Almost-Polynomial Ratio ETH-Hardness of Approximating Densest $k$-Subgraph (1611.05991v2)

Abstract: In the Densest $k$-Subgraph problem, given an undirected graph $G$ and an integer $k$, the goal is to find a subgraph of $G$ on $k$ vertices that contains maximum number of edges. Even though the state-of-the-art algorithm for the problem achieves only $O(n^{1/4 + \varepsilon})$ approximation ratio (Bhaskara et al., 2010), previous attempts at proving hardness of approximation, including those under average case assumptions, fail to achieve a polynomial ratio; the best ratios ruled out under any worst case assumption and any average case assumption are only any constant (Raghavendra and Steurer, 2010) and $2^{\Omega(\log^{2/3} n)}$ (Alon et al., 2011) respectively. In this work, we show, assuming the exponential time hypothesis (ETH), that there is no polynomial-time algorithm that approximates Densest $k$-Subgraph to within $n^{1/(\log \log n)^c}$ factor of the optimum, where $c > 0$ is a universal constant independent of $n$. In addition, our result has "perfect completeness", meaning that we prove that it is ETH-hard to even distinguish between the case in which $G$ contains a $k$-clique and the case in which every induced $k$-subgraph of $G$ has density at most $1/n^{-1/(\log \log n)^c}$ in polynomial time. Moreover, if we make a stronger assumption that there is some constant $\varepsilon > 0$ such that no subexponential-time algorithm can distinguish between a satisfiable 3SAT formula and one which is only $(1 - \varepsilon)$-satisfiable (also known as Gap-ETH), then the ratio above can be improved to $n^{f(n)}$ for any function $f$ whose limit is zero as $n$ goes to infinity (i.e. $f \in o(1)$).

• The paper establishes an almost-polynomial hardness bound for approximating DkS under the ETH and its stronger Gap-ETH variant.
• It proves that no polynomial-time algorithm can achieve an approximation ratio better than n^(1/(log log n)^c) unless the ETH fails.
• The research introduces new PCP-based reduction techniques, broadening the theoretical frontiers of approximation limits in optimization problems.

Insights into the Complexity of Approximating Densest $k$-Subgraph

The paper "Almost-Polynomial Ratio ETH-Hardness of Approximating Densest $k$-Subgraph" by Pasin Manurangsi investigates the computational limitations in approximating the Densest $k$-Subgraph (D$k$S) problem, a prominent topic in theoretical computer science. It is a profound advancement in understanding the problem's complexity, providing an almost-polynomial hard approximation guarantee under the Exponential Time Hypothesis (ETH) and its stronger variant, Gap-ETH.

Problem Context and Prior Work

In the D$k$S problem, we are tasked with finding a subgraph of $k$ vertices within a given undirected graph that maximizes the edge count. The problem is inherently challenging and has motivated a range of approximation algorithms. The most notable progress in approximation algorithms includes achieving an approximation ratio of $O(n^{1/4 + \varepsilon})$ by Bhaskara et al. Despite this, proving any significant hardness of approximation has been elusive, with existing results offering only constant-factor inapproximability under worst-case assumptions.

Contributions and Key Claims

The paper's primary contribution is establishing a strong hardness of approximation result under ETH and Gap-ETH for the D$k$S problem. It proves that there is no polynomial-time algorithm that can approximate D$k$S to within a factor of $n^{1/(\loglog n)^c}$ where $c > 0$, unless the ETH is false. This is an unprecedented result, as it suggests the impossibility of achieving a polynomial approximation ratio under these hypotheses.

Additionally, the paper shows that under Gap-ETH, where it is assumed that distinguishing between fully satisfiable 3SAT instances and those satisfied only up to $(1-\varepsilon)$ requires exponential time, the hardness can be strengthened. In this case, the approximation ratio barrier can be improved to $n^{f(n)}$ for any function $f \in o(1)$, indicating the potential impossibility of achieving even nearly-subpolynomial approximations.

Implications and Theoretical Advancements

The implications of these findings are far-reaching. They decisively push the boundaries of what is known about approximation hardness beyond prior results that failed to surpass constant factor assumptions. Manurangsi’s results place D$k$S alongside some of the most challenging optimization problems in terms of approximability, expanding the theoretical landscape shaped by hypothesis-driven complexity assumptions like ETH and Gap-ETH.

This work also enriches the toolkit for exploring hardness proofs via new techniques inspired by recent advancements in PCP theory, notably nearly-linear size PCPs, which enable sophisticated subexponential reductions. Furthermore, the approach could potentially illuminate other complex CSP problems, extending this hardness paradigm beyond D$k$S, encompassing additional challenging tasks within both worst-case and average-case analysis frameworks.

Speculation on Future Directions

The conclusions of this paper motivate numerous avenues for further exploration. Future research might well explore if these almost-polynomial ratios can translate to practical computing scenarios and how they influence the broader understanding of approximation algorithms. Investigating methods to possibly bypass these complexity barriers in special cases or restricted graph classes could be a promising direction, potentially leading to practical solutions where theoretical hardness is mitigated by domain-specific characteristics.

Moreover, this work invites a re-investigation of related combinatorial problems through the lens of enhanced complexity hypotheses. If the connection between D$k$S inapproximability and strong ETH assumptions holds, it may suggest new barriers for approximating a broader scope of optimization problems, thereby significantly shaping future theoretical investigations in computer science.

In summary, Manurangsi's research provides formidable insights into the approximation landscape of D$k$S, establishing critical hardness boundaries and imparting new directions for algorithmic research under the lens of tight complexity-theoretical constraints.