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In the {\sc Test Cover} problem we are given a hypergraph $H=(V, \mathcal{E})$ with $|V|=n, |\mathcal{E}|=m$, and we assume that $\mathcal{E}$ is a test cover, i.e. for every pair of vertices $x_i, x_j$, there exists an edge $e \in \mathcal{E}$ such that $|{x_i,x_j}\cap e|=1$. The objective is to find a minimum subset of $\mathcal{E}$ which is a test cover. The problem is used for identification across many areas, and is NP-complete. From a parameterized complexity standpoint, many natural parameterizations of {\sc Test Cover} are either $W[1]$-complete or have no polynomial kernel unless $coNP\subseteq NP/poly$, and thus are unlikely to be solveable efficiently. However, in practice the size of the edges is often bounded. In this paper we study the parameterized complexity of {\sc Test-$r$-Cover}, the restriction of {\sc Test Cover} in which each edge contains at most $r \ge 2$ vertices. In contrast to the unbounded case, we show that the following below-bound parameterizations of {\sc Test-$r$-Cover} are fixed-parameter tractable with a polynomial kernel: (1) Decide whether there exists a test cover of size $n-k$, and (2) decide whether there exists a test cover of size $m-k$, where $k$ is the parameter. In addition, we prove a new lower bound $\lceil \frac{2(n-1)}{r+1} \rceil$ on the minimum size of a test cover when the size of each edge is bounded by $r$. {\sc Test-$r$-Cover} parameterized above this bound is unlikely to be fixed-parameter tractable; in fact, we show that it is para-NP-complete, as it is NP-hard to decide whether an instance of {\sc Test-$r$-Cover} has a test cover of size exactly $\frac{2(n-1)}{r+1}$.

An oriented graph is a directed graph without directed 2-cycles. Poljak and Turz\'{i}k (1986) proved that every connected oriented graph $G$ on $n$ vertices and $m$ arcs contains an acyclic subgraph with at least $\frac{m}{2}+\frac{n-1}{4}$ arcs. Raman and Saurabh (2006) gave another proof of this result and left it as an open question to establish the parameterized complexity of the following problem: does $G$ have an acyclic subgraph with least $\frac{m}{2}+\frac{n-1}{4}+k$ arcs, where $k$ is the parameter? We answer this question by showing that the problem can be solved by an algorithm of runtime $(12k)!n^{O(1)}$. Thus, the problem is fixed-parameter tractable. We also prove that there is a polynomial time algorithm that either establishes that the input instance of the problem is a Yes-instance or reduces the input instance to an equivalent one of size $O(k^2)$.

We study the boundary of tractability for the Max-Cut problem in graphs. Our main result shows that Max-Cut above the Edwards-Erd\H{o}s 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 m/2 + (n-1)/4 + k in time 2^O(k)n^4, or decides that no such cut exists. This answers a long-standing open question from parameterized complexity that has been posed several 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.

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 $r_1,..., r_m$ be the number of literals in the clauses of $F$. Then $asat(F)=\sum_{i=1}^m (1-2^{-r_i})$ is the expected number of clauses satisfied by a random truth assignment (the truth values to the variables are distributed uniformly and independently). It is well-known that, in polynomial time, one can find a truth assignment satisfying at least $asat(F)$ clauses. In the parameterized problem MaxSat-AA, we are to decide whether there is a truth assignment satisfying at least $asat(F)+k$ clauses, where $k$ is the parameter. We prove that MaxSat-AA is para-NP-complete and, thus, MaxSat-AA is not fixed-parameter tractable unless P$=$NP. This is in sharp contrast to MaxLin2-AA which was recently proved to be fixed-parameter tractable by Crowston et al. (arXiv:1104.1135v3). In fact, we consider a more refined version of {\sc MaxSat-AA}, {\sc Max-$r(n)$-Sat-AA}, where $r_j\le r(n)$ for each $j$. Alon {\em et al.} (SODA 2010) proved that if $r=r(n)$ is a constant, then {\sc Max-$r$-Sat-AA} is fixed-parameter tractable. We prove that {\sc Max-$r(n)$-Sat-AA} is para-NP-complete for $r(n)=\lceil \log n\rceil.$ We also prove that assuming the exponential time hypothesis, {\sc Max-$r(n)$-Sat-AA} is not in XP already for any $r(n)\ge \log \log n +\phi(n)$, where $\phi(n)$ is any unbounded strictly increasing function. This lower bound on $r(n)$ cannot be decreased much further as we prove that {\sc Max-$r(n)$-Sat-AA} is (i) in XP for any $r(n)\le \log \log n - \log \log \log n$ and (ii) fixed-parameter tractable for any $r(n)\le \log \log n - \log \log \log n - \phi(n)$, where $\phi(n)$ is any unbounded strictly increasing function. The proof uses some results on {\sc MaxLin2-AA}.

In the problem {\sc Min-DESC}, we are given a digraph $D$ and an integer $k$, and asked if there exists a set $A'$ of at most $k$ arcs in $D$, such that if we remove the arcs of $A'$, in the resulting digraph every strong component is Eulerian. {\sc Min-DESC} is NP-hard; Cechl\'{a}rov\'{a} and Schlotter (IPEC 2010) asked if the problem is fixed-parameter tractable when parameterized by $k$. We consider the subproblem of{\sc Min-DESC} when $D$ is a tournament. We show that this problem is fixed-parameter tractable with respect to $k$.

In the Max Lin-2 problem we are given a system $S$ of $m$ linear equations in $n$ variables over $\mathbb{F}_2$ in which Equation $j$ is assigned a positive integral weight $w_j$ 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 $W=w_1+... +w_m$; $W/2$ is a tight lower bound on the optimal solution of Max Lin-2. Mahajan et al. (J. Comput. Syst. Sci. 75, 2009) stated the following parameterized version of Max Lin-2: decide whether there is an assignment of values to the variables that satisfies equations of total weight at least $W/2+k$, where $k$ is the parameter. They asked whether this parameterized problem is fixed-parameter tractable, i.e., can be solved in time $f(k)(nm)^{O(1)}$, where $f(k)$ is an arbitrary computable function in $k$ only. Their question remains open, but using some probabilistic inequalities and, in one case, a Fourier analysis inequality, Gutin et al. (IWPEC 2009) proved that the problem is fixed-parameter tractable in three special cases. In this paper we significantly extend two of the three special cases using only tools from combinatorics. We show that one of our results can be used to obtain a combinatorial proof that another problem from Mahajan et al. (J. Comput. Syst. Sci. 75, 2009), Max $r$-SAT above the Average, is fixed-parameter tractable for each $r\ge 2.$ Note that Max $r$-SAT above the Average has been already shown to be fixed-parameter tractable by Alon et al. (SODA 2010), but the paper used the approach of Gutin et al. (IWPEC 2009).