We study a refined framework of parameterized complexity theory where the parameter dependence of fixed-parameter tractable algorithms is not arbitrary, but restricted by a function in some familyF . For every family F of functions, this yields a notion of F -fixed parameter tractability. If F is the class of all polynomially bounded functions, then F -fixed parameter tractability coincides wit...

In MaxSat, we are given a CNF formula F with n variables and m clauses and asked to find a truth assignment satisfying the maximum number of clauses. Let r1, . . . , rm be the number of literals in the clauses of F . Then asat(F ) = ∑m i=1(1 − 2 −ri) is the expected number of clauses satisfied by a random truth assignment (the truth values to the variables are distributed uniformly and independ...

In the Max Lin-2 problem we are given a system S of m linear equations in n variables over F2 in which Equation j is assigned a positive integral weight wj for each j. We wish to find an assignment of values to the variables which maximizes the total weight of satisfied equations. This problem generalizes Max Cut. The expected weight of satisfied equations is W/2, where W = w1 + · · ·+ wm; W/2 ...

Once having classified an NP-hard problem fixed-parameter tractable with respect to a certain parameter, the race for the most efficient fixed-parameter algorithm starts. Herein, the attention usually focuses on improving the running time factor exponential in the considered parameter, and, in case of kernelization algorithms, to improve the bound on the kernel size. Both from a practical as we...

The fixed-parameter tractability of graph isomorphism is an open problem with respect to a number of natural parameters, such as tree-width, genus and maximum degree. We show that graph isomorphism is fixed-parameter tractable when parameterized by the tree-depth of the graph. We also extend this result to a parameter generalizing both tree-depth and max-leaf-number by deploying new variants of...

For many fixed-parameter problems that are trivially soluable in polynomial time, such as (k-)DOMINATING SET, essentially no better algorithm is presently known than the one which tries all possible solutions. Other problems, such as (k-)FEEDBACK VERTEX SET, exhibit fixed-parameter tractability: for each fixed k the problem is soluable in time bounded by a polynomial of degree c, where c is a c...

Repair operators are often used for constraint handling in constrained combinatorial optimization. We investigate the (1+1) EA equipped with a tailored jump-and-repair operation that can be to probabilistically repair infeasible offspring graph problems. Instead of evolving candidate solutions entire graph, we expand genotype allow develop parallel feasible solution together growing subset inst...

Is it fixed-parameter tractable to find an out-directed spanning tree of a given digraph with the maximum possible k of leaves? More precisely, consider a digraph D. An out-tree of D is a subtree in which all vertices but one have in-degree exactly 1. The vertices of out-degree 0 are called leaves. Let `(D) denote the maximum number of leaves of any out-tree in D. An out-branching of D is an ou...

We present a new algorithm that can output the rank-decomposition of width at most k of a graph if such exists. For that we use an algorithm that, for an input matroid represented over a fixed finite field, outputs its branch-decomposition of width at most k if such exists. This algorithm works also for partitioned matroids. Both these algorithms are fixed-parameter tractable, that is, they run...

We introduce h-modularity, a structural parameter of CNF formulas, and present algorithms that render the decision problem SAT and the model counting problem #SAT fixed-parameter tractable when parameterized by h-modularity. The new parameter is defined in terms of a partition of clauses of the given CNF formula into strongly interconnected communities which are sparsely interconnected with eac...