Satisfying More Than Half of a System of Linear Equations Over GF(2): A Multivariate Approach

In the parameterized problem MaxLin2-AA[k ], we are given a system with variables x1,…,xnx1,…,xn consisting of equations of the form ∏i∈Ixi=b∏i∈Ixi=b, where xi,b∈{−1,1}xi,b∈{−1,1} and I⊆[n]I⊆[n], each equation has a positive integral weight, and we are to decide whether it is possible to simultaneously satisfy equations of total weight at least W/2+kW/2+k, where W is the total weight of all equations and k is the parameter (it is always possible for k=0k=0). We show that MaxLin2-AA[k ] has a kernel with at most View the MathML sourceO(k2logk) variables and can be solved in time 2O(klogk)(nm)O(1)2O(klogk)(nm)O(1). This solves an open problem of Mahajan et al. (2006). The problem Max-r-Lin2-AA[k,rk,r] is the same as MaxLin2-AA[k] with two differences: each equation has at most r variables and r is the second parameter. We prove that Max-r-Lin2-AA[k,rk,r] has a kernel with at most (2k−1)r(2k−1)r variables

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

On the Parameterized Complexity and Kernelization of the Workflow Satisfiability Problem

A workflow specification defines a set of steps and the order in which those steps must be executed. Security requirements may impose constraints on which groups of users are permitted to perform subsets of those steps. A workflow specification is said to be satisfiable if there exists an assignment of users to workflow steps that satisfies all the constraints. An algorithm for determining whether such an assignment exists is important, both as a static analysis tool for workflow specifications, and for the construction of run-time reference monitors for workflow management systems. Finding such an assignment is a hard problem in general, but work by Wang and Li in 2010 using the theory of parameterized complexity suggests that efficient algorithms exist under reasonable assumptions about workflow specifications. In this paper, we improve the complexity bounds for the workflow satisfiability problem. We also generalize and extend the types of constraints that may be defined in a workflow specification and prove that the satisfiability problem remains fixed-parameter tractable for such constraints. Finally, we consider preprocessing for the problem and prove that in an important special case, in polynomial time, we can reduce the given input into an equivalent one, where the number of users is at most the number of steps. We also show that no such reduction exists for two natural extensions of this case, which bounds the number of users by a polynomial in the number of steps, provided a widely-accepted complexity-theoretical assumption holds