Compression via Matroids: A Randomized Polynomial Kernel for Odd Cycle Transversal

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 \BigOh(4^kkmn) 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 \BigOh(4^k), 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. 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 \BigOh(\sqrt{\log n})-approximation (Agarwal et al., STOC 2005), we get a reduction of the instance to size \BigOh(k^{4.5}), implying a randomized polynomial kernelization.Comment: Minor changes to agree with SODA 2012 version of the pape

Covering Small Independent Sets and Separators with Applications to Parameterized Algorithms

We present two new combinatorial tools for the design of parameterized algorithms. The first is a simple linear time randomized algorithm that given as input a $d$-degenerate graph $G$ and an integer $k$, outputs an independent set $Y$, such that for every independent set $X$ in $G$ of size at most $k$, the probability that $X$ is a subset of $Y$ is at least $\left({(d+1)k \choose k} \cdot k(d+1)\right)^{-1}$.The second is a new (deterministic) polynomial time graph sparsification procedure that given a graph $G$, a set $T = \{\{s_1, t_1\}, \{s_2, t_2\}, \ldots, \{s_\ell, t_\ell\}\}$ of terminal pairs and an integer $k$, returns an induced subgraph $G^\star$ of $G$ that maintains all the inclusion minimal multicuts of $G$ of size at most $k$, and does not contain any $(k+2)$-vertex connected set of size $2^{{\cal O}(k)}$. In particular, $G^\star$ excludes a clique of size $2^{{\cal O}(k)}$ as a topological minor. Put together, our new tools yield new randomized fixed parameter tractable (FPT) algorithms for Stable $s$-$t$ Separator, Stable Odd Cycle Transversal and Stable Multicut on general graphs, and for Stable Directed Feedback Vertex Set on $d$-degenerate graphs, resolving two problems left open by Marx et al. [ACM Transactions on Algorithms, 2013]. All of our algorithms can be derandomized at the cost of a small overhead in the running time.Comment: 35 page

Parameterization Above a Multiplicative Guarantee

Parameterization above a guarantee is a successful paradigm in Parameterized Complexity. To the best of our knowledge, all fixed-parameter tractable problems in this paradigm share an additive form defined as follows. Given an instance (I,k) of some (parameterized) problem ? with a guarantee g(I), decide whether I admits a solution of size at least (at most) k+g(I). Here, g(I) is usually a lower bound (resp. upper bound) on the maximum (resp. minimum) size of a solution. Since its introduction in 1999 for Max SAT and Max Cut (with g(I) being half the number of clauses and half the number of edges, respectively, in the input), analysis of parameterization above a guarantee has become a very active and fruitful topic of research. We highlight a multiplicative form of parameterization above a guarantee: Given an instance (I,k) of some (parameterized) problem ? with a guarantee g(I), decide whether I admits a solution of size at least (resp. at most) k ? g(I). In particular, we study the Long Cycle problem with a multiplicative parameterization above the girth g(I) of the input graph, and provide a parameterized algorithm for this problem. Apart from being of independent interest, this exemplifies how parameterization above a multiplicative guarantee can arise naturally. We also show that, for any fixed constant ?>0, multiplicative parameterization above g(I)^(1+?) of Long Cycle yields para-NP-hardness, thus our parameterization is tight in this sense. We complement our main result with the design (or refutation of the existence) of algorithms for other problems parameterized multiplicatively above girth

Directed Feedback Vertex Set is Fixed-Parameter Tractable

We resolve positively a long standing open question regarding the fixed-parameter tractability of the parameterized Directed Feedback Vertex Set problem. In particular, we propose an algorithm which solves this problem in $O(8^kk!*poly(n))$.Comment: 14 page