Bi-Criteria Approximations for Vertex Deletion Problems and d-Hitting Set

Abstract

We study bi-criteria approximation algorithms for vertex deletion problems in the (k,W) setting, where both the solution size and total weight are bounded simultaneously. Given a graph G, a weight function w:V → ℚ^+, a size bound k, and a weight budget W, a bi-criteria (a,b)-approximation algorithm either certifies that no solution of size at most k and weight at most W exists, or returns a solution of size at most ak and weight at most bW. Parameterizing by the solution size k - rather than the weight budget W - allows our algorithms to handle arbitrary positive rational weights without any lower bound assumption, addressing a fundamental limitation of prior W-parameterized approaches.
We obtain two families of results. For general vertex deletion problems Π-Deletion admitting a polynomial-time weighted α-approximation, we obtain a polynomial-time (α(λ+1),α(1+1/(λ)))-approximation for any λ > 0, a randomized FPT improvement for problems admitting a sampling step, and a deterministic FPT version for problems with bounded obstruction size. For (k,W)-d-Hitting Set, which captures vertex deletion problems with obstruction size at most d, we design a polynomial-time (d,d)-approximation, a parameterized family of ((1-ε)d, d)-approximations improving the size factor below d, and two algorithms that simultaneously push both factors below d: a ((d+1)/2,(d+1)/2)-approximation and a more refined (d-γ,d-γ)-approximation for any γ ∈ (0,(d-1)/2). All algorithms work with arbitrary positive rational weights and are parameterized by the solution size k.
To demonstrate the broad applicability of our framework, we instantiate our results on six well-studied vertex deletion problems: Cluster Vertex Deletion, FVS in Tournaments, Split Vertex Deletion, Feedback Vertex Set, d-Path Vertex Cover, and Pathwidth-One Vertex Deletion. In fact, our general results apply to any vertex deletion problem admitting a polynomial-time weighted approximation algorithm, and the six problems serve as representative examples spanning a range of obstruction structures - from bounded-size obstructions to unbounded ones. For (k,W) setting of Feedback Vertex Set and Pathwidth-One Vertex Deletion, we establish new sampling steps enabling the FPT approximation results. For Pathwidth-One Vertex Deletion, we additionally prove a polynomial-time 3-approximation for the weighted version on general graphs.