A Birthday Repetition Theorem and Complexity of Approximating Dense CSPs (1607.02986v1)

Abstract: A $(k \times l)$-birthday repetition $\mathcal{G}^{k \times l}$ of a two-prover game $\mathcal{G}$ is a game in which the two provers are sent random sets of questions from $\mathcal{G}$ of sizes $k$ and $l$ respectively. These two sets are sampled independently uniformly among all sets of questions of those particular sizes. We prove the following birthday repetition theorem: when $\mathcal{G}$ satisfies some mild conditions, $val(\mathcal{G}^{k \times l})$ decreases exponentially in $\Omega(kl/n)$ where $n$ is the total number of questions. Our result positively resolves an open question posted by Aaronson, Impagliazzo and Moshkovitz (CCC 2014). As an application of our birthday repetition theorem, we obtain new fine-grained hardness of approximation results for dense CSPs. Specifically, we establish a tight trade-off between running time and approximation ratio for dense CSPs by showing conditional lower bounds, integrality gaps and approximation algorithms. In particular, for any sufficiently large $i$ and for every $k \geq 2$, we show the following results: - We exhibit an $O(q^{1/i})$-approximation algorithm for dense Max $k$-CSPs with alphabet size $q$ via $O_k(i)$-level of Sherali-Adams relaxation. - Through our birthday repetition theorem, we obtain an integrality gap of $q^{1/i}$ for $\tilde\Omega_k(i)$-level Lasserre relaxation for fully-dense Max $k$-CSP. - Assuming that there is a constant $\epsilon > 0$ such that Max 3SAT cannot be approximated to within $(1-\epsilon)$ of the optimal in sub-exponential time, our birthday repetition theorem implies that any algorithm that approximates fully-dense Max $k$-CSP to within a $q^{1/i}$ factor takes $(nq)^{\tilde \Omega_k(i)}$ time, almost tightly matching the algorithmic result based on Sherali-Adams relaxation.

• The paper presents the Birthday Repetition Theorem which generalizes parallel repetition by proving an exponential decay in game value with repeated plays.
• It applies this framework to dense CSPs, establishing novel hardness of approximation results and clear trade-offs between time complexity and approximation ratios.
• The work demonstrates conditional lower bounds and integrality gaps that offer actionable insights for designing efficient approximation algorithms in combinatorial optimization.

An Expert Overview of "A Birthday Repetition Theorem and Complexity of Approximating Dense CSPs"

The paper by Pasin Manurangsi and Prasad Raghavendra presents significant contributions to computational complexity theory, specifically in the approximation of dense Constraint Satisfaction Problems (CSPs). It introduces a theoretical framework known as the Birthday Repetition Theorem and leverages this to investigate the complexity of approximating dense CSPs.

Key Contributions

Birthday Repetition Theorem

The authors develop the concept of birthday repetition for two-prover games, which generalizes the parallel repetition theorem. A $(k \times l)$-birthday repetition involves transforming a base two-prover game $\cG$ into a new game $\cG^{k \times l}$, where provers are given random sets of questions from $\cG$. The birthday repetition theorem established in this paper shows that, under mild assumptions, the value of $\cG^{k \times l}$ diminishes exponentially in $\Omega(kl/n)$, where $n$ represents the number of potential questions in the game. This result provides a complete solution to a conjecture in the area proposed by Aaronson, Impagliazzo, and Moshkovitz.

Application to Dense CSPs

The paper utilizes the birthday repetition theorem to derive new hardness of approximation results for dense CSPs. It establishes a trade-off relationship between the time complexity and the approximation ratio for these problems by proving conditional lower bounds. Notably, the authors demonstrate several results for Max $k$-CSPs, including:

1. An $O(q^{1/i})$-approximation algorithm using Sherali-Adams level relaxation for dense Max $k$-CSPs.
2. Presentation of an integrality gap of $q^{1/i}$ for a particular level Lasserre relaxation.
3. Under the conjecture that Max 3SAT cannot be approximated efficiently in sub-exponential time, they show that approximating fully-dense Max $k$-CSP to within a $q^{1/i}$ factor requires $(nq)^{\tilde \Omega_k(i)}$ time.

Implications and Future Directions

Theoretical and Practical Implications

The introduction of the birthday repetition theorem not only provides a novel analysis tool for evaluating game repetitions but also influences the understanding of approximation complexity on dense CSPs. Through this framework, the authors bridge gaps in theoretical limitations and practical algorithm design, offering a more nuanced understanding of problem hardness in CSPs.

From a practical perspective, the insights gained can guide the development of efficient approximation algorithms for dense CSP-related problems in computational areas such as scheduling, optimization, and AI.

Future Research

The work opens several avenues for further research, including:

• Exploring the implications of the birthday repetition theorem on other combinatorial optimization problems and expanding its applicability in quantum computing scenarios.
• Investigating the precise dependency of approximation complexity on parameters $\epsilon$ and $k$ and determining tight bounds where gaps exist.
• Examining the algorithmic potential of these theoretical findings to develop more robust approximation algorithms for CSPs with large alphabet sizes or complex combinatorial structures.

In conclusion, this paper's blend of sophisticated theoretical advancements with implications for computational complexity showcases a path forward in the paper of dense CSPs and other challenge areas in theoretical computer science. While addressing some longstanding conjectures, it also raises questions prompting continued exploration and innovation in complexity theory.