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

Abstract

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 sometimes 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 (STOC 2013) and several techniques in PCPs literature, we improve hardness results of some connectivity problems that are in the form k σ , for some (very) small constant σ > 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). Our results are as follows.

Approximation resistance from pairwise independent subgroups

“…Recently, Laekhanukit gave randomized reductions from 2P1R-Game to the following network connectivity problems: Rooted k-Connectivity, Vertex-Connectivity Survivable Network Design, and Vertex-Connectivity k-Route Cut [49]. His hardness results improve those in [18,14,19], and our Theorem 1.8 further strengthens his results.…”

“…His hardness results improve those in [18,14,19], and our Theorem 1.8 further strengthens his results. See [49] for details.…”

A note on degree vs gap of Min-Rep Label Cover and improved inapproximability for connectivity problems

“…For Min-Rep, the typical way to show hardness of approximation with factor g is via proving hardness of approximating of Max-Rep of factor O(g 2 ). This was indeed also the route taken in [Lae14]; due to the square loss in parameter from Max-Rep, for a gap of g in Min-Rep, the alphabet size and the maximum degree now become O(g 4 log 2 g) and O(g 2 log 2 g) respectively.…”

“…The above three problems are shown to be hard to approximate to within a factor of k δ for some small (implicit) constant δ > 0 in [CLNV14,CK12,CMVZ16]. In [Lae14], Laekhanukit formulated the reduction in terms of Min-Rep Label Cover and observed that one can make δ explicit if the degree and the alphabet size can be made explicit polynomials of the gap. More precisely, the following theorem, which is a restatement of Theorem 3.1 in [Lae14], captures the dependency between the alphabet size, the degree and the resulting connectivity parameter.…”

O (log 2 k / log log k )-approximation algorithm for directed Steiner tree

“…Cheriyan et al [7] showed that -DST admits no 2 log 1−ε n -approximation algorithm, for any ε > 0, unless NP ⊆ DTIME(2 polylog(n) ). Laekhanukit [23] showed that the problem admits no 1/2−ε -approximation for any constant ε > 0, unless NP = ZPP. Nevertheless, the negative results do not rule out the possibility of achieving reasonable approximation factors for small values of .…”