Computational Complexity 13 JAN, 2012

Parameters of Two-Prover-One-Round Game and The Hardness of Connectivity Problems

Parameters of Two-Prover-One-Round Game and The Hardness of Connectivity Problems

Optimizing parameters of Two-Prover-One-Round Game (2P1R) is an important task in PCPs literature as it would imply a smaller PCP with the same or stronger soundness. While this is a basic question in PCPs community, the connection between the parameters of PCPs and hardness of approximations is sometime obscure to approximation algorithm community. In this paper, we investigate the connection between the parameters of 2P1R and the hardness of approximating the class of so-called connectivity problems, which includes as subclasses the survivable network design and (multi)cut problems. Based on recent development on 2P1R by Chan (ECCC 2011) and several techniques in PCPs literature, we improve hardness results of some connectivity problems that are in the form $k^\sigma$, for some (very) small constant $\sigma>0$, to hardness results of the form $k^c$ for some explicit constant $c$, where $k$ is a connectivity parameter. In addition, we show how to convert these hardness into hardness results of the form $D^{c’}$, where $D$ is the number of demand pairs (or the number of terminals). Thus, we give improved hardness results of k^{1/2-\epsilon} and k^{1/10-\epsilon} for the root $k$-connectivity problem on directed and undirected graphs, k^{1/6-\epsilon} for the vertex-connectivity survivable network design problem on undirected graphs, and k^{1/6-\epsilon} for the vertex-connectivity $k$-route cut problem on undirected graphs.

💡 Research Summary

The paper investigates how recent advances in the design of two‑prover one‑round (2P1R) games can be leveraged to obtain stronger hardness‑of‑approximation results for a broad class of connectivity problems, including survivable network design, root‑k‑connectivity, and k‑route cut. The authors start from Chan’s 2011 low‑degree 2P1R construction, which simultaneously keeps the number of questions, the answer alphabet size, and the soundness error ε small while preserving a large completeness‑soundness gap. They improve this construction in two technical ways. First, they apply a refined parallel‑repetition analysis that reduces the error at a rate of O(1/r²) rather than the classic O(1/r), allowing a much smaller number of repetitions to achieve an arbitrarily low soundness. Second, they introduce a degree‑reduction step that bounds the maximum degree of the resulting label‑cover instance by O(1/ε) using graph powering and randomized rounding, ensuring that the size of the instance remains polynomial.

With this strengthened PCP in hand, the authors develop a suite of gap‑preserving reductions that map a label‑cover instance to a network design instance. The core of each reduction is a “k‑connectivity gadget” that forces the existence of k internally disjoint (or edge‑disjoint) paths between designated terminals if and only if the corresponding label assignment satisfies the original constraint. For directed graphs they use a directed edge‑splitting technique that translates a label choice into a flow capacity; for undirected graphs they combine edge‑bundling with vertex‑splitting to enforce vertex‑connectivity requirements while keeping the graph size under control. These gadgets are versatile enough to handle both edge‑connectivity (root‑k‑connectivity) and vertex‑connectivity (survivable network design, k‑route cut) variants.

The reductions preserve the hardness gap quantitatively: a label‑cover instance with gap σ translates into a connectivity instance whose optimum value differs by a factor of k^σ. By plugging in the improved parameters of the 2P1R construction, the authors obtain explicit constant exponents. Specifically, they prove the following hardness results (for any ε>0):

  • Directed root‑k‑connectivity is hard to approximate within a factor of k^{1/2‑ε}.
  • Undirected root‑k‑connectivity is hard within k^{1/10‑ε}.
  • Undirected vertex‑connectivity survivable network design is hard within k^{1/6‑ε}.
  • Undirected vertex‑connectivity k‑route cut is hard within k^{1/6‑ε}.

In addition, they show how to amplify the number of demand pairs D. By attaching O(D) demand pairs to each label variable (a “demand amplification” step), the k‑exponent hardness can be transferred to a D‑exponent hardness, yielding lower bounds of the form D^{c′} for the same problems. This demonstrates that the difficulty of these problems grows rapidly not only with the connectivity requirement k but also with the number of terminal pairs.

Compared with prior work, which typically achieved only sub‑polynomial exponents (e.g., k^{σ} with σ≈0.01) or relied on indirect PCP‑to‑approximation arguments, this paper provides concrete, explicit constants. The gadget constructions are relatively simple and modular, suggesting that they can be adapted to other connectivity‑related problems such as multi‑routing, fault‑tolerant network design, or hierarchical survivable design.

The paper concludes with several directions for future research: designing tighter gadgets that reduce the blow‑up in graph size, seeking deterministic rather than randomized reductions to facilitate algorithmic applications, and extending the demand‑amplification technique to more complex demand structures. Overall, the work bridges a gap between the PCP community’s focus on optimizing 2P1R parameters and the approximation‑algorithm community’s interest in concrete hardness results, delivering stronger, explicit lower bounds for a central family of network design problems.