Solving Connectivity Problems Parameterized by Treedepth in Single-Exponential Time and Polynomial Space

A breakthrough result of Cygan et al. (FOCS 2011) showed that connectivity problems parameterized by treewidth can be solved much faster than the previously best known time ?^*(2^{?(twlog tw)}). Using their inspired Cut&Count technique, they obtained ?^*(?^tw) time algorithms for many such problems. Moreover, they proved these running times to be optimal assuming the Strong Exponential-Time Hypothesis. Unfortunately, like other dynamic programming algorithms on tree decompositions, these algorithms also require exponential space, and this is widely believed to be unavoidable. In contrast, for the slightly larger parameter called treedepth, there are already several examples of matching the time bounds obtained for treewidth, but using only polynomial space. Nevertheless, this has remained open for connectivity problems. In the present work, we close this knowledge gap by applying the Cut&Count technique to graphs of small treedepth. While the general idea is unchanged, we have to design novel procedures for counting consistently cut solution candidates using only polynomial space. Concretely, we obtain time ?^*(3^d) and polynomial space for Connected Vertex Cover, Feedback Vertex Set, and Steiner Tree on graphs of treedepth d. Similarly, we obtain time ?^*(4^d) and polynomial space for Connected Dominating Set and Connected Odd Cycle Transversal

Multidimensional Binary Vector Assignment problem: standard, structural and above guarantee parameterizations

In this article we focus on the parameterized complexity of the Multidimensional Binary Vector Assignment problem (called \BVA). An input of this problem is defined by $m$ disjoint sets $V^1, V^2, \dots, V^m$, each composed of $n$ binary vectors of size $p$. An output is a set of $n$ disjoint $m$-tuples of vectors, where each $m$-tuple is obtained by picking one vector from each set $V^i$. To each $m$-tuple we associate a $p$ dimensional vector by applying the bit-wise AND operation on the $m$ vectors of the tuple. The objective is to minimize the total number of zeros in these $n$ vectors. mBVA can be seen as a variant of multidimensional matching where hyperedges are implicitly locally encoded via labels attached to vertices, but was originally introduced in the context of integrated circuit manufacturing. We provide for this problem FPT algorithms and negative results ($ETH$-based results, $W$[2]-hardness and a kernel lower bound) according to several parameters: the standard parameter $k$ i.e. the total number of zeros), as well as two parameters above some guaranteed values.Comment: 16 pages, 6 figure

Beyond Max-Cut: \lambda-Extendible Properties Parameterized Above the Poljak-Turz\'{i}k Bound

Poljak and Turz\'ik (Discrete Math. 1986) introduced the notion of \lambda-extendible properties of graphs as a generalization of the property of being bipartite. They showed that for any 0<\lambda<1 and \lambda-extendible property \Pi, any connected graph G on n vertices and m edges contains a subgraph H \in {\Pi} with at least \lambda m+ (1-\lambda)/2 (n-1) edges. The property of being bipartite is 1/2-extendible, and thus this bound generalizes the Edwards-Erd\H{o}s bound for Max-Cut. We define a variant, namely strong \lambda-extendibility, to which the bound applies. For a strongly \lambda-extendible graph property \Pi, we define the parameterized Above Poljak- Turz\'ik (APT) (\Pi) problem as follows: Given a connected graph G on n vertices and m edges and an integer parameter k, does there exist a spanning subgraph H of G such that H \in {\Pi} and H has at least \lambda m + (1-\lambda)/2 (n - 1) + k edges? The parameter is k, the surplus over the number of edges guaranteed by the Poljak-Turz\'ik bound. We consider properties {\Pi} for which APT (\Pi) is fixed- parameter tractable (FPT) on graphs which are O(k) vertices away from being a graph in which each block is a clique. We show that for all such properties, APT (\Pi) is FPT for all 0<\lambda<1. Our results hold for properties of oriented graphs and graphs with edge labels. Our results generalize the result of Crowston et al. (ICALP 2012) on Max-Cut parameterized above the Edwards-Erd\H{o}s bound, and yield FPT algorithms for several graph problems parameterized above lower bounds, e.g., Max q-Colorable Subgraph problem. Our results also imply that the parameterized above-guarantee Oriented Max Acyclic Digraph problem is FPT, thus solving an open question of Raman and Saurabh (Theor. Comput. Sci. 2006).Comment: 23 pages, no figur

A new model for the theta divisor of the cubic threefold

In this paper we give a birational model for the theta divisor of the intermediate Jacobian of a generic cubic threefold $X$. We use the standard realization of $X$ as a conic bundle and a $4-$dimensional family of plane quartics which are totally tangent to the discriminant quintic curve of such a conic bundle structure. The additional data of an even theta characteristic on the curves in the family gives us a model for the theta divisor