The Parameterized Complexity of Domination-type Problems and Application to Linear Codes

We study the parameterized complexity of domination-type problems. (sigma,rho)-domination is a general and unifying framework introduced by Telle: a set D of vertices of a graph G is (sigma,rho)-dominating if for any v in D, |N(v)\cap D| in sigma and for any $v\notin D, |N(v)\cap D| in rho. We mainly show that for any sigma and rho the problem of (sigma,rho)-domination is W[2] when parameterized by the size of the dominating set. This general statement is optimal in the sense that several particular instances of (sigma,rho)-domination are W[2]-complete (e.g. Dominating Set). We also prove that (sigma,rho)-domination is W[2] for the dual parameterization, i.e. when parameterized by the size of the dominated set. We extend this result to a class of domination-type problems which do not fall into the (sigma,rho)-domination framework, including Connected Dominating Set. We also consider problems of coding theory which are related to domination-type problems with parity constraints. In particular, we prove that the problem of the minimal distance of a linear code over Fq is W[2] for both standard and dual parameterizations, and W[1]-hard for the dual parameterization. To prove W[2]-membership of the domination-type problems we extend the Turing-way to parameterized complexity by introducing a new kind of non deterministic Turing machine with the ability to perform `blind' transitions, i.e. transitions which do not depend on the content of the tapes. We prove that the corresponding problem Short Blind Multi-Tape Non-Deterministic Turing Machine is W[2]-complete. We believe that this new machine can be used to prove W[2]-membership of other problems, not necessarily related to dominationComment: 19 pages, 2 figure

Exploiting c\mathbf{c}-Closure in Kernelization Algorithms for Graph Problems

A graph is c-closed if every pair of vertices with at least c common neighbors is adjacent. The c-closure of a graph G is the smallest number such that G is c-closed. Fox et al. [ICALP '18] defined c-closure and investigated it in the context of clique enumeration. We show that c-closure can be applied in kernelization algorithms for several classic graph problems. We show that Dominating Set admits a kernel of size k^O(c), that Induced Matching admits a kernel with O(c^7*k^8) vertices, and that Irredundant Set admits a kernel with O(c^(5/2)*k^3) vertices. Our kernelization exploits the fact that c-closed graphs have polynomially-bounded Ramsey numbers, as we show

Barrier Coverage with Non-uniform Lengths to Minimize Aggregate Movements

Given a line segment I=[0,L], the so-called barrier, and a set of n sensors with varying ranges positioned on the line containing I, the barrier coverage problem is to move the sensors so that they cover I, while minimising the total movement. In the case when all the sensors have the same radius the problem can be solved in O(n log n) time (Andrews and Wang, Algorithmica 2017). If the sensors have different radii the problem is known to be NP-hard to approximate within a constant factor (Czyzowicz et al., ADHOC-NOW 2009). We strengthen this result and prove that no polynomial time rho^{1-epsilon}-approximation algorithm exists unless P=NP, where rho is the ratio between the largest radius and the smallest radius. Even when we restrict the number of sensors that are allowed to move by a parameter k, the problem turns out to be W[1]-hard. On the positive side we show that a ((2+epsilon)rho+2/epsilon)-approximation can be computed in O(n^3/epsilon^2) time and we prove fixed-parameter tractability when parameterized by the total movement assuming all numbers in the input are integers

Polynomial Kernels for Generalized Domination Problems

In this paper, we study the parameterized complexity of a generalized domination problem called the [${\sigma}, {\rho}$] Dominating Set problem. This problem generalizes a large number of problems including the Minimum Dominating Set problem and its many variants. The parameterized complexity of the [${\sigma}, {\rho}$] Dominating Set problem parameterized by treewidth is well studied. Here the properties of the sets ${\sigma}$ and ${\rho}$ that make the problem tractable are identified [1]. We consider a larger parameter and investigate the existence of polynomial sized kernels. When ${\sigma}$ and ${\rho}$ are finite, we identify the exact condition when the [${\sigma}, {\rho}$] Dominating Set problem parameterized by vertex cover admits polynomial kernels. Our lower and upper bound results can also be extended to more general conditions and provably smaller parameters as well.Comment: 19 pages, 6 figure

Longest Common Subsequence on Weighted Sequences

We consider the general problem of the Longest Common Subsequence (LCS) on weighted sequences. Weighted sequences are an extension of classical strings, where in each position every letter of the alphabet may occur with some probability. Previous results presented a PTAS and noticed that no FPTAS is possible unless P=NP. In this paper we essentially close the gap between upper and lower bounds by improving both. First of all, we provide an EPTAS for bounded alphabets (which is the most natural case), and prove that there does not exist any EPTAS for unbounded alphabets unless FPT=W[1]. Furthermore, under the Exponential Time Hypothesis, we provide a lower bound which shows that no significantly better PTAS can exist for unbounded alphabets. As a side note, we prove that it is sufficient to work with only one threshold in the general variant of the problem