A Tight Approximation Algorithm for the Cluster Vertex Deletion Problem

Abstract

We give the first 2-approximation algorithm for the cluster vertex deletion problem. This is tight, since approximating the problem within any constant factor smaller than 2 is UGC-hard. Our algorithm combines the previous approaches, based on the local ratio technique and the management of true twins, with a novel construction of a "good" cost function on the vertices at distance at most 2 from any vertex of the input graph.As an additional contribution, we also study cluster vertex deletion from the polyhedral perspective, where we prove almost matching upper and lower bounds on how well linear programming relaxations can approximate the problem.9: c H (v) := 0, ∀u = v, c H (u) := c H ′ (u) 10: return c H 11: end if 12: else {N 2 (v 0 ) = ∅}

A simple 7/3-approximation algorithm for feedback vertex set in tournaments

“…As further evidence, for the related problem of cluster vertex deletion [1], we showed that one round of Sherali-Adams has an integrality gap of 5/2, and for every ǫ > 0 there exists r ∈ N such that r rounds of Sherali-Adams has integrality gap at most 2 + ǫ. Indeed, this work can be seen as unifying the approaches of [9] and some of the polyhedral results of [1].…”

“…Therefore, for NP-hard optimization problems (such as FVST), one should not expect that SA r (P ) = P I for some constant r. However, as we will see, good approximations of P I can be extremely useful if we want to approximately optimize over P I . Indeed, despite some recent results [1,5,7,10,13], we feel that the Sherali-Adams hierarchy is underutilized in the design of approximation algorithms, and hope that our work will inspire further applications.…”

“…The fact that the integrality gap of SA 1 for FVST is at most 7/3 follows from Theorem 1. For the other inequality, note that for every tournament T , the all 3 7vector is feasible for SA 1 (T ) (by setting x uv = 1 7 for all uv). On the other hand, it is easy to show via the probabilistic method that for a random n-node tournament T , OPT(T, 1 T ) = n − O(log n) with high probability.…”

A tight approximation algorithm for the cluster vertex deletion problem

“…This result is unconditional: it does not rely on P = NP nor the Unique Games Conjecture. We refer the reader to the full version of our paper [4] for precise definitions and for the proofs of Theorems 3 and 4.…”

“…2.4 we prove Theorem 2. Due to space constraints we omit some proofs and details, for which we refer to the full version of our paper [4]. In particular, [4] contains the proof that Theorem 2 implies Theorem 1, which can be proven similarly as in [10, Proof of Theorem 1], as well as a complexity analysis of Algorithm 1.…”