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 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 2012) and several techniques in PCPs literature, we improve hardness results of some connectivity problems that are in the form k , for some (very) small constant σ > 0, to hardness results of the form k 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 Dc , where D is the number of demand pairs (or the number of terminals). Our results are as follows. 1. For the rooted k-connectivity problem, we have hardness of  k1/2− on directed graphs. k1/10− on undirected graphs. D1/4− on both directed and undirected graphs. This improves upon the best known hardness of k by Cheriyan et al. (SODA 2012). 2. For the vertex-connectivity survivable network design problem, we have hardness of { k1/6− on undirected graphs D1/4− on both directed and undirected graphs. This improves upon the best known hardness of Ω(k) by Chakraborty et al. (STOC 2008). 3. For the vertex-connectivity k-route cut problem on undirected graphs, we have hardness of { k1/6− D1/4− This improves upon the best known hardness of k by Chuzhoy et al. (SODA 2012). ∗School of Computer Science, McGill University, Montreal, QC, Canada. E-mail: [email protected] This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) grant no. 288334 and 429598, by European Research Council (ERC) Starting Grant no. 279352, by Dr&Mrs M.Leong fellowship. and by Harold H Helm fellowship.