Fpt-approximation Algorithm Research Articles

On the Parameterized Approximability of Contraction to Classes of Chordal Graphs

A graph operation that contracts edges is one of the fundamental operations in the theory of graph minors. Parameterized Complexity of editing to a family of graphs by contracting k edges has recently gained substantial scientific attention, and several new results have been obtained. Some important families of graphs, namely, the subfamilies of chordal graphs, in the context of edge contractions, have proven to be significantly difficult than one might expect. In this article, we study the F -Contraction problem, where F is a subfamily of chordal graphs, in the realm of parameterized approximation. Formally, given a graph G and an integer k , F -Contraction asks whether there exists X ⊆ E(G) such that G/X ∈ F and | X | ≤ k . Here, G/X is the graph obtained from G by contracting edges in X . We obtain the following results for the F - Contraction problem: • Clique Contraction is known to be FPT . However, unless NP⊆ coNP/ poly , it does not admit a polynomial kernel. We show that it admits a polynomial-size approximate kernelization scheme ( PSAKS ). That is, it admits a (1 + ε)-approximate kernel with O ( k f(ε)) vertices for every ε > 0. • Split Contraction is known to be W[1]-Hard . We deconstruct this intractability result in two ways. First, we give a (2+ε)-approximate polynomial kernel for Split Contraction (which also implies a factor (2+ε)- FPT -approximation algorithm for Split Contraction ). Furthermore, we show that, assuming Gap-ETH , there is no (5/4-δ)- FPT -approximation algorithm for Split Contraction . Here, ε, δ > 0 are fixed constants. • Chordal Contraction is known to be W[2]-Hard . We complement this result by observing that the existing W[2]-hardness reduction can be adapted to show that, assuming FPT ≠ W[1] , there is no F(k) - FPT -approximation algorithm for Chordal Contraction . Here, F(k) is an arbitrary function depending on k alone. We say that an algorithm is an h(k) - FPT -approximation algorithm for the F -Contraction problem, if it runs in FPT time, and on any input (G, k) such that there exists X ⊆ E(G) satisfying G/X ∈ F and | X | ≤ k , it outputs an edge set Y of size at most h(k) ċ k for which G/Y is in F .

How to Keep an Eye on Small Things

We present new results on two types of guarding problems for polygons. For the first problem, we present an optimal linear time algorithm for computing a smallest set of points that guard a given shortest path in a simple polygon having [Formula: see text] edges. We also prove that in polygons with holes, there is a constant [Formula: see text] such that no polynomial-time algorithm can solve the problem within an approximation factor of [Formula: see text], unless P=NP. For the second problem, we present a [Formula: see text]-FPT algorithm for computing a shortest tour that sees [Formula: see text] specified points in a polygon with [Formula: see text] holes. We also present a [Formula: see text]-FPT approximation algorithm for this problem having approximation factor [Formula: see text]. In addition, we prove that the general problem cannot be polynomially approximated better than by a factor of [Formula: see text], for some constant [Formula: see text], unless P [Formula: see text]NP.

A constant FPT approximation algorithm for hard-capacitated k-means

Hard-capacitated k-means (HCKM) is one of the fundamental problems remaining open in combinatorial optimization and engineering. In HCKM, one is required to partition a given n-point set into k disjoint clusters with known capacity so as to minimize the sum of within-cluster variances. It is known to be at least APX-hard, and most of the work on it has been done from a meta heuristic or bi-criteria approximation perspective. To the best our knowledge, no constant approximation algorithm or existence proof of such an algorithm is known. As our main contribution, we propose an FPT(k) approximation algorithm with constant performance guarantee for HCKM in this paper.

Increasing the lifetime of sensor networks by a data dissemination model based on a new approximation algorithm

Grouping the sensor nodes into clusters can improve the overall scalability and network lifetime of a Wireless Sensor Network (WSN). In a clustered WSN, the Cluster Heads (CHs) bear more traffic load than normal sensor nodes. This is because they not only collect data from all the sensor nodes in their cluster but also aggregate it and send it to the base station, called the sink. Due to this excess traffic load, they die sooner than normal sensor nodes and thus, minimizing the load of the CHs is an important problem called Load Balanced Clustering Problem (LBCP). In this paper, we propose a cluster-based routing protocol for WSNs. It solves LBCP with an fpt-approximation algorithm that has an approximation factor of 1.1, which means it is significantly more precise than the previous approximation factors reported for this problem. The proposed protocol uses an energy-aware routing algorithm to find the optimal routing tree that connects the CHs to the sink. The routing algorithm specifies certain paths for transmitting data to the sink and changes them at certain times, in order to balance the energy consumption of the nodes and increase the network lifetime. The simulation results show that the proposed protocol has a better performance compared to a number of other similar protocols.

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: • Assuming W[1] ≠ FPT, there is no F ( k )- FPT-approximation algorithm for k -DomSet. • Assuming the Exponential Time Hypothesis (ETH), there is no F ( k )-approximation algorithm for k -DomSet that runs in T ( k ) ⋅ n o ( k ) time. • 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 ) ⋅ n k − ε time. • 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 sf W[1] ≠ FPT and (log 1/4 &minus 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 Gap-ETH [Chalermsook et al., FOCS 2017]. Note that, to the best of our knowledge, no running time lower bound of the form n &delta 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.

Solving the load balanced clustering and routing problems in WSNs with an fpt-approximation algorithm and a grid structure

Clustering is an efficient technique in designing routing algorithms for Wireless Sensor Networks (WSNs), which prolongs the network lifetime and leads to scalability. However, in clustered WSNs, the cluster heads consume more energy than the other nodes and thus die sooner. Therefore, the load of sensor nodes must be balanced among the cluster heads in order to prolong the network lifetime. This problem is called load-balanced clustering problem, which is an NP-hard problem. In this paper, we propose an approximation algorithm to solve this problem with an approximation ratio of 1.1. This algorithm runs in fixed-parameter tractable time. We use a virtual grid infrastructure in the network, which makes the algorithm practical for large-scale WSNs. We also propose a routing algorithm based on this structure. The routing algorithm reduces and balances the energy consumption in the network by finding proper paths between each cluster head and the sink. The simulation results show that the proposed algorithm is practical for large-scale WSNs in addition to having a better performance compared with other similar algorithms.

A routing algorithm for wireless sensor networks based on clustering and an fpt-approximation algorithm

Clustering sensor nodes is an effective method for routing in Wireless Sensor Networks (WSNs), which maximizes the network lifetime and reduces the energy consumption. However, in a clustered-WSN, the Cluster Heads (CHs) bear a higher load compared to the other nodes, which leads to their earlier death. Therefore, minimizing the maximum load of the CHs is an important problem, which is called the Load-Balanced Clustering Problem (LBCP). LBCP is an NP-hard problem and the best-known approximation factor for this problem is 1.5. Moreover, it has been shown that there is no polynomial-time approximation algorithm that solves this problem with a better approximation factor. In this paper, we propose a Fixed Parameter Tractable (FPT) approximation algorithm with an approximation factor of 1.2 for LBCP. We also propose an energy-efficient and energy-balanced routing algorithm for routing between the CHs and the sink. The simulation results show that the proposed algorithm is practical for large-scale WSNs and performs better than the other similar algorithms.

Fixed-Parameter Complexity and Approximability of Norm Maximization

The problem of maximizing the $$p$$pth power of a $$p$$p-norm over a halfspace-presented polytope in $$\mathbb {R}^d$$Rd is a convex maximization problem which plays a fundamental role in computational convexity. Mangasarian and Shiau showed in 1986 that this problem is $$\mathbb {NP}$$NP-hard for all values $$p \in \mathbb {N}$$p?N if the dimension $$d$$d of the ambient space is part of the input. In this paper, we use the theory of parameterized complexity to analyze how heavily the hardness of norm maximization relies on the parameter $$d$$d. More precisely, we show that for $$p=1$$p=1 the problem is fixed-parameter tractable (in FPT for short) but that for all $$p >1$$p>1 norm maximization is W[1]-hard. Concerning approximation algorithms for norm maximization, we show that, for fixed accuracy, there is a straightforward approximation algorithm for norm maximization in FPT running time, but there is no FPT-approximation algorithm with a running time depending polynomially on the accuracy. As with the $$\mathbb {NP}$$NP-hardness of norm maximization, the W[1]-hardness immediately carries over to various radius computation tasks in computational convexity.

Completely inapproximable monotone and antimonotone parameterized problems

We prove that weighted circuit satisfiability for monotone or antimonotone circuits has no fixed-parameter tractable approximation algorithm with any approximation ratio function ρ, unless FPT≠W[1]. In particular, not having such an fpt-approximation algorithm implies that these problems have no polynomial-time approximation algorithms with ratio ρ(OPT) for any nontrivial function ρ.

Parameterized Complexity and Local Search Approaches for the Stable Marriage Problem with Ties

We consider the variant of the classical Stable Marriage problem where preference lists can be incomplete and may contain ties. In such a setting, finding a stable matching of maximum size is NP-hard. We study the parameterized complexity of this problem, where the parameter can be the number of ties, the maximum or the overall length of ties. We also investigate the applicability of a local search algorithm for the problem. Finally, we examine the possibilities for giving an FPT algorithm or an FPT approximation algorithm for finding an egalitarian or a minimum regret matching.

Parameterized approximation of dominating set problems

A problem open for many years is whether there is an FPT algorithm that given a graph G and parameter k, either: (1) determines that G has no k- Dominating Set, or (2) produces a dominating set of size at most g ( k ) , where g ( k ) is some fixed function of k. Such an outcome is termed an FPT approximation algorithm. We describe some results that begin to provide some answers. We show that there is no such FPT algorithm for g ( k ) of the form k + c (where c is a fixed constant, termed an additive FPT approximation), unless FPT = W [ 2 ] . We answer the analogous problem completely for the related Independent Dominating Set (IDS) problem, showing that IDS does not admit an FPT approximation algorithm, for any g ( k ) , unless FPT = W [ 2 ] .