Point Line Cover: The Easy Kernel is Essentially Tight

The input to the NP-hard Point Line Cover problem (PLC) consists of a set $P$ of $n$ points on the plane and a positive integer $k$, and the question is whether there exists a set of at most $k$ lines which pass through all points in $P$. A simple polynomial-time reduction reduces any input to one with at most $k^2$ points. We show that this is essentially tight under standard assumptions. More precisely, unless the polynomial hierarchy collapses to its third level, there is no polynomial-time algorithm that reduces every instance $(P,k)$ of PLC to an equivalent instance with $O(k^{2-\epsilon})$ points, for any $\epsilon>0$. This answers, in the negative, an open problem posed by Lokshtanov (PhD Thesis, 2009). Our proof uses the machinery for deriving lower bounds on the size of kernels developed by Dell and van Melkebeek (STOC 2010). It has two main ingredients: We first show, by reduction from Vertex Cover, that PLC---conditionally---has no kernel of total size $O(k^{2-\epsilon})$ bits. This does not directly imply the claimed lower bound on the number of points, since the best known polynomial-time encoding of a PLC instance with $n$ points requires $\omega(n^{2})$ bits. To get around this we build on work of Goodman et al. (STOC 1989) and devise an oracle communication protocol of cost $O(n\log n)$ for PLC; its main building block is a bound of $O(n^{O(n)})$ for the order types of $n$ points that are not necessarily in general position, and an explicit algorithm that enumerates all possible order types of n points. This protocol and the lower bound on total size together yield the stated lower bound on the number of points. While a number of essentially tight polynomial lower bounds on total sizes of kernels are known, our result is---to the best of our knowledge---the first to show a nontrivial lower bound for structural/secondary parameters

Finding Even Subgraphs Even Faster

Problems of the following kind have been the focus of much recent research in the realm of parameterized complexity: Given an input graph (digraph) on $n$ vertices and a positive integer parameter $k$, find if there exist $k$ edges (arcs) whose deletion results in a graph that satisfies some specified parity constraints. In particular, when the objective is to obtain a connected graph in which all the vertices have even degrees---where the resulting graph is \emph{Eulerian}---the problem is called Undirected Eulerian Edge Deletion. The corresponding problem in digraphs where the resulting graph should be strongly connected and every vertex should have the same in-degree as its out-degree is called Directed Eulerian Edge Deletion. Cygan et al. [\emph{Algorithmica, 2014}] showed that these problems are fixed parameter tractable (FPT), and gave algorithms with the running time $2^{O(k \log k)}n^{O(1)}$. They also asked, as an open problem, whether there exist FPT algorithms which solve these problems in time $2^{O(k)}n^{O(1)}$. In this paper we answer their question in the affirmative: using the technique of computing \emph{representative families of co-graphic matroids} we design algorithms which solve these problems in time $2^{O(k)}n^{O(1)}$. The crucial insight we bring to these problems is to view the solution as an independent set of a co-graphic matroid. We believe that this view-point/approach will be useful in other problems where one of the constraints that need to be satisfied is that of connectivity

Improved approximation bounds for Vector Bin Packing

In this paper we propose an improved approximation scheme for the Vector Bin Packing problem (VBP), based on the combination of (near-)optimal solution of the Linear Programming (LP) relaxation and a greedy (modified first-fit) heuristic. The Vector Bin Packing problem of higher dimension (d \geq 2) is not known to have asymptotic polynomial-time approximation schemes (unless P = NP). Our algorithm improves over the previously-known guarantee of (ln d + 1 + epsilon) by Bansal et al. [1] for higher dimensions (d > 2). We provide a {\theta}(1) approximation scheme for certain set of inputs for any dimension d. More precisely, we provide a 2-OPT algorithm, a result which is irrespective of the number of dimensions d.Comment: 15 pages, 3 algorithm

Hitting forbidden minors: Approximation and Kernelization

We study a general class of problems called F-deletion problems. In an F-deletion problem, we are asked whether a subset of at most $k$ vertices can be deleted from a graph $G$ such that the resulting graph does not contain as a minor any graph from the family F of forbidden minors. We obtain a number of algorithmic results on the F-deletion problem when F contains a planar graph. We give (1) a linear vertex kernel on graphs excluding $t$-claw $K_{1,t}$, the star with $t$ leves, as an induced subgraph, where $t$ is a fixed integer. (2) an approximation algorithm achieving an approximation ratio of $O(\log^{3/2} OPT)$, where $OPT$ is the size of an optimal solution on general undirected graphs. Finally, we obtain polynomial kernels for the case when F contains graph $\theta_c$ as a minor for a fixed integer $c$. The graph $\theta_c$ consists of two vertices connected by $c$ parallel edges. Even though this may appear to be a very restricted class of problems it already encompasses well-studied problems such as {\sc Vertex Cover}, {\sc Feedback Vertex Set} and Diamond Hitting Set. The generic kernelization algorithm is based on a non-trivial application of protrusion techniques, previously used only for problems on topological graph classes

The effect of girth on the kernelization complexity of Connected Dominating Set

In the Connected Dominating Set problem we are given as input a graph $G$ and a positive integer $k$, and are asked if there is a set $S$ of at most $k$ vertices of $G$ such that $S$ is a dominating set of $G$ and the subgraph induced by $S$ is connected. This is a basic connectivity problem that is known to be NP-complete, and it has been extensively studied using several algorithmic approaches. In this paper we study the effect of excluding short cycles, as a subgraph, on the kernelization complexity of Connected Dominating Set. Kernelization algorithms are polynomial-time algorithms that take an input and a positive integer $k$ (the parameter) and output an equivalent instance where the size of the new instance and the new parameter are both bounded by some function $g(k)$. The new instance is called a $g(k)$ kernel for the problem. If $g(k)$ is a polynomial in $k$ then we say that the problem admits polynomial kernels. The girth of a graph $G$ is the length of a shortest cycle in $G$. It turns out that Connected Dominating Set is ``hard\u27\u27 on graphs with small cycles, and becomes progressively easier as the girth increases. More specifically, we obtain the following interesting trichotomy: Connected Dominating Set (a) does not have a kernel of any size on graphs of girth $3$ or $4$ (since the problem is W[2]-hard); (b) admits a $g(k)$ kernel, where $g(k)$ is $k^{O(k)}$, on graphs of girth $5$ or $6$ but has no polynomial kernel (unless the Polynomial Hierarchy (PH) collapses to the third level) on these graphs; (c) has a cubic ($O(k^3)$) kernel on graphs of girth at least $7$. While there is a large and growing collection of parameterized complexity results available for problems on graph classes characterized by excluded minors, our results add to the very few known in the field for graph classes characterized by excluded subgraphs

Business Merchant Data Consolidator

A consolidated view of the account data of a merchant is provided. Whenever a transaction occurs, the merchant has to collect each payment through the merchant terminal. Further, existing systems do not provide option to view the receivable accounts and reconciliation view of the transactions. For this purpose, a unique identity is provided to the merchant. In every transaction the unique identity is attached to the invoice made by the merchant. This provides a consolidated notification from a platform to the merchants instead of sending multiple notifications. Further, the merchant can have a consolidated view of the receivables either in all locations or for a specific location. By providing the unique identity the merchant receives the direct credit of payment instead of collecting each payment through terminal