Parameterized Complexity of Vertex Cover Variants

Important variants of theVERTEX COVER problem (among others, CONNECTED VERTEX COVER, CAPACITATED VERTEX COVER, and MAXIMUM PARTIAL VERTEX COVER) have been intensively studied in terms of polynomial-time approximability. By way of contrast, their parameterized complexity has so far been completely open. We close this gap here by showing that, with the size of the desired vertex cover as the parameter, CONNECTED VERTEX COVER and CAPACITATED VERTEX COVER are both fixed-parameter tractable while MAXIMUM PARTIAL VERTEX COVER is W[1]-complete. This answers two open questions from the literature. The results extend to several closely related problems. Interestingly, although the considered variants of VERTEX COVER behave very similar in terms of constant factor approximability, they display a wide range of different characteristics when investigating their parameterized complexities.     

Vertex Cover Problem Parameterized Above and Below Tight Bounds

We study the well-known Vertex Cover problem parameterized above and below tight bounds. We show that two of the parameterizations (both were suggested by Mahajan et al. in J. Comput. Syst. Sci. 75(2):137–153, 2009) are fixed-parameter tractable and two other parameterizations are W[1]-hard (one of them is, in fact, W[2]-hard).     

随机图点覆盖1度顶点核化算法分析

将随机图引入参数计算领域,利用随机图统计和概率分布等特性,从全局和整体上研究参数化点覆盖问题1度点核化过程中问题的核及度分布演变的内在机制和变化规律,并得出关于随机图1度点核化强度与顶点平均度关系及随机图点覆盖问题的决策与度分布关系的两个重要推论.最后分别从MIPS和BIND提取数据进行1度核化实验和分析.初步结果表明,对随机图点覆盖问题的分析方法不仅具有理论上的意义,而且随着问题随机度的大小而对问题有不同程度的把握能力.     

Vertex Cover Kernelization Revisited

An important result in the study of polynomial-time preprocessing shows that there is an algorithm which given an instance (G,k) of Vertex Cover outputs an equivalent instance (G′,k′) in polynomial time with the guarantee that G′ has at most 2k′ vertices (and thus $\mathcal{O}((k')^{2})$ edges) with k′≤k. Using the terminology of parameterized complexity we say that k-Vertex Cover has a kernel with 2k vertices. There is complexity-theoretic evidence that both 2k vertices and Θ(k ²) edges are optimal for the kernel size. In this paper we consider the Vertex Cover problem with a different parameter, the size $\mathop{\mathrm{\mbox{\textsc{fvs}}}}(G)$ of a minimum feedback vertex set for G. This refined parameter is structurally smaller than the parameter k associated to the vertex covering number $\mathop{\mathrm{\mbox {\textsc{vc}}}}(G)$ since $\mathop{\mathrm{\mbox{\textsc{fvs}}}}(G)\leq\mathop{\mathrm{\mbox{\textsc{vc}}}}(G)$ and the difference can be arbitrarily large. We give a kernel for Vertex Cover with a number of vertices that is cubic in $\mathop{\mathrm{\mbox{\textsc{fvs}}}}(G)$ : an instance (G,X,k) of Vertex Cover, where X is a feedback vertex set for G, can be transformed in polynomial time into an equivalent instance (G′,X′,k′) such that |V(G′)|≤2k and $|V(G')| \in\mathcal{O}(|X'|^{3})$ . A similar result holds when the feedback vertex set X is not given along with the input. In sharp contrast we show that the Weighted Vertex Cover problem does not have a polynomial kernel when parameterized by the cardinality of a given vertex cover of the graph unless NP ? coNP/poly and the polynomial hierarchy collapses to the third level.     

基于最小点覆盖及多参数方法的关键蛋白识别

针对已有方法对关键蛋白识别度不高的现状,认为进一步提高识别度有两条途径：一是发现与关键蛋白关系更密切的参数,二是充分挖掘现有参数的信息并进行有效地整合。由于点覆盖在网络（图）拓扑结构上的重要地位而研究将其引入关键蛋白质的识别中：针对算法的复杂性引进参数计算的相关算法将复杂度大幅度降低的同时对蛋白质网络进行最小点覆盖分析并获得一种新的拓扑参数-点覆盖参数,相关分析表明该参数与关键蛋白有着密切的联系。进一步研究发现,参数之间相关性的大小在很大程度上预示它们所蕴含的关键蛋白信息之间互补性的强弱,根据这一发现探讨利用包括点覆盖在内的各个参数的有限信息进行有效整合,仿真结果证实该方法能明显提高关键蛋白识别度。     

Crown Structures for Vertex Cover Kernelization

Crown structures in a graph are defined and shown to be useful in kernelization algorithms for the classic vertex cover problem. Two vertex cover kernelization methods are discussed. One, based on linear programming, has been in prior use and is known to produce predictable results, although it was not previously associated with crowns. The second, based on crown structures, is newer and much faster, but produces somewhat variable results. These two methods are studied and compared both theoretically and experimentally with each other and with older, more primitive kernelization algorithms. Properties of crowns and methods for identifying them are discussed. Logical connections between linear programming and crown reductions are established. It is shown that the problem of finding an induced crown-free subgraph, and the problem of finding a crown of maximum size in an arbitrary graph, are solvable in polynomial time.     

On Making a Distinguished Vertex of Minimum Degree by Vertex Deletion

For directed and undirected graphs, we study how to make a distinguished vertex the unique minimum-(in)degree vertex through deletion of a minimum number of vertices. The corresponding NP-hard optimization problems are motivated by applications concerning control in elections and social network analysis. Continuing previous work for the directed case, we show that the problem is W[2]-hard when parameterized by the graph’s feedback arc set number, whereas it becomes fixed-parameter tractable when combining the parameters “feedback vertex set number” and “number of vertices to delete”. For the so far unstudied undirected case, we show that the problem is NP-hard and W[1]-hard when parameterized by the “number of vertices to delete”. On the positive side, we show fixed-parameter tractability for several parameterizations measuring tree-likeness. In particular, we provide a dynamic programming algorithm for graphs of bounded treewidth and a vertex-linear problem kernel with respect to the parameter “feedback edge set number”. On the contrary, we show a non-existence result concerning polynomial-size problem kernels for the combined parameter “vertex cover number and number of vertices to delete”, implying corresponding non-existence results when replacing vertex cover number by treewidth or feedback vertex set number.     

Private Approximation of Clustering and Vertex Cover

Private approximation of search problems deals with finding approximate solutions to search problems while disclosing as little information as possible. The focus of this work is on private approximation of the vertex cover problem and two well studied clustering problems – k-center and k-median. Vertex cover was considered in (Beimel, Carmi, Nissim, and Weinreb, STOC, 2006) and we improve their infeasibility results. Clustering algorithms are frequently applied to sensitive data, and hence are of interest in the contexts of secure computation and private approximation. We show that these problems do not admit private approximations, or even approximation algorithms that are allowed to leak a significant number of bits of information. For the vertex cover problem we show a tight infeasibility result: every algorithm that ρ(n)-approximates vertex-cover must leak Ω(n/ρ(n)) bits (where n is the number of vertices in the graph). For the clustering problems we prove that even approximation algorithms with a poor approximation ratio must leak Ω(n) bits (where n is the number of points in the instance). For these results we develop new proof techniques, which are simpler and more intuitive than those in Beimel et al., and yet allow for stronger infeasibility results. Our proofs rely on the hardness of the promise problem where a unique optimal solution exists (Valiant and Vazirani, Theoretical Computer Science, 1986), on the hardness of approximating witnesses for NP-hard problems (Kumar and Sivakumar, CCC, (1999) and Feige, Langberg, and Nissim, APPROX, (2000)), and on a simple random embedding of instances into bigger instances.     

最小顶点覆盖快速降阶算法

通过定义判别函数来判别顶点覆盖作用的优劣,得出一个把顶点加入到最小顶点覆盖集的一般化规则,并得出该规则在多种具体情况下的应用定理,在此基础上给出了一个快速降阶算法,该算法能确定某些顶点应该在最小顶点覆盖中,某些顶点不应该在最小顶点覆盖中,达到降低原问题的规模和求解难度的目的.该算法既可以单独使用,又可以与算法结合来达到更好的结果,文中还给出了应用实例及其分析.     

顶点覆盖问题线性内核算法

参数复杂性作为算法研究的一个重要分支近10年在国际上受到了广泛的关注,线性内核问题作为参数复杂性研究的一类重要问题被广泛研究.主要给出了顶点覆盖问题的线性内核算法,在国内首次从理论上证明了顶点覆盖问题存在线性内核.算法首先通过顶点覆盖问题的2近似算法,将图的顶点集合分成两个顶点集合A,B,进而通过一系列规约将原始图的顶点覆盖问题转换到新图的顶点覆盖问题,然后证明了新图的顶点数目至多为2k,并且2k是这个问题的下界(k为参数具体定义见文章).     

Fixed-Parameter Evolutionary Algorithms and the Vertex Cover Problem

In this paper, we consider multi-objective evolutionary algorithms for the Vertex Cover problem in the context of parameterized complexity. We consider two different measures for the problem. The first measure is a very natural multi-objective one for the use of evolutionary algorithms and takes into account the number of chosen vertices and the number of edges that remain uncovered. The second fitness function is based on a linear programming formulation and proves to give better results. We point out that both approaches lead to a kernelization for the Vertex Cover problem. Based on this, we show that evolutionary algorithms solve the vertex cover problem efficiently if the size of a minimum vertex cover is not too large, i.e., the expected runtime is bounded by O(f(OPT)?n ^c ), where c is a constant and f a function that only depends on OPT. This shows that evolutionary algorithms are randomized fixed-parameter tractable algorithms for the vertex cover problem.     

一种求解顶点覆盖问题的混合遗传算法

顶点覆盖问题是一个NP难问题,在排序、计算机网络等现实生活中有许多的应用。使用基本遗传算法进行搜索时,存在着局部搜索能力较弱的不足,本文提出了一种新的求解最小顶点覆盖问题的混合遗传算法,将基本遗传算法与局部优化策略相结合,改善遗传算法的局部搜索能力,加快求解该问题的速度。对几种典型无向图的实验证实了新方法的有效性,其整体性能优于现有的一些顶点覆盖问题遗传算法。     

Cutting Planes and the Parameter Cutwidth

We introduce the parameter cutwidth for the Cutting Planes (CP) system of Gomory and Chvátal. We provide linear lower bounds on cutwidth for two simple polytopes. Considering CP as a propositional refutation system, one can see that the cutwidth of a CNF contradiction F is always bound above by the Resolution width of F. We provide an example proving that the converse fails: there is an F which has constant cutwidth, but has Resolution width Ω(n). Following a standard method for converting an FO sentence ψ, without finite models, into a sequence of CNFs, F _ψ,n , we provide a classification theorem for CP based on the sum cutwidth plus rank. Specifically, the cutwidth + rank of F _ψ,n is bound by a constant c (depending on ψ only) iff ψ has no (infinite) models. This result may be seen as a relative of various gap theorems extant in the literature.     

Enumerate and Expand: Improved Algorithms for Connected Vertex Cover and Tree Cover

We present a new method of solving graph problems related to Vertex Cover by enumerating and expanding appropriate sets of nodes. As an application, we obtain dramatically improved runtime bounds for two variants of the Vertex Cover problem. In the case of Connected Vertex Cover, we take the upper bound from O ^*(6 ^k ) to O ^*(2.7606 ^k ) without large hidden factors. For Tree Cover, we show a complexity of O ^*(3.2361 ^k ), improving over the previous bound of O ^*((2k) ^k ). In the process, faster algorithms for solving subclasses of the Steiner tree problem on graphs are investigated. Supported by the DFG under grant RO 927/6-1 (TAPI).     

一种求解平面图的最小顶点覆盖算法

最小顶点覆盖问题是图论中经典的组合优化问题,在实际生活中有着广泛的应用价值。根据最小顶点覆盖与最大独立集在图论中事实上是属于等价问题这一特性,从最大独立集的角度出发,根据最大独立集的特性,设计了一种求解简单平面图的最大独立集算法,从而求出最小顶点覆盖。通过实验结果的比对验证算法的正确性和有效性。     

Optimizing Polynomial-Time Solutions to a Network Weighted Vertex Cover Game

Weighted vertex cover (WVC) is one of the most important combinatorial optimization problems. In this paper, we provide a new game optimization to achieve efficiency and time of solutions for the WVC problem of weighted networks. We first model the WVC problem as a general game on weighted networks. Under the framework of a game, we newly define several cover states to describe the WVC problem. Moreover, we reveal the relationship among these cover states of the weighted network and the strict Nash equilibriums (SNEs) of the game. Then, we propose a game-based asynchronous algorithm (GAA), which can theoretically guarantee that all cover states of vertices converging in an SNE with polynomial time. Subsequently, we improve the GAA by adding 2-hop and 3-hop adjustment mechanisms, termed the improved game-based asynchronous algorithm (IGAA), in which we prove that it can obtain a better solution to the WVC problem than using a the GAA. Finally, numerical simulations demonstrate that the proposed IGAA can obtain a better approximate solution in promising computation time compared with the existing representative algorithms.