On the Feasibility of Maintenance Algorithms in Dynamic Graphs

On the F easibili t y of Main tenance Algorithms in Dynamic Graphs Arnaud Casteigts 1 , Bernar d Mans 2 , and Luke Mathieson 2 1 SITE, Univ ersity of Ottaw a, 800 King Edw ard, Otta wa, Ontario K1N 6N5, Canada. casteig@site. uottawa.ca 2 Department of Computing, Macquarie Universit y , Sydney , NSW 2109, Au stralia. {bernard.m ans,luke.mathieson}@mq.edu.au Abstract. Near ubiquitous mobile computing has led to intense in terest in dy namic graph theory . This provides a new and c hallenging setting for algorithmics and complexity theory . F or an y graph-based problem, the rapid evolution of a (p ossibly d isconnected) graph ov er time naturally leads to the imp ortant complexit y question: is it b etter to calculate a new sol u tion fro m scratch or to adapt the known solution on the prior graph to quickly pro vide a solution of guaranteed quality for the changed graph? In this paper, we demonstrate that the former is the b est approac h in some cases, but that there are cases where the latter is feasi b le. W e prov e that W [1] -hardness for the parameterized appro ximation problem implies the non-existence of a main tenance alg orithm for the given ap- proxima tion ratio, even giv en time exp onential in the sol ut ion size and p olynomial in the ov er all size — i.e., even with a large amoun t of time, having a solution to a v ery similar graph do es not help in compu ting a solution to the curren t graph. T o ac h ieve this, w e formalize the idea as a maintenanc e algorithm . T o illustrate our results we show that r - Regular Subgr aph is W [1] -hard for the parameterized approximation problem and thus has no main tenance algorithm for the giv en appro xi- mation ratio. Conv ersely we show that Ver tex Cover , which is fixed- parameter tractable, has a 2 -approximate maintenance algorithm. The implications of N P -hardness and N P O -hardness are also ex plored. 1 In tro duction With the development o f sufficiently small and p ow erful hardware, mobile com- puting devices present a new a nd in teres ting netw ork environmen t in whic h the complexity a nd algor ithmics fo r even well understo o d pr oblems can c hang e rad- ically . T o mo del this sort of netw ork, v ario us notions of dynamic g raphs hav e bee n develop ed, including delay-tole ra nt [7], disruptive-toler ant , intermitt ently- c onne cte d [20], opp ortunistic , time-varying [4] and evolving [2,3]. Each of these mo dels differing a spe c ts of the dynamics under differing a ssumptions or with different applications in mind. One of the key a spec ts of dyna mic g raphs is that a gr a ph may not be connected at a n y giv en moment, or ev er, but due to th e app earance and d is appea rance of edg es, it may still be p ossible t o construct journeys (rather than paths) throug h the graph over time and space. Howev er, dependent on the a s sumptions o f the system, there may be even fundamental prop erties or tra ditiona l pr o blems which a re incomputable or definitionally am- biguous (e.g., [2]). The results presented her e are indepen den t of the mo del a s long as the change in the gra ph occur s (or can b e describ ed) in a discrete manner. Given a tra ditiona l graph pr oblem Π and a sequence of gra phs { G i } , we may reframe Π in se veral wa ys as a dynamic g raph problem; p ermanent — where we try to find a solution that ho lds at all p oints in time (e.g., a single s et of dominators that cov er all nodes in every G i ), over-time — where the solution is defined with r esp ect to the sequence as a whole (e.g., a s et of do minators such that each no de is covered in at lea st one G i ), a nd evolving — where w e co mpute a (p ossibly different) solution for each G i . Permanent and over-time problems may app ear s omewhat less general in that they re quire prior knowledge o f the graph dynamics. Ho wev er it is worth noting that there ar e relev ant in few, y et impo r tant , practical scenario s, such as with a known schedule (e.g., public trans- po rts, low-earth satellites, sensors with sleeping s c hedule ), or a known schedule prop erty (e.g., p erio dicity [8] or r e curr enc e [4] — i.e., edges that exist once are guaranteed to re- appe a r at some kno wn, b ounded, or un b ounded time). It can be observed that for so me problems, e .g ., c overing problems like V er- tex Cover , all three v aria n ts are strongly related. Given such a problem Π , one can easily c heck that a solution to p ermanent Π is a (po ssibly far from optimal) solution to evolving Π , a nd a solution to an y G i in the evolving Π is also v alid fo r over-time Π . The connexion is even stronger: the int erse ct ion of solutions to evolving Π is v alid for over-time Π , and their union is a solution to p ermanent Π . F rom this p ersp ective, the evolving v ariant a ppea r s quite cen tr a l, and the p ermanent or over-time Π actually form upper a nd low er b ounds for each solution in the evolving Π . In this paper, w e fo cus on the evolving Π and lo ok at finding the solution for each point in time. The dynamic co n text is a n int er esting algor ithmic and c omplexity setting. Although a na ï ve appro ach would simply compute a new solution for every G i , it is p ossible that given a relatively limited a mo un t of change in the gr aph at each step we ma y leverage the previous solution to allow quic k computation of the new solution. This leads to the idea of a maintenanc e algorithm (q.v., Section 2). Moreover this a pproach fits well with the re a l world inspira tion, where along with a poss ibly rapid pace o f change, there is often no c e ntralised control. If the previous solution can be used to compute the new solution, this suggests a certain lo ca lization, which would b e idea l for mobile infra structureless net works. Co nsidering that dynamic gr aphs can be per manently disco nnected (while still o ffering connectedness over time), we initially consider w e ll- known problems that are not a m big uous o r undefined in a disconnecte d setting: r - Regular Subgraph and Ver tex C over . A solution must b e well defined for each (p ossibly) partitioned co mpo nent and trivia l solutio ns m us t exist for tr iv ial cases (e.g., single no des). As we are mainly in terested in maintaining a solution of some guaranteed quality for ea ch particular problem, our results can b e summarized as follows: 2 – we prov e that W [1] -hardness for the parameterized approximation problem implies the non-existence o f a maintenance a lgorithm for the given approxi- mation r atio, even giv en time exp o nent ia l in the solution siz e and po lynomial in the ov er all size, – we sho w that r -Regular Subgraph and sev era l v ariants hav e no param- eterized approximation algo rithms thus ha ve no F P T -time maintenance al- gorithm, and conv erse ly , – we show that problems that are fixed-para meter tr a ctable may ha ve an ap- proximate maintenance alg orithm by proving that this is indeed the case for Ver tex Co ver , – we establish a co mplexity clas s ification for m a in tena nce alg orithms that provides a strong rela tionship with para meterized approximation co mplex- it y [12]. Similar ideas regarding dyna mic alg orithms and complexit y ha ve b een ex- plored b efore, but from no tably different p ositions. T ypica lly previo us work has fo c us sed on p olynomia l or lo garithmic time pro blems, and did not include the consideration of locality that is cen tr al to ma ny practical applications for dy- namic complexity (cen tr al ov ersight and co mm unication is ne ither guaranteed nor generally desir able in a dynamic net work), Section 2 gives the relev ant def- initions. Howev er many interesting results and ideas are a pplicable in this con- text. Patnaik & Immerman [15] consider dynamic co mplexit y fro m a descriptive complexity theor y p ers pective, defining D ynFO , a class of dyna mic problems that are expressible in first order logic. W eber & Sch wen tick [19] build up on this, a g ain concentrating on a descriptive complexity appro ach. Holm, de Lich t- enberg & Thorup [10] give a s eries of results that can readily b e interpreted as maintenance alg orithms in our cont ex t. Their results for Connectivity , 2- Edge and Biconnectivity r ely on the maintenance under edge deletion and addition o f a s o lution for Minimum Sp anning Forest , giving p olylog arith- mic running times for a ll pro blems, but with no b ound on lo ca l it y . Miltersen et al. [14] presen t another, similar appro ach wher e the dynamism is ac hieved at a lower level by perturbing individual bits in the input. They also fo cus on problems o f p olynomial complexit y showing, in our context, that pr o blems such as the Circuit V alue P r obl em and Propositional Horn Sa tisfiability hav e no p olylogar ithmic maintenance a lgorithms but that interestingly there ex- ist o ther P -complete problems tha t do. Ausiello, Bonifaci & Escoffier [1] discuss a differen t mo del, where there is only int er est in using an existing optimal solu- tion to so lve (to some degree o f appr oximation) a p erturb ed instance (the mo del is generally called r e optimization ). This approach is in some ways an unsuitable per sp e ctiv e for our co n text, as we do not expect at any p oint to be able to com- pute a n o ptimal so lution, how ever their negative results in par ticular ca rry to our setting, such as Min Coloring having no P -time main tenance a lgorithm for approximation r atio 4 3 − ε for an y ε . The r est of the pa per is organized a s follo ws. W e define our no tation and provide our main results in Section 2 . W e pr ovide some basic and necessa ry background on parameterize d complexit y and parameterized approximation in 3 Section A. W e provide the main pro ofs of our results and of results on para me- terized approximation complexity for Ver tex Co ver , r -Regular Subgraph and some generaliza tions of r -Regular Subgraph in Section 3. W e give some concluding remarks in Section 4. 2 The F easibilit y of Main t enance The key difference in the dynamic setting with regar ds to co mputation is that as the graph is contin ually c ha ng ing, we ar e r equired to con tinually reco mpute solutions for the problem. Hence the st yle of algo rithm we are concerned with is somewhat differen t. In this case we would ideally lik e an algorithm that can exploit the known so lution to the previous graph, which is so mewhat similar to the current gra ph, to pro duce a solution for the current gra ph that main tains a desired solution quality (whether optimal or within some approximation b ound) in a time faster than it w ould take to c o mpletely r ecompute the solution. W e formalize the idea as a maintenanc e algorithm . Definition 1 (Maintena nce Algorithm). Given a se quenc e of gr aphs { G i } wher e the e diting distanc e b etwe en G i and G i +1 is 1 , A is a maintenanc e algo- rithm for pr oblem Π if given a solution S i for Π on gr aph G i , A c an c ompute a solution S i +1 for Π on gr aph G i +1 that pr eserves the quality of the solution. The e diting distanc e b etw een t wo gra phs is the n umber of e diting op er ations that hav e to b e p erfor med to obtain one from the other. Typically the editing op erations we will b e in terested in are some subset of edge addition, edge dele- tion, vertex a ddition and vertex deletion. W e set the editing distance here to 1 in order to rema in sufficiently general in establishing ne gative results. As the graph changes rela tiv ely quickly , we would like the co mputation to be done within a limited computation time (preferably a small constant). The notion of a b ounde d maintenance algor ithm formalizes this bound. Definition 2 (Bounded Maintenanc e Al g orithm). A maintenanc e algo- rithm is b ounde d if ther e is a fun ction f such that at e ach step, the c omputation p erforme d at e ach vertex is time b ounde d by f . W e r efer to su ch an algorithm as a f -maintenanc e alg orithm. Also, because in a practical setting the no des model indepen den t devices, we would like the computation to be done as lo ca lly as po ssible, i.e., the s ta te of each vertex being based o nly on the s tate of its r -neighborho o d, with r ≪ diameter ( G ) . Definition 3 ( r -Lo cal Main tenance Algorithm). A m aintenanc e algori t hm is r -lo c al if the state of e ach vertex is c ompute d b ase d only on the states of vertic es within distanc e r . W e r efer to such an algorithm as a r -lo c al maintenanc e algorithm. 4 F or example a 1 -lo cal O (1) -maintenance algor ithm a llows each vertex to p er- form a constan t n umber of computational steps at each iteration o f the a lgorithm, knowing only the states of its direct neig h b or s . It is with these computational restrictions that knowledge o f a prior solution and the conce pt of a main tena nce algorithm can be most useful. Indeed there are many prio r results that fit into this framework, thoug h previo usly there has b een little consideration of the lo- cality . F or example, Lemma 1 ([14]). Undirected Forest Accessibility has a d -lo c al O (log n ) - maintenanc e algorithm wher e d is the diameter of the gr aph and n is the numb er of vertic es. Lemma 2 ([10]). Minimum Sp anning F orest has a d -lo c al O (log 4 n ) - maintenanc e algo rithm wher e d is the diameter of the gr aph. Lemma 3 ([17]). Directed Rea chibility has a d -lo c al O ( m + n log n ) - maintenanc e algori t hm wher e d is the di ameter of t he gr aph, m is the numb er of e dges and n is t he nu mb er of vertic es. Ausiello, Bo nifaci & Escoffier [1] e x amine the so rt of ha rder pro blems that we are interested in (rather than P -time pro blems), how ever their setting r equires a n initially optimal solution and only cov ers a single pertur ba tion, rather than the highly dynamic environmen t exp ected from the application domain we consider. Of course there are limits to the exis tence o f main tenance algor ithms: Theorem 1. F or any c omputable function f and p olynomial p , if a pr oblem Π is W [ t ] -har d for p ar ameter k and any t ≥ 1 , Π has no O ( f ( k ) p ( n )) -maintenanc e algorithm u nless W [ t ] = F P T wher e n is the size of the input. Pr o of. Ass ume that Π has such an alg orithm. Then let ( G, k ) b e an instance of Π and G 0 be the trivia l instance defined by taking the completely disconnected gra ph on | V ( G ) | vertices with trivial optimal solution S 0 . Let G 0 , . . . , G | E ( G ) | be a sequence of graphs g enerated by adding the edges of G one by one in arbitrar y order to G 0 , resulting in G | E ( G ) | = G . As Π has a main tena nce a lgorithm beginning with G 0 the algor ithm can b e applied to each pair G i , G j where j = i + 1 with so lution S i to o btain solution S j . Then if S | E ( G ) | is a witness that ( G, k ) is a Yes -instance, the a lgorithm answers Yes (or returns S | E ( G ) | in the case of a sear ch problem) and No other wis e. Then we ha ve an algor ithm that p erforms O ( f ( k ) p ( n )) op era tions for each vertex at ea ch iteration. As there are at mo st n 2 iterations, the ov erall algor ithm has running time O ( n 3 f ( k ) p ( n )) , and hence Π is fixe d- parameter tr actable, which co n tr a dicts the W [ t ] -hardness of Π with pa rameter k . By the same arg ument, we can also obtain a classical complexity a nalogue. Corollary 1. F or any p olynomial p , if a pr oblem Π is N P -har d, Π has no O ( p ( n )) -maintenanc e a lgorithm that optimal ly solves Π u nless P = N P wher e n is the size of the input. 5 Similarly , classical approximation results are pres erved. Corollary 2. F or any p olynomial p , if a pr oblem Π is N P O -PB-har d, Π has no O ( p ( n )) -maintenanc e algo rithm that solves Π within an appr oximation factor of O ( n 1 − ε ) for any ε > 0 unless P = N P wher e n is the size of the input. If, given a para meter ized problem with parameter k , the b ound is O ( f ( k ) p ( n )) where f is a computable function, p is a polyno mial in n , the size of the input, w e deno te the a sso ciated b ounded maintenance a lgorithm as a n ( r - lo cal) F P T -maintenance algo rithm. The complexity results given in Section 3 then give the follo wing results. Theorem 2. g ( k ) -Appro x-Ver tex Deletion to Regular Subgraph has no F P T -maintenanc e algorithm unless W [1] = F P T . Theorem 3. g ( k ) -Appro x-Deletion to Regular Subgraph has no F P T - maintenanc e algo rithm un less W [1] = F P T . Theorem 4. g ( k ) -Appro x-Weighted Degree Constrained Deletion has no F P T - m aintenanc e algorithm unless W [1] = F P T . Similar statements can b e ma de regarding c -Add-Appro x-Domina ting Set and g ( k ) -Appr ox-Independent Domina ting Set using their resp ective approximation har dness results: Lemma 4 ([6]). c -Add-Appro x-Domina ting Set h as no F P T -maintenanc e algorithm un - less W [2] = F P T . Lemma 5 ([6]). g ( k ) -Appr ox-Independent Domina ting Set has n o F P T - maintenanc e algo rithm un less W [1] = F P T . Independently of the results ab ov e, we obs e r ve that if the b ound on the maint e na nce algorithm is further res tricted, then in certain cases no approxima- tion ratio can be maint a ined. More precisely the appr oximation ratio will b e a function of the n umber of iterations of the ma intenance algorithm, up to trivia l bo unds on the quality of the solution. Given a (w.l.o.g) minimization pro blem Π in N P O with the following proper ties: 1. The decision coun ter part of Π is N P -hard. 2. Π has a O (1 ) -maint ena nce algorithm A . W e make the following claims: Lemma 6. Ther e is a step in A that is divergen t . Where diver gent means that if the size o f the optimal solution decreases, the size of the solution given by A does not decrease, and similarly if the s ize o f the optimal solution do es not change, the size of the solution given by A increases. 6 Pr o of. F ollows immediately from the N P -har dnes s of the decision co un ter pa rt of Π . Let γ i be the s ize o f the solution given as input from instance I i (per haps as the result of a previous application of A ) and γ ∗ i be the size of the optimal solution for I i . Let I i +1 be the instance after altera tions and γ i +1 be the size of the solution given b y A and γ ∗ i +1 be the size of the optimal solution. Lemma 7. Given an ap pr oximation r atio A and an instanc e wi t h solution of size γ i = A · γ ∗ i , then A c annot guar ante e that γ i +1 ≤ A · γ ∗ i +1 . Pr o of. Let the change from I i to I i +1 be such that it induces A to take a divergent step. Assume without loss of g enerality that γ i +1 = γ i + 1 and γ ∗ i +1 = γ ∗ i Then γ i +1 γ ∗ i +1 = γ i + 1 γ ∗ i = A · γ ∗ i + 1 γ ∗ i = A + 1 γ ∗ i It is easy to see that a similar res ult o ccurs with the other po ssibilities for divergence. Given a problem Π with a long diver gent se quenc e this problem ma y be am- plified. A long diver gent se quen c e is a set of instances wher e the changes betw een each instance induce divergent steps in A . F or example g iven the completely dis- connected gra ph and the problem Domina ting Set , the initial solution is to take all vertices in the do mina ting set (which is o ptimal), then one by one we add edges to obtain a star g raph. Using an a dv er sarial approach we add the edges suc h that the centre o f the star is not in the do minating set, but all other vertices are, taking n − 1 vertices where only 1 is needed. Lemma 8. Given a long diver gent se quenc e of length d , we have γ d +1 /γ ∗ d +1 ≥ 1 + d/γ ∗ d +1 . Pr o of. At each step we are for ced to incr ease the size of the s olution relative to the optimal by at least one. Then by step d (instance I d +1 ) we hav e γ d +1 ≥ γ 1 + d how ever γ ∗ d +1 ≤ γ ∗ 1 . Therefore we at least have: γ d +1 γ ∗ d +1 ≥ γ 1 + d γ ∗ 1 = γ ∗ 1 + d γ ∗ 1 = 1 + d γ ∗ 1 ≥ 1 + d γ ∗ d +1 Then for the dominating set ex ample w e hav e the approximation r atio of O ( n ) . In g e ner al, g iven a pro blem with a lo ng divergen t sequence o f length O ( n ) , we obtain a similar appr oximation ratio. 3 Problems in Dynamic Graphs Given a gra ph, the pr oblem of obtaining an r -re g ular subgraph with a mini- m um num b er of excluded vertices (known v ar iously as r -Regular Subgraph , 7 k -Almost r -Regular Subgraph and Ver tex Deletion to Regular Sub- graph ) is W [1] -har d [13]. The prop erty of having a b ounded degree or regular graph is pa rticularly interesting for r outing pur po ses. Although it is W [1] -ha rd, for a dynamic gra ph, an approximation would be s ufficien t as we exp e c t the graph to change rapidly , so computing a reaso nable solution quickly is more ef- fectiv e than computing an exact solution slowly . In this section howev er we show that this problem has no par ameterized approximation unless W [1] = F P T . F ormally w e define the pr oblem as: g ( k ) -Appro x-Ver tex Deletion to Regular Subgraph Instanc e: A graph G = ( V , E ) , integers k and r . Par ameter: k . Question: Is there a set V ′ ⊂ V with | V ′ | ≤ k such that the subg r aph G ′ = G [ V \ V ′ ] is r regular? This problem has many p ossible gener alizations, we note tw o in par ticular: g ( k ) -Appro x-Deletion to Regular Subgraph Instanc e: A graph G = ( V , E ) , integers k and r . Par ameter: k . Question: Is there a set D ⊂ V ∪ E with | D | ≤ k suc h that the s ubgraph G ′ = ( V \ D , E \ D ) is r regular ? g ( k ) -Appro x-Weighted Degree Constrained Deletion Instanc e: A gra ph G = ( V , E ) , integers k and r , a weight function ρ : V ∪ E → [1 , k + 1] and a degree list function δ : V → 2 { 0 ,...,r } . Par ameter: k . Question: Is there a set D ⊂ V ∪ E with P d ∈ D ρ ( d ) ≤ k such that for every vertex v in the subg r aph G ′ = ( V \ D , E \ D ) we hav e P e ∈ N G ′ ( v ) ρ ( e ) ∈ δ ( v ) ? The reduction w e use is from the following problem: Strongl y Regular Mul ticolored Clique Instanc e: A gra ph G = ( V , E ) with V = ⊎ i ∈ [ k ] V i such tha t | V i | = s a nd for all v ∈ V i we hav e d ( v ) | V j = d for all i , j , a nd an in teger k . Par ameter: k . Question: Is there a cliq ue V ′ of size k suc h that V ′ has one vertex from each V i ? In the remainder of this section w e present the hardnes s re du ction that demonstrates that g ( k ) -Appr ox-Ver tex De l etion to Re gul ar Subgraph and subsequently the more general problems are W [1] -hard, and therefore hav e no F P T - maint ena nce alg orithms. W e then pro ve that problems that are fixed- parameter trac ta ble may have an approximate maintenance algo rithm by proving that this is indeed the cas e for Ver tex Co ver . 8 3.1 Hardness Re sults Lemma 9. g ( k ) -Appro x-Ver tex Del etion to R egular Subgraph is W [1] -har d. Pr o of. The underlying exact problem Ver tex Deletion to Regular Sub- graph was shown to b e W [1] -hard in [13] with a reduction than can b e used with little mo dification to show hardness for the approximation pro blem. The reduction is from Str ongl y Regular Mul ticolored Clique . Let ( G, k ) of Strongl y Regular Mul ticolored Clique where V ( G ) = ⊎ i ∈ [ k ] V i is partitioned int o k co lor cla sses wher e | V i | = s fo r all i and each vertex has d neighbours in each color class. W e co nstruct an instance ( G ′ , k ′ ) of g ( k ) - Appr ox-Ver tex Deleti o n to Regular Subgraph by setting k ′ = k +  k 2  , creating a vertex copy V ′ i for e a ch co lo r class V i and a s et P ij of vertices for each pair V i , V j of colo r clas ses, which will con tro l edge selec tio n. W e make ea ch V ′ i a complete graph. F or ev er y pair of vertices u , v with u ∈ V i and v ∈ V j , i 6 = j , let u ′ and v ′ be the corre s po nding vertices in V ′ i and V ′ j . If u v ∈ E ( G ) w e add t wo vertices u ′ v ′ , v ′ u ′ to P ij with the edges u ′ u ′ v ′ , u ′ v ′ v ′ u ′ and v ′ v ′ u ′ . F or each pair of ver- tices u v , u ′ v ′ in P ij where u a nd u ′ are in the same V l ( l = i , j ) and u 6 = u ′ we add the edge u v u ′ v ′ . W e choose r to b e greater than max { ( s − 1) + d ( k − 1) , 2 + ( s − 1) d } such that r is also of opp os ite par it y to s , the smallest such r suffices. Then for each V ′ i we a dd a set of vertices V ′′ i where e a ch vertex in V ′′ i has a n edge to each vertex in V ′ i such tha t the degr ee of each v er tex in V ′ i is r + 1 . W e do similarly for each P ij with a set P ′ ij . W e then increase the degree of ea ch v er tex in each V ′′ i and each P ′ ij to r + 1 by taking an set of r + 1 clique, breaking an edge in each a nd a ttaching it at these tw o points to the vertex b eing adjusted. F or each of these vertices, if the total num ber of vertices in the attached cliques is less than g ( k ′ ) w e aug ment them by adding new r + 1 cliques by breaking an edge in the old and new clique a nd reattaching them to ea c h other. Claim. If G has a prop erly colored clique of size at least k , then we can delete at most k ′ vertices to make G ′ r -r egular. This claim follo ws a s in [1 3]. Note in particular that G has a clique of size a t most k (as ther e ar e only k color clas s es and w e m ust use at leas t k ′ vertices, one from e ach V ′ i and tw o from eac h P ij to ma ke the gra ph r -r egular. The vertices chosen for deletio n co r resp ond to the v er tices and edges of the clique. Claim. If G ′ can b e made r -regular by the deletion of at most g ( k ′ ) vertices, then G has a prop erly co lored clique of size at least k . Note that we must delete at least o ne vertex from each V ′ i and tw o from each P ij . If we dele te any v ertices from the degr ee adjustmen t comp on e nts of the graph, V ′′ i , P ′ ij or the adjustmen t c liques, then w e must delete all suc h vertices, which would require mo r e than g ( k ) deletio ns . Therefore the k ′ vertices must be the only vertices deleted. As with [13], these m ust corr esp ond to a prop erly colored k -clique in G . 9 Note a lso that if w e allow edge deletion as well as vertex dele tio n, we obta in no adv antage by deleting any edges, therefore any deletions must be of vertices only . This gives the following corollar y . Corollary 3. g ( k ) -Appro x-Deletion to Regular Subgraph is W [1] -har d. By s ubproblem con ta inment, the more genera l version where we allo w each vertex to have a list o f p ossible final degr ees, rather than simple degree r is also W [1] -ha rd. Corollary 4. g ( k ) -Appro x-Weighted Degree Constrained Deletion is W [1] -har d when the p ossible e diting op er ations ar e vertex deletion or vertex and e dge deletion. 3.2 A Maintenanc e Alg orithm for V ertex Co v er Thu s w e may conclude that maintenance is no t fea sible (in the complexity guar - antee sense) ev en if w e hav e po ly no mial time at each step (in fact we may relax this p erhaps to fixed-para meter tractable time, dep endent o n the num b er of steps), if the para meterized approximation pr oblem is W [ t ] - hard for any t ≥ 1 . How ever if the parameteriz e d a pproximation pro blem is fixed-para meter tractable, then we may have some hope. In a broad sense ma n y kernelisation algorithms may b e r ein ter pr eted a s ma in tenance a lg orithms, e.g., Lemma 10 ([16]). k -Max Cut has an F P T -time 2 k -lo c al m ainten anc e algo- rithm. Lemma 11 (A ttribute to Buss i n [5]). Ver tex Cover has an F P T -time 2 k 2 -lo c al maintenanc e algorithm. Note that these exa mples were not chosen as necessar ily the most efficient algorithms for these problem, but b ecause they employ simple, loca l reduction rules. How ever in b oth cases (and all such adaptations) the algorithm do es not use any prior solution, so p erha ps is not the most effective main tenance strat- egy . No netheless they do give g oals for the p erforma nce of any maintenance algorithm. By allowing an approximate so lution, we can develop potentially muc h sim- pler, faster and mor e lo cal maintenance a lg orithms. W e demons trate with the following case, Ver tex Cover , which is fixed-parameter tracta ble, therefore fixed-parameter approximable a nd has a simple greedy 2 -approximation. F ur- thermore there is ev idence that a 2 - approximation is in so me sense the b est po ssible ratio [11]. In this case, the following maintenance algorithm is efficien t and of tolerable approximation qualit y . Lemma 12. Ther e is a 1-lo c al maintenanc e algorithm for Maximal Ma tch- ing and henc e t her e is a 1-lo c al maintenanc e algorithm for Ver tex Cover with an app r oximation factor of 2 . 10 Pr o of. The algor ithm exploits the result that any maximal matching is a 2 - approximation for the size o f the minim um v ertex cov er. Given a gra ph G , with vertex cov er S , and a set o f edge deletions and ad- ditions T , the algor ithm pro ceeds b y main taining a matc hing at the site of the deletions and additions. W e may assume that a t each completed step in the algorithm (as this alg orithm can b e used to g e ner ate the initial solution) each non-isolated vertex in the vertex cov er has a p air-vertex , its partner in the match- ing. Then for each elemen t t of T , if t is a deletion that separates a v er tex from its pair -vertex, and the v ertex has any neighbour no t in the cov er, w e c ho ose arbitrarily one of these neighbour s to be its new pair -vertex. If it has no such neighbours, the vertex is r emov ed from the cov er. Otherwise if t is an addition that connects t wo vertices no t in the cov er , they are b oth added to the cov er and are noted as each others pair-vertex. Then a t an y p oint in the algorithm ea ch vertex in the cov er has an adjacen t pair-vertex (i.e., the edg e b et ween the tw o is a matc hing e dg e), and there is no edge that has neither endp oint in the cov er. Therefor e the implicit matching is maximal. Thus we immediately hav e our 2 -approximation. By way o f contrast, Max Cut admits a simple 1 2 approximation algo- rithm [1 8] (though this is far from the b est r atio) which is ea sily transla ted in to a simple ma in tenance algorithm how ever the lo cality seems to be the diam- eter of the graph in the worst case. This suggests that there is p otentially some separation pos sible betw een “lo cal” and “non-lo cal” problems that is not cap- tured by regula r co mplexit y classes. This idea has b een explored from a different per sp e ctiv e b y Milteren et al. [14] who show a split o f P -complete problems by the class incr - P O LY LO GT I M E . 4 Conclusion W e show that W [1] -hardness pr ecludes maint ena nce, and indeed given sufficient restriction on the stepwise co mputation no maintenance algor ithm can g uarantee any but the most trivial approximation ratios. How ever, as demonstrated with Ver tex Cover , some pro blems do admit fa s t, effectiv e approximate mainte- nance alg orithms. It is not clear whether there is a neat class ification of these problems, how e ver fixed-para meter tra ctabilit y for the r elated par ameterized (approximation) problem(s) is at least a necessa ry condition. The most lik ely situation is that there is s o me prop er subset of the pr oblems in F P T that admit a maintenance algor ithm, as the structure of a maint ena nce algorithm seems to rely on some as yet undefined lo ca lizing pr o per t y which is unlikely to b e iden tica l to the proper t y of being fixed-pa r ameter tra ctable. If indeed there is a coherent underlying prop erty that a llows maintenance, then p erha ps it would also b e po s- sible to construct maintenanc e-pr eserving r e ductions (similar to the incr emental r e ductions of [1 4]), whic h w o uld a llow a s impler and more robust demonstr ation of the p ossibility o r impossibility of main tena nce for a giv en problem. 11 References 1. G. Ausiello, V. Bonifaci, and B. Escoffier. Complexity and appro x imation in re- optimization. In S. B. Co op er and A. Sorbi, editors, Computability in Context: Computation and L o gic in the R e al W orld , pages 101–1 30. W orld Scientific, 2011. 2. B.-M. Bui- X uan, A. 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The ory of Computing Systems , 40(4):355–37 7, 2007. 20. Z. Zh ang. Routing in intermitten tly connected mobile ad ho c net works and dela y toleran t n et works: Overview and chal lenges. I EEE Communic ations Surveys and T utorials , 8(1–4):24–37 , 2006. 12 A P arameterized Appro ximation Complexit y In this section, we only pro vide the necessary definitions and prop erties of P a- rameterized Complexity and Parameterized Approximation Complexity . W e re - fer the reader to [5,9] a nd [12] r esp e c tiv ely for mor e deta ils. A.1 Basic Par ame terized Compl e xit y P a r ameterized Co mplexity explo r es the complexity of com binator ial problems using pa rameters as indep endent measure s of str ucture in ad dition to the overall size o f the input. An instance ( I , k ) of a parameterized problem co nsists o f the input I , corresp onding to the input of a classical problem and an in teger parameter k , a sp ecial part of the input indep endent from | I | . A pro blem is fixe d-p ar ameter t ra ctable (or in F P T ) if there is a n alg orithm that solves e ach instance ( I , k ) of the problem in time b ounded b y f ( k ) · | I | O (1) where f is a computable function dep enden t only on k . Conv ersely hardness for any class in the W -hierar ch y pr ovides evidence that a pr oblem is not fixed-para meter tractable. Har dness for such class e s is typically established by p ar ameterize d r e duction , the Parameterized Complexity r eduction scheme where given an instance ( I , k ) of problem Π 1 an insta nce ( I ′ , k ′ ) of problem Π 2 is computed in time b ounded by f ( k ) · | I | O (1) , with k ′ ≤ g ( k ′ ) for some computable function g and ( I , k ) is a Yes -instance if and only if ( I ′ , k ′ ) is a Yes -instance. A.2 P arameterized Appro xim ation Complexity The a dditional mea sure embo died in the para meter also provides a n alter native po ssibilit y for approximation problems. Of pa rticular in terest is appro x ima ting the cost of the solution, wher e the par ameter k is the desir ed cos t o f the solution. Given parameter ized problem Π with an a dditional optimizatio n ob jective (ei- ther min or max ), the general co s t minimization (maximization) para meter ized problem for Π is: g ( k ) -Appro x- Π Instanc e: A instance I of Π , an integer k . Par ameter: k . Output: Either No , asser ting that there is no solution o f siz e a t most (at least) k for I , or a solution of size at most (at least) g ( k ) . Of cours e for minimization problems the approximation is only int er esting if g ( k ) ≥ k , and vice versa for max imization. F or spec ific functions g ( k ) we obtain the following tw o interesting subcas es: c -Add-Appro x- Π Instanc e: A instance I of Π , an integer k . Par ameter: k . Output: Either No , asser ting that there is no solution o f siz e a t most (at least) k for I , or a solution of size at most k + c (at lea st k − c ). 13 c -Mul t-Appr ox- Π Instanc e: A instance I of Π , an integer k . Par ameter: k . Output: Either No , asser ting that there is no solution o f siz e a t most (at least) k for I , or a solution of size at most ck (at least k /c ). Then W [ t ] -ha rdness for any t ≥ 1 for such an approximation problem gives evidence that there is no fixed-parameter tractable alg orithm that can produce a solution of size b ounded by g ( k ) with k as the parameter . 14