Parameterized Algorithm Of Running Time Research Articles

Dealing with several parameterized problems by random methods

In this paper, we apply random methods to deal with several parameterized problems. For the Parameterized Weighted P3-Packing problem, by randomly partitioning the vertices in given graph, a tripartite graph can be obtained. We prove that the Parameterized Weighted P3-Packing problem can be solved in polynomial time on tripartite graphs. Based on the algorithm on tripartite graphs, a randomized parameterized algorithm of running time O⁎(32k) is given for the Parameterized Weighted P3-Packing problem. For the Parameterized Weighted Load Coloring problem, by randomly partitioning the vertices in given graph into two parts and studying the structure properties of the connected components in two parts, a randomized parameterized algorithm of running time O⁎(11.32k) is presented. For the Parameterized Claw-free Edge Deletion problem on Diamond-free Graphs, by combining random with branching methods, a parameterized algorithm of running time O⁎(2.42k) is given.

Parameterized algorithms for Edge Biclique and related problems

Maximum Edge Biclique and related problems have wide applications in management science, bioinformatics, etc. In this paper, we study parameterized algorithms for Parameterized Edge Biclique problem, Parameterized Edge Biclique Packing problem, Parameterized Biclique Edge Deletion problem, and Parameterized Bipartite Biclique Clustering problem. For the Parameterized Edge Biclique problem, the current best result is of running time O⁎(2k), and we give a parameterized algorithm of running time O(nk2.5k⌈k⌉), where k is the parameter and n is the number of vertices in the given graph. For the Parameterized Edge Biclique Packing problem, based on randomized divide-and-conquer technique, a parameterized algorithm of running time O⁎(k⌈k⌉logk4(2k−1)t)) is given, where k is the parameter and t is the number of bicliques in the solution. We study the Parameterized Biclique Edge Deletion problem on bipartite graphs and general graphs, and give parameterized algorithms of running time O⁎(2k) and O⁎(3k), respectively. For the Parameterized Bipartite Biclique Clustering problem, based on modular decomposition method, a kernel of size O(k2) and a parameterized algorithm of running time O(2.42k(n+m)) are presented, where k is the parameter, n is the number of vertices, and m is the number of edges in the given graph.

Partition on trees with supply and demand: Kernelization and algorithms

Network reconfiguration is an important research topic in the planning and operation of power distribution networks. In this paper, we study the partition problem on trees with supply and demand from parameterized computation perspective. We analyze the relationship between supply nodes and demand nodes, and give four reduction rules, which result in a kernel of size O(k2) for the problem. Based on branching technique, a parameterized algorithm of running time O⁎(2.828k) is presented.

Randomized parameterized algorithms for $$P_2$$ P 2 -Packing and Co-Path Packing problems

In this paper, we study the Parameterized $$P_2$$ P 2 -Packing problem and Parameterized Co-Path Packing problem from random perspective. For the Parameterized $$P_2$$ P 2 -Packing problem, based on the structure analysis of the problem and using random partition technique, a randomized parameterized algorithm of running time $$O^*(6.75^k)$$ O ? ( 6 . 75 k ) is obtained, improving the current best result $$O^*(8^k)$$ O ? ( 8 k ) . For the Parameterized Co-Path Packing problem, we firstly study the kernel and randomized algorithm for the degree-bounded instance, where each vertex in the instance has degree at most three. A kernel of size $$20k$$ 20 k and a randomized algorithm of running time $$O^*(2^k)$$ O ? ( 2 k ) are given for the Parameterized Co-Path Packing problem with bounded degree constraint. By applying iterative compression technique and based on the randomized algorithm for degree bounded problem, a randomized algorithm of running time $$O^*(3^k)$$ O ? ( 3 k ) is given for the Parameterized Co-Path Packing problem.

An [formula omitted]-time parameterized algorithm for the 3-set packing problem

A parameterized algorithm for the well-known NP-complete problem 3- set packing is presented. The algorithm is based on the following new approaches: (1) an efficient algorithm for problem instances in which one element is known for each of the 3-sets in a solution packing (i.e., type-1 instances); (2) an efficient algorithm for problem instances in which at least two elements are known for each of the 3-sets in a solution packing (i.e., type-2 instances); and (3) an effective decomposition of a general instance of 3 -set packing into two instances of type-1 and type-2, respectively, whose base sets are disjoint. This technique results in an improved parameterized algorithm of running time O ∗ ( 3.5 3 3 k ) for the 3 -set packing problem, improving the previous best upper bound O ∗ ( 4.6 1 3 k ) .

An improved kernelization for [formula omitted]-packing

The P 2 -packing problem is a generalized matching problem and is known to be NP-hard. A kernelization algorithm for the parameterized P 2 -packing problem is a polynomial time algorithm that on an instance ( G , k ) of the problem produces another instance ( G ′ , k ′ ) such that k ′ ⩽ k and that ( G , k ) is a yes-instance if and only if ( G ′ , k ′ ) is a yes-instance. The graph G ′ in the reduced instance is called a kernel. We provide new structural analysis and develop a new kernelization algorithm for the problem that produces a kernel of at most 7 k vertices. This improves the previous best kernelization algorithm that produces a kernel of 15 k vertices. Based on the new kernel, we present a parameterized algorithm of running time O ∗ ( 17.66 k ) for the problem, improving the previous best upper bound O ∗ ( 32 k ) .

Parameterized algorithms for weighted matching and packing problems

The weighted m - d matching and m - set packing problems ( m ≥ 3 ) are usually solved through approximation algorithms. In this paper, we define the parameterized versions of these problems and study their algorithms. For the weighted m - set packing problem, we develop a parameterized algorithm of running time O ( 12 . 8 m k n O ( 1 ) ) , which is based on the recently improved color-coding technology and dynamic programming. By refining the techniques, we also develop a more efficient parameterized algorithm of running time O ( 12 . 8 ( m − 1 ) k n O ( 1 ) ) for the weighted m - d matching problem, which is a restricted version of the weighted m - set packing problem.

Improved Parameterized Set Splitting Algorithms: A Probabilistic Approach

In this paper, we study parameterized algorithms for the set splitting problem, for both weighted and unweighted versions. First, we develop a new and effective technique based on a probabilistic method that allows us to develop a simpler and more efficient deterministic kernelization algorithm for the unweighted set splitting problem. We then propose a randomized algorithm for the weighted set splitting problem that is based on a new subset partition technique and has its running time bounded by O *(2 k ), which is significantly better than that of the previous best deterministic algorithm (which only works for the simpler unweighted set splitting problem) of running time O *(2.65 k ). We also show that our algorithm can be de-randomized, which leads to a deterministic parameterized algorithm of running time O *(4 k ) for the weighted set splitting problem and gives the first proof that the problem is fixed-parameter tractable.

Using Nondeterminism to Design Efficient Deterministic Algorithms

In this paper we illustrate how nondeterminism can be used conveniently and effectively in designing efficient deterministic algorithms. In particular, our method gives a parameterized algorithm of running time O((5.7 k)k n) for the 3-D matching problem, which significantly improves the previous algorithm by Downey et al. The algorithm can be generalized to yield an improved algorithm for the r-D matching problem for any positive integer r. The method can also be employed in designing deterministic algorithms for other optimization problems as well.