C S Karthik

On the parameterized complexity of approximating dominating set

We study the parameterized complexity of approximating the k-Dominating Set (domset) problem where an integer k and a graph G on n vertices are given as input, and the goal is to find a dominating set of size at most F(k) · k whenever the graph G has a dominating set of size k. When such an algorithm runs in time T(k)poly(n) (i.e., FPT-time) for some computable function T, it is said to be an F(k)-FPT-approximation algorithm for k-domset. Whether such an algorithm exists is listed in the seminal book of Downey and Fellows (2013) as one of the ”most infamous” open problems in Parameterized Complexity. This work gives an almost complete answer to this question by showing the non-existence of such an algorithm under W[1]≠FPT and further providing tighter running time lower bounds under stronger hypotheses. Specifically, we prove the following for every computable functions T, F and every constant ε > 0: (i) Assuming W[1]≠FPT, there is no F(k)-FPT-approximation algorithm for k-domset, (ii) Assuming the Exponential Time Hypothesis (ETH), there is no F(k)-approximation algorithm for k-domset that runs in T(k)no(k) time, (iii) Assuming the Strong Exponential Time Hypothesis (SETH), for every integer k ≥ 2, there is no F(k)-approximation algorithm for k-domset that runs in T(k)nk − ε time, (iv) Assuming the k-sum Hypothesis, for every integer k ≥ 3, there is no F(k)-approximation algorithm for k-domset that runs in T(k) n⌈ k/2 ⌉ − ε time. Previously, only constant ratio FPT-approximation algorithms were ruled out under W[1]≠FPT and (log1/4 − ε k)-FPT-approximation algorithms were ruled out under ETH [Chen and Lin, FOCS 2016]. Recently, the non-existence of an F(k)-FPT-approximation algorithm for any function F was shown under gapETH [Chalermsook et al., FOCS 2017]. Note that, to the best of our knowledge, no running time lower bound of the form nδ k for any absolute constant δ > 0 was known before even for any constant factor inapproximation ratio. Our results are obtained by establishing a connection between communication complexity and hardness of approximation, generalizing the ideas from a recent breakthrough work of Abboud et al. [FOCS 2017]. Specifically, we show that to prove hardness of approximation of a certain parameterized variant of the label cover problem, it suffices to devise a specific protocol for a communication problem that depends on which hypothesis we rely on. Each of these communication problems turns out to be either a well studied problem or a variant of one; this allows us to easily apply known techniques to solve them.

Building Efficient and Compact Data Structures for Simplicial Complexes

The Simplex Tree (ST) is a recently introduced data structure that can represent abstract simplicial complexes of any dimension and allows efficient implementation of a large range of basic operations on simplicial complexes. In this paper, we show how to optimally compress the ST while retaining its functionalities. In addition, we propose two new data structures called the Maximal Simplex Tree and the Simplex Array List. We analyze the compressed ST, the Maximal Simplex Tree, and the Simplex Array List under various settings.

Did the train reach its destination: The complexity of finding a witness

Recently, Dohrau et al. studied a zero-player game on switch graphs and proved that deciding the termination of the game is in NP

An Efficient Representation for Filtrations of Simplicial Complexes

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On the Sensitivity Conjecture for Disjunctive Normal Forms

coNP. In this short paper, we show that the search version of this game on switch graphs, i.e., the task of finding a witness of termination (or of non-termination) is in PLS.

On the Complexity of Closest Pair via Polar-Pair of Point-Sets

A filtration over a simplicial complex K is an ordering of the simplices of K such that all prefixes in the ordering are subcomplexes of K. Filtrations are at the core of Persistent Homology, a major tool in Topological Data Analysis. To represent the filtration of a simplicial complex, the entire filtration can be appended to any data structure that explicitly stores all the simplices of the complex such as the Hasse diagram or the recently introduced Simplex Tree [Algorithmica’14]. However, with the popularity of various computational methods that need to handle simplicial complexes, and with the rapidly increasing size of the complexes, the task of finding a compact data structure that can still support efficient queries is of great interest. This direction has been recently pursued for the case of maintaining simplicial complexes. For instance, Boissonnat et al. [Algorithmica’17] considered storing the simplices that are maximal with respect to inclusion and Attali et al. [IJCGA’12] considered storing the simplices that block the expansion of the complex. Nevertheless, so far there has been no data structure that compactly stores the filtration of a simplicial complex, while also allowing the efficient implementation of basic operations on the complex. In this article, we propose a new data structure called the Critical Simplex Diagram (CSD), which is a variant of the Simplex Array List [Algorithmica’17]. Our data structure allows one to store in a compact way the filtration of a simplicial complex and allows for the efficient implementation of a large range of basic operations. Moreover, we prove that our data structure is essentially optimal with respect to the requisite storage space. Finally, we show that the CSD representation admits fast construction algorithms for Flag complexes and relaxed Delaunay complexes.