Linear-Time FPT Algorithms via Network Flow

In the area of parameterized complexity, to cope with NP-Hard problems, we introduce a parameter k besides the input size n, and we aim to design algorithms (called FPT algorithms) that run in O(f(k)n^d) time for some function f(k) and constant d. Though FPT algorithms have been successfully designed for many problems, typically they are not sufficiently fast because of huge f(k) and d. In this paper, we give FPT algorithms with small f(k) and d for many important problems including Odd Cycle Transversal and Almost 2-SAT. More specifically, we can choose f(k) as a single exponential (4^k) and d as one, that is, linear in the input size. To the best of our knowledge, our algorithms achieve linear time complexity for the first time for these problems. To obtain our algorithms for these problems, we consider a large class of integer programs, called BIP2. Then we show that, in linear time, we can reduce BIP2 to Vertex Cover Above LP preserving the parameter k, and we can compute an optimal LP solution for Vertex Cover Above LP using network flow. Then, we perform an exhaustive search by fixing half-integral values in the optimal LP solution for Vertex Cover Above LP. A bottleneck here is that we need to recompute an LP optimal solution after branching. To address this issue, we exploit network flow to update the optimal LP solution in linear time.Comment: 20 page

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 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 ratio for Length-Bounded Cut matches the best approximation ratio up to a constant. Previously, the best hardness ratio was 1.1377 for Length-Bounded Cut and 2 for Shortest Path Interdiction. * For any constant k >= 2 and epsilon > 0, we show that Directed Multicut with k source-sink pairs is hard to approximate within a factor k - epsilon. 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 - epsilon, matching the trivial 2-approximation algorithm. * Assuming a variant of the UGC (implied by another variant of Bansal and Khot), we prove that it is hard to approximate Resource Minimization Fire Containment within any constant factor. Previously, the best hardness ratio was 2. For directed layered graphs with b layers, our hardness ratio Omega(log b) matches the best approximation algorithm. Our results are based on a general method of converting an integrality gap instance to a length-control dictatorship test for variants of the s-t cut problem, which may be useful for other problems

Fixed-parameter tractability of multicut parameterized by the size of the cutset

Given an undirected graph $G$, a collection $\{(s_1,t_1),..., (s_k,t_k)\}$ of pairs of vertices, and an integer $p$, the Edge Multicut problem ask if there is a set $S$ of at most $p$ edges such that the removal of $S$ disconnects every $s_i$ from the corresponding $t_i$. Vertex Multicut is the analogous problem where $S$ is a set of at most $p$ vertices. Our main result is that both problems can be solved in time $2^{O(p^3)}... n^{O(1)}$, i.e., fixed-parameter tractable parameterized by the size $p$ of the cutset in the solution. By contrast, it is unlikely that an algorithm with running time of the form $f(p)... n^{O(1)}$ exists for the directed version of the problem, as we show it to be W[1]-hard parameterized by the size of the cutset