prove the subexponential-time inapproximability of these problems, or can weaker assumptions suffice?

Key findings

• Under the Exponential Time Hypothesis, any r-approximation algorithm for maximum independent set requires at least 2^{n^{1−ε}/r^{1+ε}} time for any r > some constant, nearly matching the 2^{n/r} upper bound.
• The k-hypergraph pricing problem admits no polynomial-time min{k^{1−ε}, n^{1/2−ε}}-approximation algorithm under ETH, nearly matching the best-known upper bound of min{O(k), Õ(√n)}.
• The induced matching problem on d-degree-bounded bipartite graphs is hard to approximate within n^{1/2−ε} in polynomial time, assuming ETH.
• The k-hypergraph pricing problem admits an O(n^δ)-approximation in quasi-polynomial time for any δ > 0, showing a significant gap between polynomial and quasi-polynomial time approximability.
• A constant-factor approximation algorithm for k-hypergraph pricing exists in time O((log nm)^n poly(n,m)), achieving an approximation ratio of O(α²/(α−1)) for any α > 1.
• The paper demonstrates a rare case where quasi-polynomial time enables a substantial improvement in approximation ratio, highlighting a unique approximability class.