Improved Hardness for Cut, Interdiction, and Firefighter Problems

We study variants of the classic s-t cut problem and prove the following improved hardness results assuming the Unique Games Conjecture (UGC). • For any constant k ≥ 2 and ǫ > 0, we show that Directed Multicutwith k source-sink pairs is hard to approximate within a factor k − ǫ. This matches the trivial k-approximation algorithm. By a simple reduction, our result for k = 2 implies that Directed Multiway Cut with two terminals (also known as s-t Bicut) is hard to approximate within a factor 2− ǫ, matching the trivial 2-approximation algorithm. Previously, the best hardness factor for these problems (for constant k) was 1.5− ǫ [EVW13, CM16] under the UGC. • For Length-Bounded Cut and Shortest Path Interdiction, we show that both problems are hard to approximate within any constant factor, even if we allow bicriteria approximation. If we want to cut vertices or the graph is directed, our hardness factor for LengthBounded Cut matches the best approximation ratio up to a constant. Previously, the best hardness factor was 1.1377 for Length-Bounded Cut [BEH10] and 2 for Shortest Path Interdiction [KBB07]. • Assuming a variant of the UGC (implied by another variant of Bansal and Khot [BK09]), we prove that it is hard to approximate Resource Minimization Fire Containmentwithin any constant factor. Previously, the best hardness factor was 2 [KM10]. Our results are based on a general method of converting an integrality gap instance to a lengthcontrol dictatorship test for variants of the s-t cut problem, which may be useful for other problems. Supported by the Samsung Scholarship, the Simons Award for Graduate Students in TCS, and Venkat Guruswami’s NSF CCF-1115525. [email protected]