Compression via matroids: a randomized polynomial kernel for odd cycle transversal

• Authors: Stefan Kratsch;Magnus Wahlström
• Venue: Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
• Year: 2012

The Odd Cycle Transversal problem (OCT) asks whether a given graph can be made bipartite by deleting at most k of its vertices. In a breakthrough result Reed, Smith, and Vetta (Operations Research Letters, 2004) gave a O(4kkmn) time algorithm for it, the first algorithm with polynomial runtime of uniform degree for every fixed k. It is known that this implies a polynomial-time compression algorithm that turns OCT instances into equivalent instances of size at most O(4k), a so-called kernelization. Since then the existence of a polynomial kernel for OCT, i.e., a kernelization with size bounded polynomially in k, has turned into one of the main open questions in the study of kernelization. Despite the impressive progress in the area, including the recent development of lower bound techniques (Bodlaender et al., ICALP 2008; Fortnow and Santhanam, STOC 2008) and meta-results on kernelizations for graph problems on planar and other sparse graph classes (Bodlaender et al., FOCS 2009; Fomin et al., SODA 2010), the existence of a polynomial kernel for OCT has remained open, even when the input is restricted to be planar. This work provides the first (randomized) polynomial kernelization for OCT. We introduce a novel kernelization approach based on matroid theory, where we encode all relevant information about a problem instance into a matroid with a representation of size polynomial in k. For OCT, the matroid is built to allow us to simulate the computation of the iterative compression step of the algorithm of Reed, Smith, and Vetta, applied (for only one round) to an approximate odd cycle transversal which it is aiming to shrink to size k. The process is randomized with one-sided error exponentially small in k, where the result can contain false positives but no false negatives, and the size guarantee is cubic in the size of the approximate solution. Combined with an O(√ log n)-approximation (Agarwal et al., STOC 2005), we get a reduction of the instance to size O(k4.5), implying a randomized polynomial kernelization. Interestingly, the known lower bound techniques can be seen to exclude randomized kernels that produce no false negatives, as in fact they exclude even co-nondeterministic kernels (Dell and van Melkebeek, STOC 2010). Therefore, our result also implies that deterministic kernels for OCT cannot be excluded by the known machinery.