Abstract
Recent empirical evaluations of exact algorithms for Feedback Vertex Set have demonstrated the efficiency of a highest-degree branching algorithm with a degree-based pruning. In this paper, we prove that this empirically fast algorithm runs in \(O(3.460^k n)\) time, where k is the solution size. This improves the previous best \(O(3.619^k n)\)-time deterministic algorithm obtained by Kociumaka and Pilipczuk (Inf Process Lett 114:556–560, 2014. https://doi.org/10.1016/j.ipl.2014.05.001).
Highlights
Feedback Vertex Set (FVS) is a problem of finding the minimum-size vertex deletion set to make the input graph a forest, which is one of the Karp’s 21 NP-complete problems [18]
It is known that this problem is fixed-parameter tractable (FPT) parameterized by the solution size k [2, 10]; i.e., we can find a deletion set of size k in O∗(f (k))1 time for some function f
The current fastest deterministic FPT algorithm for FVS is a branching algorithm combined with the iterative compression technique [20] which runs in O∗(3.619k) time
Summary
Feedback Vertex Set (FVS) is a problem of finding the minimum-size vertex deletion set to make the input graph a forest, which is one of the Karp’s 21 NP-complete problems [18]. The current fastest algorithm is the randomized cut-and-count dynamic programming, the result of the challenge suggests that branching is the best choice in practice This is not so surprising; because the theoretical analysis of branching algorithms is difficult, the proved upper bound of the running time is not so tight, and the worst-case instances for branching algorithms are usually very rare in practice. The theoretically proved running time of other branching algorithms (without the degree-based pruning) are O∗(4k) for the LP-guided branching [16] and O∗(3.619k) for the iterative-compression branching [20] These affairs motivated us to refine the analysis of the highest-degree branching with the degree-based pruning. 1: if k < 0 return No 2: if G is a forest return Yes. 3: Apply reduction rules 1–6. 4: Apply the degree-based pruning. 5: Apply the highest-degree branching
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