Parameterized Inapproximability for Steiner Orientation by Gap Amplification

In the $k$-Steiner Orientation problem, we are given a mixed graph, that is, with both directed and undirected edges, and a set of $k$ terminal pairs. The goal is to find an orientation of the undirected edges that maximizes the number of terminal pairs for which there is a path from the source to the sink. The problem is known to be W[1]-hard when parameterized by k and hard to approximate up to some constant for FPT algorithms assuming Gap-ETH. On the other hand, no approximation factor better than $O(k)$ is known. We show that $k$-Steiner Orientation is unlikely to admit an approximation algorithm with any constant factor, even within FPT running time. To obtain this result, we construct a self-reduction via a hashing-based gap amplification technique, which turns out useful even outside of the FPT paradigm. Precisely, we rule out any approximation factor of the form $(\log k)^{o(1)}$ for FPT algorithms (assuming FPT $\ne$ W[1]) and $(\log n)^{o(1)}$ for~purely polynomial-time algorithms (assuming that the class W[1] does not admit randomized FPT algorithms). Moreover, we prove $k$-Steiner Orientation to belong to W[1], which entails W[1]-completeness of $(\log k)^{o(1)}$-approximation for $k$-Steiner Orientation This provides an example of a natural approximation task that is complete in a parameterized complexity class. Finally, we apply our technique to the maximization version of directed multicut - Max $(k,p)$-Directed Multicut - where we are given a directed graph, $k$ terminals pairs, and a budget $p$. The goal is to maximize the number of separated terminal pairs by removing $p$ edges. We present a simple proof that the problem admits no FPT approximation with factor $O(k^{\frac 1 2 - \epsilon})$ (assuming FPT $\ne$ W[1]) and no polynomial-time approximation with ratio $O(|E(G)|^{\frac 1 2 - \epsilon})$ (assuming NP $\not\subseteq$ co-RP).

[1]  Bingkai Lin,et al.  The Parameterized Complexity of k-Biclique , 2014, SODA.

[2]  Alan Siegel,et al.  On Universal Classes of Extremely Random Constant-Time Hash Functions , 1995, SIAM J. Comput..

[3]  Jason Li,et al.  On the Fixed-Parameter Tractability of Capacitated Clustering , 2022, ICALP.

[4]  Rolf Niedermeier,et al.  Exploiting bounded signal flow for graph orientation based on cause-effect pairs , 2011, Algorithms for Molecular Biology.

[5]  Prasad Raghavendra,et al.  A Birthday Repetition Theorem and Complexity of Approximating Dense CSPs , 2016, ICALP.

[6]  Ken-ichi Kawarabayashi,et al.  A nearly 5/3-approximation FPT Algorithm for Min-k-Cut , 2020, SODA.

[7]  DinurIrit The PCP theorem by gap amplification , 2007 .

[8]  Jaroslaw Byrka,et al.  Constant factor FPT approximation for capacitated k-median , 2018, ESA.

[9]  Prasad Raghavendra,et al.  Beating the Random Ordering is Hard: Inapproximability of Maximum Acyclic Subgraph , 2008, 2008 49th Annual IEEE Symposium on Foundations of Computer Science.

[10]  Nimrod Megiddo,et al.  On orientations and shortest paths , 1989 .

[11]  Anupam Gupta,et al.  Faster Exact and Approximate Algorithms for k-Cut , 2018, 2018 IEEE 59th Annual Symposium on Foundations of Computer Science (FOCS).

[12]  Pasin Manurangsi,et al.  Almost-polynomial ratio ETH-hardness of approximating densest k-subgraph , 2016, STOC.

[13]  Moni Naor,et al.  Small-bias probability spaces: efficient constructions and applications , 1990, STOC '90.

[14]  Subhash Khot,et al.  Optimal Long Code Test with One Free Bit , 2009, 2009 50th Annual IEEE Symposium on Foundations of Computer Science.

[15]  Pasin Manurangsi,et al.  Inapproximability of Maximum Edge Biclique, Maximum Balanced Biclique and Minimum k-Cut from the Small Set Expansion Hypothesis , 2017, ICALP.

[16]  Roded Sharan,et al.  An Algorithm for Orienting Graphs Based on Cause-Effect Pairs and Its Applications to Orienting Protein Networks , 2008, WABI.

[17]  Irit Dinur,et al.  The PCP theorem by gap amplification , 2006, STOC.

[18]  Guy Kortsarz,et al.  Steiner Forest Orientation Problems , 2012, ESA.

[19]  Sanjeev Khanna,et al.  Polynomial flow-cut gaps and hardness of directed cut problems , 2007, STOC '07.

[20]  Rajesh Chitnis,et al.  A Tight Lower Bound for Planar Steiner Orientation , 2019, Algorithmica.

[21]  Noga Alon,et al.  Improved approximation for directed cut problems , 2007, STOC '07.

[22]  Carsten Lund,et al.  Proof verification and the hardness of approximation problems , 1998, JACM.

[23]  Dániel Marx Completely Inapproximable Monotone and Antimonotone Parameterized Problems , 2010, 2010 IEEE 25th Annual Conference on Computational Complexity.

[24]  Esther M. Arkin,et al.  A note on orientations of mixed graphs , 2002, Discret. Appl. Math..

[25]  Nicolas Bousquet,et al.  Multicut is FPT , 2010, STOC '11.

[26]  Pasin Manurangsi,et al.  Parameterized Approximation Algorithms for Directed Steiner Network Problems , 2017, ESA.

[27]  Subhash Khot,et al.  On the power of unique 2-prover 1-round games , 2002, Proceedings 17th IEEE Annual Conference on Computational Complexity.

[28]  Amit Kumar,et al.  Tight FPT Approximations for $k$-Median and k-Means , 2019, ICALP.

[29]  Saket Saurabh,et al.  Parameterized Complexity and Approximability of Directed Odd Cycle Transversal , 2017, SODA.

[30]  U. Vazirani Randomness, adversaries and computation (random polynomial time) , 1986 .

[31]  Magnus Wahlström,et al.  Directed Multicut is W[1]-hard, Even for Four Terminal Pairs , 2015, SODA.

[32]  Russell Impagliazzo,et al.  Which problems have strongly exponential complexity? , 1998, Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280).

[33]  Pasin Manurangsi,et al.  On the parameterized complexity of approximating dominating set , 2017, Electron. Colloquium Comput. Complex..

[34]  Larry Carter,et al.  Universal Classes of Hash Functions , 1979, J. Comput. Syst. Sci..

[35]  Prasad Raghavendra,et al.  Sdp gaps and ugc hardness for multiway cut, 0-extension, and metric labeling , 2008, STOC.

[36]  Roded Sharan,et al.  Improved Orientations of Physical Networks , 2010, WABI.

[37]  Pasin Manurangsi,et al.  Parameterized Intractability of Even Set and Shortest Vector Problem from Gap-ETH , 2018, Electron. Colloquium Comput. Complex..

[38]  Luca Trevisan,et al.  From Gap-ETH to FPT-Inapproximability: Clique, Dominating Set, and More , 2017, 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS).

[39]  Irit Dinur,et al.  Mildly exponential reduction from gap 3SAT to polynomial-gap label-cover , 2016, Electron. Colloquium Comput. Complex..

[40]  Anupam Gupta Improved results for directed multicut , 2003, SODA '03.

[41]  G. Fertin,et al.  On the Complexity of two Problems on Orientations of Mixed Graphs , 2012 .