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A novel parameterised approximation algorithm for minimum vertex cover

A novel parameterised approximation algorithm for minimum vertex cover
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  This article appeared in a journal published by Elsevier. The attachedcopy is furnished to the author for internal non-commercial researchand education use, including for instruction at the authors institutionand sharing with colleagues.Other uses, including reproduction and distribution, or selling orlicensing copies, or posting to personal, institutional or third partywebsites are prohibited.In most cases authors are permitted to post their version of thearticle (e.g. in Word or Tex form) to their personal website orinstitutional repository. Authors requiring further informationregarding Elsevier’s archiving and manuscript policies areencouraged to visit:  Author's personal copy Theoretical Computer Science 511 (2013) 85–108 Contents lists available at ScienceDirect Theoretical Computer Science  journal homepage: A novel parameterised approximation algorithm for minimum vertex cover ✩ Ljiljana Brankovic a , Henning Fernau b, ∗ a School of Electrical Engineering and Computer Science, The University of Newcastle, Callaghan, NSW 2308, Australia b Fachbereich 4, Abteilung Informatik, Universität Trier, D-54286 Trier, Germany a r t i c l e i n f o Keywords: Parameterised algorithmsApproximation algorithmsVertex cover a b s t r a c t Parameterised approximation is a relatively new but growing field of interest. Itmerges two ways of dealing with NP-hard optimisation problems, namely polynomialapproximationandexactparameterised(exponential-time)algorithms.Weexemplifythisidea by designing and analysing parameterised approximation algorithms for  minimumvertex cover . More specifically, we provide a simple algorithm that works on anyapproximation ratio of the form  2 l + 1 l + 1  ,  l  =  1 , 2 ,... , and has complexity that outperformspreviously published algorithms based on sophisticated exact parameterised algorithms.Inparticular,for l  =  1(factor-1 . 5approximation)ouralgorithmrunsintime O ∗ ( 1 . 0883 k ) ,where parameter  k  ≤  23 τ  , and  τ   is the size of a minimum vertex cover. Additionally,we present an improved polynomial-time approximation algorithm for graphs of averagedegree at most four and a limited number of vertices with degree less than two.Crown Copyright © 2012 Published by Elsevier B.V. All rights reserved. 1. Introduction 1.1.  minimum vertex cover : A hard problem and how to deal with it  Given a graph  G  =  ( V  , E  ) , where  V   is the set of vertices and  E   the set of edges of   G , a  vertex cover C   of   G  is a subset of   V  whose removal leaves only isolated vertices. The problem of finding a minimum vertex cover in an arbitrary graph, called minimum vertex cover  or  minvc  for short, has long attracted the attention of an army of theoretical computer scientistsaroundtheglobe. minvc isamongthefirstgraphproblemseverproventobeNP-hard,beingonthefamouslistofKarp[39],and it has remained one of the most studied NP-hard graph problems in complexity theory ever since. In particular, it hasbeen the paradigmatic test-bed problem for the development of parameterised algorithms, being among the first problemspresented in any introduction to that field. Minimum vertex cover can be formulated as a decision problem as follows. Problem name:  vertex cover (VC) Given:  A graph  G  =  ( V  , E  ) , a non-negative integer  k Parameter:  k Output:  Is there a  vertex cover C   ⊆  V   such that | C  | ≤  k ? ✩ Anextendedabstractofthispaperwaspresentedatthe21st‘‘InternationalSymposiumAlgorithmsandComputation’’(ISAAC2010),JejuIsland,Korea(Brankovic and Fernau (2010) [12]). ∗  Corresponding author. Tel.: +49 651 201 2827. E-mail addresses: (L. Brankovic), (H. Fernau). 0304-3975/$ – see front matter Crown Copyright © 2012 Published by Elsevier B.V. All rights reserved.doi:10.1016/j.tcs.2012.12.003  Author's personal copy 86  L. Brankovic, H. Fernau / Theoretical Computer Science 511 (2013) 85–108 Both the polynomial-time approximation approach and the fixed-parameter approach to  VC  have their advantagesand limitations, as detailed in Section 1.2 below. It is therefore of interest to combine both approaches, aiming at betterapproximation factors by allowing FPT-time. We refer the reader to the survey by D. Marx [42] and to the conceptualframework introduced by L. Cai and X. Huang [13] for more details on how notions from parameterised complexity can ∼ dmarx/papers/marx-dagstuhl11091-approx.pdf .ThisseemstobetheonlyreferencewheretheW[1]-hard famous  (Topological) Bandwidth  problem was shown to be constant-factor fixed-parameter approximable.Notice that  Minimum Bandwidth  is unlikely to be constant-factor approximable in polynomial-time; for more detailssee [24].In this paper, we follow the definition by Y. Chen et al. 1 [19], which in our context can be phrased as follows. Givena graph  G  and a parameter  k , a  parameterised approximation algorithm with (constant) approximation ratio  ρ  for  minimumvertex cover  produces a vertex cover  C   such that  | C  | ≤  ρ | C  ∗ |  at least if   | C  ∗ | ≤  k . More precisely, our algorithms willbehave as follows. •  If our algorithms answers NO, then | C  ∗ |  >  k , that is, graph  G  does not have a vertex cover of size  k . •  If our algorithm produces a vertex cover  C  , then | C  | ≤  ρ  min { k , | C  ∗ |} .We give an alternative description in terms of the size | C  ∗ | of some minimum vertex cover  C  ∗ : •  If  | C  ∗ | ≤  k , our algorithm produces a vertex cover  C   such that | C  | ≤  ρ | C  ∗ | . •  If   k  <  | C  ∗ | ≤  ρ k , our algorithm can either answer NO, or produce a vertex cover  C   of size | C  | ≤  ρ k . •  If  | C  ∗ |  > ρ k , our algorithm answers NO.In all cases, our algorithm runs in time  O ∗ (  f  ( k ))  for some function  f  , where  O ∗  notation suppresses polynomial factors, i.e. ,  O ∗ (  f  ( k ))  is a shorthand for  O (  f  ( k ) poly ( n )) .Generally speaking, parameterised approximation is an emerging research area that appears very promising, especiallyfor those problems where under certain complexity-theoretic assumptions no further advancements in approximationand/or parameterised algorithms alone are to be expected. For example, it has been shown [13] that MAX SNP completeproblems (such as  minimum vertex cover  restricted to subcubic graphs) admit a fixed-parameter approximation scheme,while no polynomial-time approximation scheme (PTAS) can be expected for such problems. On the other hand, W[P]-hardminimisation problems do not admit constant-factor fixed-parameter approximation algorithms unless FPT equals W[P][25], which is considered to be quite unlikely. 1.2. Earlier results on  VC We report on the main results in each of the three approaches to the vertex cover problem: polynomial-time approxi-mation, exact parameterised algorithms and parameterised (exponential-time) approximation.1.  Polynomial-timeapproximation. Despiteallefforts,thebestknownconstantfactorapproximationalgorithmforgeneralgraphs is still (basically) a factor 2 approximation. More specifically, G. Karakostas [38] derived an approximation factorof 2 − Θ (  1 √  log n ) . I. Dinur and S. Safra [23] showed that there is no polynomial-time approximation algorithm for  minvc achievinganapproximationratiobetterthan10 √  5 − 21  ≈  1 . 36067,unlessP  =  NP.Moreover,assumingthattheUniqueGames Conjecture is true, no approximation factor of the form 2 − ε  is possible for any  ε >  0, as shown by S. Khot andO. Regev [40]. There are better results for some specific classes of graphs, most notably graphs with bounded degree(2  −  2 ∆  [34] and 2  −  5 ∆ + 3  +  ε ,  ε >  0 [8]), and 4-colourable graphs (1 . 5) [34]. A good (although somewhat out-dated)survey is contained in the third chapter of D. Hochbaum’s book [35]. The above mentioned results for degree-boundedgraphswereimprovedbyE. Halperin[33]towardsaperformancefactorof  ( 2 − ( 1 − o ( 1 ))) · 2lnln ∆ ln ∆  .Evenforgraphswithmaximumdegree3noapproximationfactorarbitrarilyclosetoone( i.e. ,noPTAS)canbeachievedinpolynomial-time,asshown by P. Alimonti and V. Kann [3]. Another interesting result is due to T. Imamura and K. Iwama [36] who showed that graphs  G  =  ( V  , E  )  of maximum degree  ∆  ∈  Ω  | V  |·  loglog ( | V  | ) log ( | V  | )   and average degree  d a v  g   can be approximatedwith a factor of   21 + da v  g  2 ∆ provided that | E  | ≤  ∆ · ( | V  |− ∆ ) . This yields factor- 43  approximations for dense regular graphs( i.e. ,  d a v  g   =  ∆ ) and an approximation factor of   32  provided that  d a v  g   ≥  23 ∆  for dense graphs.For certain graph classes like planar graphs, polynomial-time approximation schemes are known [6]. Some classesof graphs even have polynomial-time solutions,  e.g. , bipartite graphs, where  minimum vertex cover  can be solved bymatching techniques due to the Theorem of König and Egerváry. One of the oldest online sources of this theorem seemsto be a paper of T. Gallai [31]. 1A long version can be obtained from the homepage of M. Grohe in Berlin ∼ grohe/pub/chegrogru06.pdf .  Author's personal copy L. Brankovic, H. Fernau / Theoretical Computer Science 511 (2013) 85–108  87 For the ease of reference, we make explicit the following three results that are the most relevant to this paper: Lemma 1.  [8] There exists a polynomial-time factor   7 / 6 + ε  approximation algorithm for   minimum vertex cover  for any graph with maximum degree  3  and any  ε >  0 . Lemma 2.  There exist polynomial-time approximation algorithms for   minimum vertex cover  for any graph with maximumdegree  4  that guarantee factors of   3 / 2  [34] or even of   97  + ε , for any  ε >  0  [8]. Lemma 3.  [32] For an arbitrary graph with average degree d a v  g  , there exists a factor   ( 4 d a v  g   + 1 )/( 2 d a v  g   + 3 )  approximationalgorithm for   minimum vertex cover , providing that the size of a minimum vertex cover is at least  | V  | / 2 . 2.  Exact parameterised algorithms. Asmentioned, VC istheparadigmatictest-bedproblemforthistypeofalgorithms.Forgeneral graphs, several parameterised algorithms offer running times of about  O ∗ ( 1 . 28 k )  [15,16,18,37,44]. For special classesofgraphs,betterrunningtimeboundsareknown.Forexample,forcubicgraphswehave O ∗ ( 1 . 194 k ) [17]andevenbetter claims [50]. For graphs of bounded genus,  O ∗ ( c  √  k )  can be obtained [21]. Such running times cannot be obtainedfor general graphs unless the Exponential Time Hypothesis fails [14].3.  Exact moderately exponential-time algorithms.  Here, the task is to solve  minimum vertex cover , or equivalently, dueto Gallai’s identity [31],  Maximum Independent Set , in running times O ∗ ( c  n )  on graphs of order  n . A sequence of paperssuggested branching algorithms with  c   ≈  1 . 2; the reader is referred to [29,30,41,46,47] and the references therein. 4.  Parameterised and exponential-time approximation.  More recently, the idea of using parameterised algorithms forobtaining better approximation factors at the cost of exponential-time (best confined to a small entity called theparameter) was proposed in several independent papers in 2006, as surveyed by D. Marx in [42]. The focus of thesepapers is either on non-approximability or on problems where possibly new qualities of approximation can be achievedby investing exponential-time. Since  minvc  turns out to be quite a nice problem already, both from the parameterisedcomplexity point of view and from the standpoint of approximation, a combination of both lines of attack on thisparticularNP-hardproblemwasnotventuredthere.However,asnoconstantfactorimprovementsoverthewell-knownfactor-2approximationalgorithmsaretobeexpectedfor minvc usingpolynomial-timeonly,investingFPT-timeappearswarranted.Thebenefitsofallowingexponentialtimetoimprovetheapproximationfactorshavebeenstudiedbyseveralauthors.For example, E. Dantsin et al. [20] showed how to transform an existing approximation algorithm for MAX SAT intoan algorithm with a better approximation factor by allowing exponential time. In relation to our paper, the article byN. Bourgeois et al. [11] is the most relevant one, as it shows that if there exists an exact exponential time algorithmfor computing minimum vertex cover that runs in time  O ∗ (γ  n ) , then an approximation factor of 2  −  ρ , for  ρ  ∈  ( 0 , 1 ] can be obtained with running time  O ∗ (γ  ρ n ) . Furthermore, any parameterised algorithm for  VC  running in time  O ∗ (δ k ) that produces an associated feasible solution can be used to obtain a parameterised approximation algorithm withapproximationratio ρ  andrunningtime O ∗ (δ ρ k ) .AsimilarresulthasbeenindependentlyobtainedbyM.Fellows,A.Kulik,F. Rosamond and H. Shachnai [27] via so-called fidelity preserving transformations.An interesting alternative view on the matter is offered by V. Vassilevska and R. Williams and S. L. M. Woo [49] whosuggest algorithms that either provide better approximation ratios than any polynomial-time approximation algorithmor better running times than any exact algorithm can do (under certain complexity assumptions). We do not follow thisapproach here, but recommend it for future research. 1.3. Main contributions and organisation of the paper  In this paper, we present the first genuine parameterised approximation algorithm for  minimum vertex cover . Our O ∗ ( 1 . 09 k )  factor- 32  approximation algorithm is simple, yet faster than the  O ∗ (δ . 5 k )  =  O ∗ ( 1 . 13 k )  algorithm by N. Bourgeoiset al. that incorporates a quite sophisticated exact parameterised O ∗ ( 1 . 28 k )  algorithm for  VC . The main difference betweenour algorithm and other approaches (both the one of N. Bourgeois et al. and the one of M. Fellows et al.) lies in the fact thatweexploitthetargetapproximationfactorineachbranchingstep,asopposedtotakingtheexact(parameterised)algorithmas a black box. This way, we not only obtain far better running times, but our branching algorithms are distinguishedby their simplicity. It is also the first such algorithm that introduces local-ratio techniques known from polynomial-timeapproximation, see [7], into parameterised algorithmics, combining them with the search tree analysis itself, althoughsimilar ideas are the basis of the mentioned results due to M. Fellows et al. We generalise this factor- 32  approximationalgorithm to any constant factor of the form  2 l + 1 l + 1  . Actually, it is also possible to obtain any other constant factor less than2, and we elaborate on this in the Conclusion. We also obtain an improved polynomial-time approximation algorithms forgraphs of average degree at most four. This outset also explains the organisation of the paper.The considerable improvements that we obtain in comparison with other approaches are due to the following mainingredients of our algorithms:1. We develop new and powerful approximation-preserving reduction rules that may be of independent interest, also inthe context of polynomial-time approximation. Actually, the announced polynomial-time approximation result that wepresent in this paper profits greatly from this idea.  Author's personal copy 88  L. Brankovic, H. Fernau / Theoretical Computer Science 511 (2013) 85–108 2. Our algorithm branches on a pair of vertices, rather than on a single vertex. The vertices are selected in such a way so asto ensure that together they have at least two neighbours more than any one of them separately, which adds greatly tothe efficiency of the algorithm. We note that this novel strategy is not applicable to the exact parameterised case.3. We interleave so-called worsening steps with exact branching steps in order to speed up the branching. In this way,we can exploit these worsening steps to create better branching situations. By way of contrast, the fidelity preservingtransformations of Fellows et al. put these worsening steps at the very beginning of a parameterised approximationalgorithm, using any exact parameterised in a second step as a black box. Conversely, Dantsin et al. first do some (exact)branchingandthenapplytheknownpolynomial-timeapproximationalgorithmsineachoftheleavesofthesearchtree.Bourgeois et al. (independently) suggested introducing a worsening step (by selecting an edge every time a vertex isput into the independent set) to obtain a factor-0 . 5 approximation for  Maximum Independent Set . However, their mainresult related to ours ( i.e. , to  Minimum Vertex Cover ) relies on a different technique, splitting the graph into pieces andthen to solving the problem on these pieces.4. Thevalidityoftheapproachisjustifiedbyargumentsreminiscentofthelocal-ratiotechniqueknownfromapproximationtheory. 1.4. Some bits of graph theory We consider simple undirected graphs and we use  G  =  ( V  , E  )  to denote a graph  G  with the vertex set  V   and the edgeset  E  , where  E   can be seen as a set consisting of two-element vertex sets, modelling the symmetric  adjacency relation  on  V  .Two edges  e ,  f   with  e ∩  f   ̸  = ∅ are called  incident  . We use  ∆ ( G )  and  δ( G )  to denote the maximum and minimum degree in G ,respectively,and d G (v) todenotethedegreeofthevertex v  ∈  V   within G ,suppressingthesubscript G ifunderstoodfromthe context. The neighbourhood of the vertex  v  is denoted by  N  G (v)  and contains all the vertices at distance exactly 1 from v , so that  d G (v)  = | N  G (v) | . Given a graph  G  and two nonadjacent vertices in  G ,  u  and  v , those neighbours of   u  that are notalso neighbours of   v  are called  private neighbours of u with respect to  v , referring to the set  N  ( u ) \ N  (v) . C   ⊆  V   is a  vertex cover   of   G  =  ( V  , E  )  if every edge  e  ∈  E   has at least one of its endpoints in  C  . The size of the smallestvertex cover of   G  is also known as the  vertex cover number   of   G  and is denoted by  τ( G ) . The complement of a vertex coveris known as an  independent set  ,  i.e. , a set  I   of vertices of   G  such that no two vertices  u ,v  ∈  I   are connected with an edge.Likewise, a set  M   of edges is called  independent   or a  matching   if, for any pair of edges  e ,  f   ∈  M  ,  e ∩  f   = ∅ ,  i.e. , no two edgesin  M   share a vertex.If   G  =  ( V  , E  )  is a graph and  U   ⊆  V  , then  G [ U  ]  denotes the graph  induced by U  ,  i.e. ,  G [ U  ] =  ( U  , F  )  with  F   = { e  ∈  E   | e  ⊆  U  } .Wewilloftenneedtheoperationof  removing   or deleting   verticesor,moregenerally,asetofvertices D fromagraph(together with the incident edges), referring to  G − D  =  G [ V   \ D ] . Note that  C   is a vertex cover of   G  if and only if   G − C   hasno edges.A  path  is a sequence  v 0 ,...,v ℓ  of vertices such that each two consecutive vertices are connected with an edge,  i.e. , foreach  i  =  1 , 2 ,...,ℓ  there exists an edge  e i  = { v i − 1 ,v i } ∈  E  ; we will also refer to such a path as a path between  v 0  and  v ℓ . If  ℓ  = |{ v 1 ,...,v ℓ }| ,thenthepathiscalled simple .If  v 0  =  v ℓ ,thenasimplepathiscalleda cycle .Acycleoflengththreeisalsoknownasa triangle .Agraph G iscalled triangle-free ifthereisnotriple (  x ,  y ,  z  ) ofverticessuchthat  x ,  y ,  z  ,  x isatriangle.Weuse dist G ( u ,v)  to denote the  distance  between two vertices  u  and  v ,  i.e. , the length  ℓ  of the shortest path between  u  and  v .The following two Lemmas are crucial for the algorithm analysis that we present in Sections 3 and 4. Lemma 4.  Consider a connected graph G with maximum degree  ∆ ( G ) . Let G ′  be a graph obtained from G by deleting one or more vertices together with their incident edges. If G ′  contains any edges at all, then there exists a vertex y in G ′  such that  0  <  d G ′ (  y ) <  ∆ ( G ) . Proof.  Consider a connected graph  G  =  ( V  , E  )  with maximum degree  ∆ ( G ) . Let  G ′  =  ( V  ′ , E  ′ )  be a graph obtained from  G by deleting  X   ⊆  V   (together with the incident edges) and let  Y   be the set of neighbours of vertices from  X   in  G  that are notcontained in  X  . Then  d G ′ (  y ) <  ∆ ( G )  for all  y  ∈  Y  , as each vertex in  Y   has a neighbour in  X   in graph  G . If   d G ′ (  y )  =  0 for all  y  ∈  Y  , then the only neighbours of   Y  -vertices in  G  were from  X  . Since  G  is connected, there are no other vertices in  G  apartfrom vertices in  X   and  Y  , and thus  V  ′  =  Y   and  E  ′  = ∅ .   Lemma 5.  In a triangle-free graph with minimum degree  δ  ≥  2 , any vertex  v  has a vertex u at distance two. Proof.  Let  G  =  ( V  , E  )  be a triangle-free graph with minimum degree  δ  ≥  2. Consider some vertex  v  ∈  V  . Since  d (v)  ≥  2,we have that  N  (v)  ̸  = ∅ . Consider any  x  ∈  N  (v) . As  d (  x )  ≥  2, it follows that there exists some vertex  u  ∈  N  (  x ) \{ v } . Sincethere are no triangles in  G , it follows that  u  / ∈  N  (v) . Hence, dist (v, u )  =  2.   We occasionally need the following generalisation of the previous lemma: Lemma 6.  In a triangle-free graph with minimum degree  δ  ≥  2 , to any vertex  v  we can associate a matching M  v , | M  v | ≥  δ − 1 ,such that each matching edge xy  ∈  M  v  obeys  |{  x ,  y } ∩  N  (v) | = |{  x ,  y } ∩  N  [ v ]| =  1 . This matching can be greedily found in polynomial-time.
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