Robert Crowston

Simultaneously Satisfying Linear Equations Over F_2: MaxLin2 and Max-r-Lin2 Parameterized Above Average.

In the parameterized problem MaxLin2-AA[

Max-cut parameterized above the edwards-erdős bound

k

Parameterized complexity of maxsat above average

], we are given a system with variables x_1,...,x_n consisting of equations of the form Product_{i in I}x_i = b, where x_i,b in {-1, 1} and I is a nonempty subset of {1,...,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+k, where W is the total weight of all equations and k is the parameter (if k=0, the possibility is assured). We show that MaxLin2-AA[k] has a kernel with at most O(k^2 log k) variables and can be solved in time 2^{O(k log k)}(nm)^{O(1)}. This solves an open problem of Mahajan et al. (2006). The problem Max-r-Lin2-AA[k,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 a theorem on Max-

Parameterized study of the test cover problem

r

Fixed-Parameter Tractability of Workflow Satisfiability in the Presence of Seniority Constraints

-Lin2-AA[k,r] which implies that Max-r-Lin2-AA[k,r] has a kernel with at most (2k-1)r variables, improving a number of results including one by Kim and Williams (2010). The theorem also implies a lower bound on the maximum of a function f that maps {-1,1}^n to the set of reals and whose Fourier expansion (which is a multilinear polynomial) is of degree r. We show applicability of the lower bound by giving a new proof of the Edwards-Erdos bound (each connected graph on n vertices and m edges has a bipartite subgraph with at least m/2 +(n-1)/4 edges) and obtaining a generalization.

Max-Cut Parameterized Above the Edwards-Erdős Bound

We study the boundary of tractability for the Max-Cut problem in graphs. Our main result shows that Max-Cut above the Edwards-Erdős bound is fixed-parameter tractable: we give an algorithm that for any connected graph with n vertices and m edges finds a cut of size

Parameterizations of Test Cover with Bounded Test Sizes

\frac{m}{2} + \frac{n-1}{4} + k

Parameterized Complexity of Satisfying Almost All Linear Equations over

in time 2O(k)·n4, or decides that no such cut exists. This answers a long-standing open question from parameterized complexity that has been posed a number of times over the past 15 years. Our algorithm is asymptotically optimal, under the Exponential Time Hypothesis, and is strengthened by a polynomial-time computable kernel of polynomial size.