Balanced Judicious Bipartition is Fixed-Parameter Tractable

The family of judicious partitioning problems, introduced by Bollob\u27as and Scott to the field of extremal combinatorics, has been extensively studied from a structural point of view for over two decades. This rich realm of problems aims to counterbalance the objectives of classical partitioning problems such as Min Cut, Min Bisection and Max Cut. While these classical problems focus solely on the minimization/maximization of the number of edges crossing the cut, judicious (bi)partitioning problems ask the natural question of the minimization/maximization of the number of edges lying in the (two) sides of the cut. In particular, Judicious Bipartition (JB) seeks a bipartition that is "judicious" in the sense that neither side is burdened by too many edges, and Balanced JB also requires that the sizes of the sides themselves are "balanced" in the sense that neither of them is too large. Both of these problems were defined in the work by Bollob\u27as and Scott, and have received notable scientific attention since then. In this paper, we shed light on the study of judicious partitioning problems from the viewpoint of algorithm design. Specifically, we prove that BJB is FPT (which also proves that JB is FPT)

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

Induced Matching below Guarantees: Average Paves the Way for Fixed-Parameter Tractability

In this work, we study the Induced Matching problem: Given an undirected graph $G$ and an integer $\ell$, is there an induced matching $M$ of size at least $\ell$? An edge subset $M$ is an induced matching in $G$ if $M$ is a matching such that there is no edge between two distinct edges of $M$. Our work looks into the parameterized complexity of Induced Matching with respect to "below guarantee" parameterizations. We consider the parameterization $u - \ell$ for an upper bound $u$ on the size of any induced matching. For instance, any induced matching is of size at most $n / 2$ where $n$ is the number of vertices, which gives us a parameter $n / 2 - \ell$. In fact, there is a straightforward $9^{n/2 - \ell} \cdot n^{O(1)}$-time algorithm for Induced Matching [Moser and Thilikos, J. Discrete Algorithms]. Motivated by this, we ask: Is Induced Matching FPT for a parameter smaller than $n / 2 - \ell$? In search for such parameters, we consider $MM(G) - \ell$ and $IS(G) - \ell$, where $MM(G)$ is the maximum matching size and $IS(G)$ is the maximum independent set size of $G$. We find that Induced Matching is presumably not FPT when parameterized by $MM(G) - \ell$ or $IS(G) - \ell$. In contrast to these intractability results, we find that taking the average of the two helps -- our main result is a branching algorithm that solves Induced Matching in $49^{(MM(G) + IS(G))/ 2 - \ell} \cdot n^{O(1)}$ time. Our algorithm makes use of the Gallai-Edmonds decomposition to find a structure to branch on