Property Testing on k-Vertex-Connectivity of?Graphs

We present an algorithm for testing the k-vertex-connectivity of graphs with the given maximum degree. The time complexity of the algorithm is independent of the number of vertices and edges of graphs. Fixed degree bound d, a graph G with n vertices and a maximum degree at most d is called ε-far from k-vertex-connectivity when at least $\frac{\epsilon dn}{2}$ edges must be added to or removed from G to obtain a k-vertex-connected graph with a maximum degree at most d. The algorithm always accepts every graph that is k-vertex-connected and rejects every graph that is ε-far from k-vertex-connectivity with a probability of at least 2/3. The algorithm runs in $O(d(\frac{c}{\epsilon d})^{k}\log\frac {1}{\epsilon d})$ time (c>1 is a constant) for (k?1)-vertex-connected graphs, and in $O(d(\frac{ck}{\epsilon d})^{k}\log\frac{k}{\epsilon d})$ time (c>1 is a constant) for general graphs. It is the first constant-time k-vertex-connectivity testing algorithm for general k≥4.     

Tight Bounds for Distributed Minimum-Weight Spanning Tree Verification

This paper introduces the notion of distributed verification without preprocessing. It focuses on the Minimum-weight Spanning Tree (MST) verification problem and establishes tight upper and lower bounds for the time and message complexities of this problem. Specifically, we provide an MST verification algorithm that achieves simultaneously $\tilde{O}(m)$ messages and $\tilde{O}(\sqrt{n} + D)$ time, where m is the number of edges in the given graph G, n is the number of nodes, and D is G’s diameter. On the other hand, we show that any MST verification algorithm must send $\tilde{\varOmega}(m)$ messages and incur $\tilde{\varOmega}(\sqrt{n} + D)$ time in worst case. Our upper bound result appears to indicate that the verification of an MST may be easier than its construction, since for MST construction, both lower bounds of $\tilde{\varOmega}(m)$ messages and $\tilde{\varOmega}(\sqrt{n} + D)$ time hold, but at the moment there is no known distributed algorithm that constructs an MST and achieves simultaneously $\tilde{O}(m)$ messages and $\tilde{O}(\sqrt{n} + D)$ time. Specifically, the best known time-optimal algorithm (using ${\tilde{O}}(\sqrt {n} + D)$ time) requires O(m+n ^3/2) messages, and the best known message-optimal algorithm (using ${\tilde{O}}(m)$ messages) requires O(n) time. On the other hand, our lower bound results indicate that the verification of an MST is not significantly easier than its construction.     

Resource augmentation for uniprocessor and multiprocessor partitioned scheduling of sporadic real-time tasks

Although the earliest-deadline-first (EDF) policy is known to be optimal for preemptive real-time task scheduling in uniprocessor systems, the schedulability analysis problem has recently been shown to be $\mathit{co}\mathcal{NP}$ -hard. Therefore, approximation algorithms, and in particular, approximations based on resource augmentation have attracted a lot of attention for both uniprocessor and multiprocessor systems. Resource augmentation based approximations assume a certain speedup of the processor(s). Using the notion of approximate demand bound function (dbf), in this paper we show that for uniprocessor systems the resource augmentation factor is at most $\frac{2e-1}{e} \approx1.6322$ , where e is the Euler number. We approximate the dbf using a linear approximation when the analysis interval length of interest is larger than the relative deadline of the task. For identical multiprocessor systems with M processors and constrained-deadline task sets, we show that the deadline-monotonic partitioning (that has been proposed by Baruah and Fisher) with the approximate dbf leads to an approximation factor of $\frac{3e-1}{e}-\frac{1}{M} \approx 2.6322-\frac{1}{M}$ with respect to resource augmentation. We also show that the corresponding factor is $3-\frac{1}{M}$ for arbitrary-deadline task sets. The best known results so far were $3-\frac{1}{M}$ for constrained-deadline tasks and $4-\frac {2}{M}$ for arbitrary-deadline ones. Our tighter analysis exploits the structure of the approximate dbf directly and uses the processor utilization violations (which were ignored in all previous analysis) for analyzing resource augmentation factors. We also provide concrete input instances to show that the lower bound on the resource augmentation factor for uniprocessor systems—using the above approximate dbf—is 1.5, and the corresponding bound is 2.5 for identical multiprocessor systems with an arbitrary order of fitting and a large number of processors. Further, we also provide a polynomial-time approximation scheme (PTAS) to derive near-optimal solutions under the assumption that the ratio of the maximum relative deadline to the minimum relative deadline of tasks is a constant, which is a more relaxed assumption compared to the assumptions required for deriving such a PTAS in the past.     

Augmenting Edge-Connectivity between Vertex Subsets

Given a directed or undirected graph G=(V,E), a collection ${\mathcal{R}}=\{(S_{i},T_{i}) \mid i=1,2,\ldots,|{\mathcal{R}}|, S_{i},T_{i} \subseteq V, S_{i} \cap T_{i} =\emptyset\}$ of two disjoint subsets of V, and a requirement function $r: {\mathcal{R}} \to\mathbb{R}_{+}$ , we consider the problem (called area-to-area edge-connectivity augmentation problem) of augmenting G by a smallest number of new edges so that the resulting graph $\hat{G}$ satisfies $d_{\hat{G}}(X)\geq r(S,T)$ for all X?V, $(S,T) \in{\mathcal{R}}$ with S?X?V?T, where d _G (X) denotes the degree of a vertex set X in G. This problem can be regarded as a natural generalization of the global, local, and node-to-area edge-connectivity augmentation problems. In this paper, we show that there exists a constant c such that the problem is inapproximable within a ratio of $c\log{|{\mathcal{R}}|}$ , unless P=NP, even restricted to the directed global node-to-area edge-connectivity augmentation or undirected local node-to-area edge-connectivity augmentation. We also provide an ${\mathrm{O}}(\log{|{\mathcal{R}}|})$ -approximation algorithm for the area-to-area edge-connectivity augmentation problem, which is a natural extension of Kortsarz and Nutov’s algorithm (Kortsarz and Nutov, J. Comput. Syst. Sci., 74:662–670, 2008). This together with the negative result implies that the problem is ${\varTheta}(\log{|{\mathcal{R}}|})$ -approximable, unless P=NP, which solves open problems for node-to-area edge-connectivity augmentation in Ishii et al. (Algorithmica, 56:413–436, 2010), Ishii and Hagiwara (Discrete Appl. Math., 154:2307–2329, 2006), Miwa and Ito (J. Oper. Res. Soc. Jpn., 47:224–243, 2004). Furthermore, we characterize the node-to-area and area-to-area edge-connectivity augmentation problems as the augmentation problems with modulotone and k-modulotone functions.     

The Rank-Width of Edge-Coloured Graphs

A C-coloured graph is a graph, that is possibly directed, where the edges are coloured with colours from the set C. Clique-width is a complexity measure for C-coloured graphs, for finite sets C. Rank-width is an equivalent complexity measure for undirected graphs and has good algorithmic and structural properties. It is in particular related to the vertex-minor relation. We discuss some possible extensions of the notion of rank-width to C-coloured graphs. There is not a unique natural notion of rank-width for C-coloured graphs. We define two notions of rank-width for them, both based on a coding of C-coloured graphs by ${\mathbb{F}}^{*}$ -graphs— $\mathbb {F}$ -coloured graphs where each edge has exactly one colour from $\mathbb{F}\setminus \{0\},\ \mathbb{F}$ a field—and named respectively $\mathbb{F}$ -rank-width and $\mathbb {F}$ -bi-rank-width. The two notions are equivalent to clique-width. We then present a notion of vertex-minor for $\mathbb{F}^{*}$ -graphs and prove that $\mathbb{F}^{*}$ -graphs of bounded $\mathbb{F}$ -rank-width are characterised by a list of $\mathbb{F}^{*}$ -graphs to exclude as vertex-minors (this list is finite if $\mathbb{F}$ is finite). An algorithm that decides in time O(n ³) whether an $\mathbb{F}^{*}$ -graph with n vertices has $\mathbb{F}$ -rank-width (resp. $\mathbb{F}$ -bi-rank-width) at most k, for fixed k and fixed finite field $\mathbb{F}$ , is also given. Graph operations to check MSOL-definable properties on $\mathbb{F}^{*}$ -graphs of bounded $\mathbb{F}$ -rank-width (resp. $\mathbb{F}$ -bi-rank-width) are presented. A specialisation of all these notions to graphs without edge colours is presented, which shows that our results generalise the ones in undirected graphs.     

Leader election using loneliness detection

We consider the problem of leader election (LE) in single-hop radio networks with synchronized time slots for transmitting and receiving messages. We assume that the actual number n of processes is unknown, while the size u of the ID space is known, but is possibly much larger. We consider two types of collision detection: strong (SCD), whereby all processes detect collisions, and weak (WCD), whereby only non-transmitting processes detect collisions. We introduce loneliness detection (LD) as a key subproblem for solving LE in WCD systems. LD informs all processes whether the system contains exactly one process or more than one. We show that LD captures the difference in power between SCD and WCD, by providing an implementation of SCD over WCD and LD. We present two algorithms that solve deterministic and probabilistic LD in WCD systems with time costs of ${\mathcal{O}(\log \frac{u}{n})}$ and ${\mathcal{O}(\min( \log \frac{u}{n}, \frac{\log (1/\epsilon)}{n}))}$ , respectively, where ${\epsilon}$ is the error probability. We also provide matching lower bounds. Assuming LD is solved, we show that SCD systems can be emulated in WCD systems with factor-2 overhead in time. We present two algorithms that solve deterministic and probabilistic LE in SCD systems with time costs of ${\mathcal{O}(\log u)}$ and ${\mathcal{O}(\min ( \log u, \log \log n + \log (\frac{1}{\epsilon})))}$ , respectively, where ${\epsilon}$ is the error probability. We provide matching lower bounds.     

A new PAC bound for intersection-closed concept classes

For hyper-rectangles in $\mathbb{R}^{d}$ Auer (1997) proved a PAC bound of $O(\frac{1}{\varepsilon}(d+\log \frac{1}{\delta}))$ , where $\varepsilon$ and $\delta$ are the accuracy and confidence parameters. It is still an open question whether one can obtain the same bound for intersection-closed concept classes of VC-dimension $d$ in general. We present a step towards a solution of this problem showing on one hand a new PAC bound of $O(\frac{1}{\varepsilon}(d\log d + \log \frac{1}{\delta}))$ for arbitrary intersection-closed concept classes, complementing the well-known bounds $O(\frac{1}{\varepsilon}(\log \frac{1}{\delta}+d\log \frac{1}{\varepsilon}))$ and $O(\frac{d}{\varepsilon}\log \frac{1}{\delta})$ of Blumer et al. and (1989) and Haussler, Littlestone and Warmuth (1994). Our bound is established using the closure algorithm, that generates as its hypothesis the intersection of all concepts that are consistent with the positive training examples. On the other hand, we show that many intersection-closed concept classes including e.g. maximum intersection-closed classes satisfy an additional combinatorial property that allows a proof of the optimal bound of $O(\frac{1}{\varepsilon}(d+\log \frac{1}{\delta}))$ . For such improved bounds the choice of the learning algorithm is crucial, as there are consistent learning algorithms that need $\Omega(\frac{1}{\varepsilon}(d\log\frac{1}{\varepsilon} +\log\frac{1}{\delta}))$ examples to learn some particular maximum intersection-closed concept classes.     

Orthogonal Graph Drawing with Flexibility Constraints

Traditionally, the quality of orthogonal planar drawings is quantified by either the total number of bends, or the maximum number of bends per edge. However, this neglects that in typical applications, edges have varying importance. In this work, we investigate an approach that allows to specify the maximum number of bends for each edge individually, depending on its importance. We consider a new problem called FlexDraw that is defined as follows. Given a planar graph G=(V,E) on n vertices with maximum degree 4 and a function $\operatorname{flex}: E \longrightarrow\mathbb{N}_{0}$ that assigns a flexibility to each edge, does G admit a planar embedding on the grid such that each edge e has at most $\operatorname{flex}(e)$ bends? Note that in our setting the combinatorial embedding of G is not fixed. FlexDraw directly extends the problem β-embeddability asking whether G can be embedded with at most β bends per edge. We give an algorithm with running-time O(n ²) solving FlexDraw when the flexibility of each edge is positive. This includes 1-embeddability as a special case and thus closes the complexity gap between 0-embeddability, which is $\mathcal{NP}$ -hard to decide, and 2-embeddability, which is efficiently solvable since every planar graph with maximum degree 4 admits a 2-embedding except for the octahedron. In addition to the polynomial-time algorithm we show that FlexDraw is $\mathcal{NP}$ -hard even if the edges with flexibility 0 induce a tree or a union of disjoint stars.     

Mining frequent subgraphs over uncertain graph databases under probabilistic semantics

Frequent subgraph mining has been extensively studied on certain graph data. However, uncertainty is intrinsic in graph data in practice, but there is very few work on mining uncertain graph data. This paper focuses on mining frequent subgraphs over uncertain graph data under the probabilistic semantics. Specifically, a measure called ${\varphi}$ -frequent probability is introduced to evaluate the degree of recurrence of subgraphs. Given a set of uncertain graphs and two real numbers ${0 < \varphi, \tau < 1}$ , the goal is to quickly find all subgraphs with ${\varphi}$ -frequent probability at least τ. Due to the NP-hardness of the problem and to the #P-hardness of computing the ${\varphi}$ -frequent probability of a subgraph, an approximate mining algorithm is proposed to produce an ${(\varepsilon, \delta)}$ -approximate set Π of “frequent subgraphs”, where ${0 < \varepsilon < \tau}$ is error tolerance, and 0 <?δ?< 1 is a confidence bound. The algorithm guarantees that (1) any frequent subgraph S is contained in Π with probability at least ((1 ? δ) /2) ^s , where s is the number of edges in S; (2) any infrequent subgraph with ${\varphi}$ -frequent probability less than ${\tau - \varepsilon}$ is contained in Π with probability at most δ/2. The theoretical analysis shows that to obtain any frequent subgraph with probability at least 1 ? Δ, the input parameter δ of the algorithm must be set to at most ${1 - 2 (1 - \Delta)^{1 / \ell_{\max}}}$ , where 0 <?Δ <?1, and ? _max is the maximum number of edges in frequent subgraphs. Extensive experiments on real uncertain graph data verify that the proposed algorithm is practically efficient and has very high approximation quality. Moreover, the difference between the probabilistic semantics and the expected semantics on mining frequent subgraphs over uncertain graph data has been discussed in this paper for the first time.     

Highly efficient remote preparation of an arbitrary three-qubit state via a four-qubit cluster state and an EPR state

An efficient protocol for remotely preparing an arbitrary three-qubit state is devised with a four-qubit cluster state and an Einstein–Podolsky–Rosen state as the shared quantum resource. Using an appropriate set of eight-qubit mutually orthogonal measurement basis, the remote three-qubit preparation is successfully completed with the probability of ${\frac{1}{8}}$ in general case. Then to achieve our concerns of improving the probability of this protocol, some special ensembles of three-qubit states are minutely investigated. As a result, it is shown that the total probability of the RSP protocol, in these particular cases, can be improved to ${\frac{1}{4}}$ and ${\frac{1}{2}}$ , respectively, or even that the RSP protocol can be realized with unit success probability.     

Efficient broadcasting in radio networks with long-range interference

We study broadcasting, also known as one-to-all communication, in synchronous radio networks with known topology modeled by undirected (symmetric) graphs, where the interference range of a node is likely exceeding its transmission range. In this model, if two nodes are connected by a transmission edge they can communicate directly. On the other hand, if two nodes are connected by an interference edge they cannot communicate directly and transmission of one node disables recipience of any message at the other node. For a network $G,$ we term the smallest integer $d$ , s.t., for any interference edge $e$ there exists a simple path formed of at most $d$ transmission edges connecting the endpoints of $e$ as its interference distance $d_I$ . In this model the schedule of transmissions is precomputed in advance. It is based on the full knowledge of the size and the topology (including location of transmission and interference edges) of the network. We are interested in the design of fast broadcasting schedules that are energy efficient, i.e., based on a bounded number of transmissions executed at each node. We adopt $n$ as the number of nodes, $D_T$ is the diameter of the subnetwork induced by the transmission edges, and $\varDelta $ refers to the maximum combined degree (formed of transmission and interference edges) of the network. We contribute the following new results: (1) We prove that for networks with the interference distance $d_I\ge 2$ any broadcasting schedule requires at least $D_T+\varOmega (\varDelta \cdot \frac{\log {n}}{\log {\varDelta }})$ rounds. (2) We provide for networks modeled by bipartite graphs an algorithm that computes $1$ -shot (each node transmits at most once) broadcasting schedules of length $O(\varDelta \cdot \log {n})$ . (3) The main result of the paper is an algorithm that computes a $1$ -shot broadcasting schedule of length at most $4 \cdot D_T + O(\varDelta \cdot d_I \cdot \log ^4{n})$ for networks with arbitrary topology. Note that in view of the lower bound from (1) if $d_I$ is poly-logarithmic in $n$ this broadcast schedule is a poly-logarithmic factor away from the optimal solution.     

Hyper-Hamiltonian laceability of balanced hypercubes

The balanced hypercube, proposed by Wu and Huang, is a new variation of hypercube. The particular property of the balanced hypercube is that each processor has a backup processor that shares the same neighborhood. A Hamiltonian bipartite graph with bipartition $V_{0}\cup V_{1}$ is said to be Hamiltonian laceable if there is a Hamiltonian path between any two vertices $x\in V_{0}$ and $y\in V_{1}$ . A graph $G$ is hyper-Hamiltonian laceable if it is Hamiltonian laceable and, for any vertex $v\in V_{i}$ , $i\in \{0,1\}$ , there is a Hamiltonian path in G–v between any pair of vertices in $V_{1-i}$ . In this paper, we mainly prove that the balanced hypercube is hyper-Hamiltonian laceable.     

Derandomized Parallel Repetition Theorems for Free Games

Raz’s parallel repetition theorem (SIAM J Comput 27(3):763–803, 1998) together with improvements of Holenstein (STOC, pp 411–419, 2007) shows that for any two-prover one-round game with value at most ${1- \epsilon}$ 1 - ? (for ${\epsilon \leq 1/2}$ ? ≤ 1 / 2 ), the value of the game repeated n times in parallel on independent inputs is at most ${(1- \epsilon)^{\Omega(\frac{\epsilon^2 n}{\ell})}}$ ( 1 - ? ) Ω ( ? 2 n ? ) , where ? is the answer length of the game. For free games (which are games in which the inputs to the two players are uniform and independent), the constant 2 can be replaced with 1 by a result of Barak et al. (APPROX-RANDOM, pp 352–365, 2009). Consequently, ${n=O(\frac{t \ell}{\epsilon})}$ n = O ( t ? ? ) repetitions suffice to reduce the value of a free game from ${1- \epsilon}$ 1 - ? to ${(1- \epsilon)^t}$ ( 1 - ? ) t , and denoting the input length of the game by m, it follows that ${nm=O(\frac{t \ell m}{\epsilon})}$ n m = O ( t ? m ? ) random bits can be used to prepare n independent inputs for the parallel repetition game. In this paper, we prove a derandomized version of the parallel repetition theorem for free games and show that O(t(m + ?)) random bits can be used to generate correlated inputs, such that the value of the parallel repetition game on these inputs has the same behavior. That is, it is possible to reduce the value from ${1- \epsilon}$ 1 - ? to ${(1- \epsilon)^t}$ ( 1 - ? ) t while only multiplying the randomness complexity by O(t) when m = O(?). Our technique uses strong extractors to “derandomize” a lemma of Raz and can be also used to derandomize a parallel repetition theorem of Parnafes et al. (STOC, pp 363–372, 1997) for communication games in the special case that the game is free.     

Stabilizing maximum matching in bipartite networks

A matching ${E_\mathcal{M}}$ of graph G =  (V, E) is a subset of the edges E, such that no vertex in V is incident to more than one edge in ${E_\mathcal{M}}$ . The matching ${E_\mathcal{M}}$ is maximum if there is no matching in G with size strictly larger than the size of ${E_\mathcal{M}}$ . In this paper, we present a distributed stabilizing algorithm for finding maximum matching in bipartite graphs based on the stabilizing PIF algorithm of Cournier et al. (Proceedings of 21st IEEE international conference on distributed computing systems, 91–98, 2001). Since our algorithm is stabilizing, it does not require initialization and withstands transient faults. The complexity of the proposed algorithm is O(d × n) rounds, where d is the diameter of the communication network and n is the number of nodes in the network. The space complexity is O(log Δ +  log d), where Δ is the largest degree of all the nodes in the communication network. In addition, an optimal version of the proposed algorithm finding maximum matching in linear time is also presented.     

Maps on a quantum logic

In this paper we will study functions G of two variables on a quantum logic L, such that for each compatible elements $a,b\in L,$ $G(a,b)=m(a\wedge b)$ or $ G(a,b)=m(a\vee b)$ or $G(a,b)=m(a\triangle b),$ where m is a state on L.     

On reducing factorization to the discrete logarithm problem modulo a composite

The discrete logarithm problem modulo a composite??abbreviate it as DLPC??is the following: given a (possibly) composite integer n??? 1 and elements ${a, b \in \mathbb{Z}_n^*}$ , determine an ${x \in \mathbb{N}}$ satisfying a ^x ?=?b if one exists. The question whether integer factoring can be reduced in deterministic polynomial time to the DLPC remains open. In this paper we consider the problem ${{\rm DLPC}_\varepsilon}$ obtained by adding in the DLPC the constraint ${x\le (1-\varepsilon)n}$ , where ${\varepsilon}$ is an arbitrary fixed number, ${0 < \varepsilon\le\frac{1}{2}}$ . We prove that factoring n reduces in deterministic subexponential time to the ${{\rm DLPC}_\varepsilon}$ with ${O_\varepsilon((\ln n)^2)}$ queries for moduli less or equal to n.     

Proper Interval Vertex Deletion

The NP-complete problem Proper Interval Vertex Deletion is to decide whether an input graph on n vertices and m edges can be turned into a proper interval graph by deleting at most k vertices. Van Bevern et al. (In: Proceedings WG 2010. Lecture notes in computer science, vol. 6410, pp. 232–243, 2010) showed that this problem can be solved in $\mathcal {O}((14k +14)^{k+1} kn^{6})$ time. We improve this result by presenting an $\mathcal {O}(6^{k} kn^{6})$ time algorithm for Proper Interval Vertex Deletion. Our fixed-parameter algorithm is based on a new structural result stating that every connected component of a {claw,net,tent,C ₄,C ₅,C ₆}-free graph is a proper circular arc graph, combined with a simple greedy algorithm that solves Proper Interval Vertex Deletion on {claw,net,tent,C ₄,C ₅,C ₆}-free graphs in $\mathcal {O}(n+m)$ time. Our approach also yields a polynomial-time 6-approximation algorithm for the optimization variant of Proper Interval Vertex Deletion.     

Error Analysis of a Compact ADI Scheme for the 2D Fractional Subdiffusion Equation

In this paper, a Crank–Nicolson-type compact ADI scheme is proposed for solving two-dimensional fractional subdiffusion equation. The unique solvability, unconditional stability and convergence of the scheme are proved rigorously. Two error estimates are presented. One is $\mathcal{O }(\tau ^{\min \{2-\frac{\gamma }{2},\,2\gamma \}}+h_1^4+h^4_2)$ in standard $H^1$ norm, where $\tau $ is the temporal grid size and $h_1,h_2$ are spatial grid sizes; the other is $\mathcal{O }(\tau ^{2\gamma }+h_1^4+h^4_2)$ in $H^1_{\gamma }$ norm, a generalized norm which is associated with the Riemann–Liouville fractional integral operator. Numerical results are presented to support the theoretical analysis.     

Finding Induced Paths of Given Parity in Claw-Free Graphs

The Parity Path problem is to decide if a given graph contains both an induced path of odd length and an induced path of even length between two specified vertices. In the related problems Odd Induced Path and Even Induced Path, the goal is to determine whether an induced path of odd, respectively even, length between two specified vertices exists. Although all three problems are NP-complete in general, we show that they can be solved in $\mathcal{O}(n^{5})$ time for the class of claw-free graphs. Two vertices s and t form an even pair in G if every induced path from s to t in G has even length. Our results imply that the problem of deciding if two specified vertices of a claw-free graph form an even pair, as well as the problem of deciding if a given claw-free graph has an even pair, can be solved in $\mathcal{O}(n^{5})$ time and $\mathcal{O}(n^{7})$ time, respectively. We also show that we can decide in $\mathcal{O}(n^{7})$ time whether a claw-free graph has an induced cycle of given parity through a specified vertex. Finally, we show that a shortest induced path of given parity between two specified vertices of a claw-free perfect graph can be found in $\mathcal {O}(n^{7})$ time.     

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.