A New View on Rural Postman Based on Eulerian Extension and Matching

Transcription

1 A New Vew on Rural Postman Based on Euleran Extenson and Matchng Manuel Sorge, René van Bevern, Rolf Nedermeer, and Mathas Weller Insttut für Softwaretechnk und heoretsche Informatk, U Berln {manuel.sorge, rene.vanbevern, rolf.nedermeer, mathas.weller}@tu-berln.de Abstract We provde a new characterzaton of the NP-hard arc routng problem Rural Postman n terms of a constraned varant of mnmum-weght perfect matchng on bpartte graphs. o ths end, we employ a parameterzed equvalence between Rural Postman and Euleran Extenson, a natural arc addton problem n drected multgraphs. We ndcate the NP-hardness of the ntroduced matchng problem. In partcular, we use t to make some partal progress towards answerng the open queston about the parameterzed complexty of Rural Postman wth respect to the number of weakly connected components n the graph nduced by the requred arcs. hs s a more than thrty years open and long-tme neglected queston wth sgnfcant practcal relevance. 1 Introducton he Rural Postman (RP) problem wth ts specal case, the Chnese Postman problem, s a famous arc routng problem n combnatoral optmzaton. Gven a drected, arcweghted graph G and a subset R of ts arcs (called requred arcs ), the task s to fnd a mnmum-cost closed walk n G that vsts all arcs of R. he manfold practcal applcatons of RP nclude snow plowng, garbage collecton, and mal delvery [1, 2, 3, 6, 7, 15]. Recently, t has been observed that RP s closely related (more precsely, parameterzed equvalent ) to the arc addton problem Euleran Extenson (EE). Here, gven a drected and arc-weghted multgraph G, the task s to fnd a mnmum-weght set of arcs to add to G such that the resultng multgraph s Euleran [10, 4]. RP and EE are NP-hard. In fact, ther mentoned parameterzed equvalence means that many algorthmc and complextytheoretc results for one of them transfer to the other. In partcular, ths gves a new vew on RP, perhaps leadng to novel approaches to attack ts computatonal hardness. A key ssue n both problems s to determne the nfluence of the number c of connected components 1 on each problem s computatonal complexty [4, 9, 11, 13, 14]. Indeed, Frederckson [9] observed that RP (and, thus, EE) s polynomal-tme solvable when c s constant. However, ths left open whether c nfluences the degree of the polynomal or Partally supported by the DFG, project AREG, NI 369/9 and project PABI, NI 369/7. Supported by the DFG, project AREG, NI 369/9. Supported by the DFG, project DARE, NI 369/11. 1 More precsely, c refers to the number of weakly connected components n the nput for EE and the number of weakly connected components n the graph nduced by the requred arcs for RP.

2 whether RP can be solved n f (c) n O(1) tme for some exponental functon f. In other words, t remaned open whether RP and EE are fxed-parameter tractable 2 wth respect to the parameter c [4]. We remark that ths parameter s presumably small n a number of applcatons [4, 7, 9], strongly motvatng to attack ths seemngly hard open queston. Our Results. In ths work, we contrbute new nsghts concernng the seemngly hard open queston whether RP (and EE) s fxed-parameter tractable wth respect to the parameter number of components. o ths end, our man contrbuton s a new characterzaton of RP n terms of a constraned varant of mnmum-weght perfect matchng on bpartte graphs. Referrng to ths problem as Conjonng Bpartte Matchng (CBM), we show ts NP-hardness and a parameterzed equvalence to RP and EE. Moreover, we show that CBM s fxed-parameter tractable 3 when restrcted to bpartte graphs where one partton set has maxmum vertex degree two. hs mples correspondng fxed-parameter tractablty results for relevant specal cases of RP and EE whch would probably have been harder to formulate and to detect usng the defntons of these problems. Indeed, we hope that CBM mght help to fnally answer the puzzlng open queston concernng the parameterzed complexty of RP wth respect to the number of components. In ths paper we consder decson problems. However, our results easly transfer to the correspondng optmzaton problems. For the sake of notatonal convenence and justfed by the known parameterzed equvalence [4], most of our results and proofs refer to EE nstead of RP. Due to space constrants, most proofs are deferred to a full verson of the paper. 2 Prelmnares and Preparatons Consder a drected multgraph G = (V, A), comprsng the vertex set V and the arc multset A. For notatonal convenence, we defne a component graph G as a clque whose vertces one-to-one correspond to the weakly connected components of G. Snce we never consder strongly connected components, we omt the adverb weakly. A walk W n G s a sequence of arcs n G such that each arc ends n the same vertex as the next arc starts n. We use V(W) and A(W) to refer to the set of vertces n whch arcs of W start or end, and the multset of arcs of W, respectvely. he frst vertex n the sequence s called the ntal vertex of the walk and the last vertex n the sequence s called the termnal vertex of the walk. A walk W n G such that A(W) s a submultset of the multset A(G) s called a tral of G. A tral n G such that every vertex n G has at most two ncdent arcs n A() s called a cycle f the ntal and termnal vertces of are equal, and path otherwse. If G s clear from the context, we omt t. We use balance(v) ndeg(v) outdeg(v) to denote the balance of a vertex v n G and I G + and IG to denote the set of all vertces v n G wth balance(v) > 0 and balance(v) < 0, respectvely. A vertex v s balanced f balance(v) = 0. 2 See Secton 2 and the lterature [5, 8, 12] for more on parameterzed complexty analyss. 3 he correspondng parameter jon set sze measures the nstance s dstance from trvalty and translates to the parameter number of components n equvalent nstances of EE and RP.

3 Our results are n the context of parameterzed complexty [5, 8, 12]. A parameterzed problem L Σ s called fxed-parameter tractable (FP) wth respect to a parameter k f (x, k) L s decdable n f (k) x O(1) tme, where f s a computable functon only dependng on k. We consder two types of parameterzed reductons between problems: A polynomalparameter polynomal-tme many-one reducton ( PPP m -reducton) from a parameterzed problem L to a parameterzed problem L s a polynomal-tme computable functon g such that (x, k) L (x, k ) L, wth (x, k ) g(x, k), and k p(k), where p s a polynomal only dependng on k. If such a reducton exsts, we wrte L PPP m L. A parameterzed urng reducton ( FP -reducton) from a parameterzed problem L to a parameterzed problem L s an algorthm that decdes (x, k) L n f (k) x O(1) tme, where queres of the form (x, g(k)) L are assumed to be decdable n O(1) tme and f, g are functons solely dependng on k. If such a reducton exsts, we wrte L FP L. If L FP L and L FP L, then we say that L and L are FP -equvalent. Note that every PPP m -reducton s a FP -reducton. Also, f L FP and L FP L, then L FP. In ths work, we consder the problem of makng a gven drected multgraph Euleran by addng arcs. A drected multgraph G s Euleran f t s connected and each vertex s balanced. An Euleran extenson E for G = (V, A) s a multset over V V such that G = (V, A E) s Euleran. Euleran Extenson (EE) Input: A drected multgraph G = (V, A), an nteger ω max, and a weght functon ω: V V [0, ω max ] { }. Queston: Is there an Euleran extenson E of G whose weght s at most ω max? In the context of EE we speak of allowed arcs a V V, f ω(a) <. 2.1 Preprocessng Routnes A polynomal-tme preprocessng for EE routne ntroduced by Dorn et al. [4] ensures that the balance of every vertex s n { 1, 0, 1}. hs smplfes the problem and helps n constructons later on. Dorn et al. [4] showed that the correspondng transformaton can be computed n O(n(n + m)) tme. In the followng, we assume that all nput nstances of EE have been transformed thusly, and hence, we assume that the followng observaton holds. Observaton 1. Let v be a vertex n a pre-processed nstance of EE. hen, balance(v) { 1, 0, 1}. We use a second preprocessng routne to make further observatons about trals n Euleran extensons. hs preprocessng s a varant of the algorthm used by Dorn et al. [4] to remove solated vertces from the nput graph. Bascally, t replaces the weght of a vertex par by the weght of the lghtest path n the graph (V, V V) wth respect to ω. Note that the resultng weght functon respects the trangle nequalty. Dorn et al. [4] showed that ths transformaton can be computed n O(n 3 ) tme. In the followng, we assume all nput nstances of EE to have gone through ths transformaton, and hence, we assume that the followng holds.

4 Observaton 2. Let ω be a weght-functon of a pre-processed nstance of EE. hen, ω respects the trangle nequalty, that s, for each x, y, z, t holds that ω(x, z) ω(x, y) + ω(y, z). In the subsequent sectons, we use ths preprocessng n parameterzed algorthms and reductons. o ths end, note that both transformatons are parameter-preservng, that s, they do not change the number of connected components. he presented transformatons lead to useful observatons regardng trals n Euleran extensons. For nstance, we often need the followng fact. Observaton 3. For any Euleran extenson E of G, there s an Euleran extenson E of at most the same weght such that any path p and any cycle c n E does not vst a connected component of G twce, except for the ntal and termnal vertex of p and c. 2.2 Advce Snce Euleran extensons have to balance every vertex, they contan paths startng n vertces wth postve balance and endng n vertces wth negatve balance. hese paths together wth cycles have to connect all connected components of the nput graph. In order to reduce the complexty of the problem, we use advce as addtonal nformaton on the structure of optmal Euleran extensons. Advce conssts of hnts whch specfy that there must be a path or cycle n an Euleran extenson that vsts connected components n a dstnct order. Hnts however do not specfy exactly whch vertces these paths or cycles vst. For an example of advce, see Fgure 1a. Formally, a hnt for a drected multgraph G = (V, A) s an undrected path or cycle t of length at least one n the component graph G together wth a flag determnng whether t s a cycle or a path. 4 Dependng on ths flag, we call the hnts cycle hnts and path hnts, respectvely. We say that a set of hnts H s an advce for the graph G f the hnts are edge-dsjont. 5 For a tral t n G, G (t) s the tral n G that s obtaned by makng t undrected and, for every connected component C of G, substtutng every maxmum length subtral t of t wth V(t ) C by the vertex n G correspondng to C. We say that a path p n the graph (V, V V) realzes a path hnt h f G (p) = h and the ntal vertex of p has postve balance and the termnal vertex has negatve balance n G. We say that a cycle c n the graph (V, V V) realzes a cycle hnt h f G (c) = h. We say that an Euleran extenson E heeds the advce H f t can be decomposed nto a set of paths and cycles that realze all hnts n H. A topc n ths work s how havng an advce helps n solvng an nstance of Euleran extenson. In order to dscuss ths, we ntroduce the followng verson of EE. Euleran Extenson wth Advce (EEA) Input: A drected multgraph G = (V, A), an nteger ω max, a weght functon ω: V V [0, ω max ] { }, and advce H. Queston: Is there an Euleran extenson E of G that s of weght at most ω max and heeds the advce H? 4 he flag s necessary because a hnt to a path n G may correspond to a cycle n G. 5 Note that there s a dfference between advce n our sense and the noton of advce n computatonal complexty theory. here, an advce apples to every nstance of a specfc length.

5 We wll see that the hard part of computng an Euleran extenson that heeds a gven advce H s to choose ntal and termnal vertces for path hnts n H. In fact, t s possble to compute optmal realzatons for all cycle hnts n any gven advce n O(n 3 ) tme. Observaton 4 ([16]). Let (G, ω max, ω, H) be an nstance of EEA. In O(n 3 ) tme we can compute an equvalent nstance (G, ω max, ω, H ) such that H does not contan a cycle hnt. Furthermore, the number of components at most decreases. In ths regard we note that we can compute an optmal realzaton of a path hnt for gven endponts n the correspondng drected multgraph. hs s possble n quadratc tme, manly usng Observaton 2, forbddng arcs contaned n one connected component, and Djkstra s algorthm. Observaton 5. Let (G, ω max, ω, H) be an nstance of EE, let h H be a path hnt and let C, C t be the connected components of G that correspond to the endponts of h. Furthermore, let (u, v) (C C t ) (I G + I G ). hen, we can compute a mnmum-weght realzaton of h wth ntal vertex u and termnal vertex v n O(n 2 ) tme. Snce we want to derve Euleran extensons from an advce and every Euleran extenson for a graph connects all of the graphs connected components, we are manly nterested n connectng advce. We say that an advce for a drected multgraph G s connectng, f ts hnts connect all vertces n G. Furthermore, f there s no connectng advce H wth H H, then H s called mnmal connectng advce. We consder the followng restrcted verson of EEA that allows only mnmal connectng advce (note that, by Observaton 4, we can assume the gven advce to be cycle-free). Euleran Extenson wth Cycle-free Mnmal Connectng Advce (EE CA) Input: A drected multgraph G = (V, A), an nteger ω max, a weght functon ω: V V [0, ω max ] { }, and mnmal connectng cycle-free advce H. Queston: Is there an Euleran extenson E of G wth weght at most ω max and heedng the advce H? We can show that each mnmal connectng cycle-free advce can be obtaned from a forest n G. Enumeratng these forests allows us to generate all such advces for a gven graph G wth c connected components n f (c) G O(1) ) tme, where f s some functon only dependng on c. Deferrng the presentaton of detals to a long verson of ths work, we state that EE s parameterzed urng reducble to EE CA [16]. Lemma 1. EE s FP -reducble to EE CA n 16c log(c) G O(1) tme. 3 Euleran Extenson and Conjonng Bpartte Matchng hs secton shows that Rural Postman (RP) s parameterzed equvalent to a matchng problem. By the parameterzed equvalence of RP and Euleran Extenson (EE) gven by Dorn et al. [4], we may concentrate on the equvalence of EE and matchng nstead. Frst we ntroduce a varant of perfect bpartte matchng. Let G be a bpartte graph, M be a matchng of the vertces n G, and let P be a vertex partton wth the cells C 1,..., C k. We call an unordered par {, j} of ntegers 1 < j k a jon and

6 a set J of such pars a jon set wth respect to G and P. We say that a jon {, j} J s satsfed by the matchng M of G f there s at least one edge e M wth e C and e C j. We say that a matchng M of G s J-conjonng wth respect to a jon set J f all jons n J are satsfed by M. If the jon set s clear from the context, we smply say that M s conjonng. Conjonng Bpartte Matchng (CBM) Input: A bpartte graph G = (V 1 V 2, E), an nteger ω max, a weght functon ω: E [0, ω max ], a partton P = {C 1,..., C k } of the vertces n G, and a jon set J. Queston: Is there a matchng M of the vertces of G such that M s perfect, M s conjonng and M has weght at most ω max? CBM can be nterpreted as a job assgnment problem wth addtonal constrants: an assgnment of workers to tasks s sought such that each worker s busy and each task s beng processed. Furthermore, every worker must be qualfed for hs or her assgned task. Both the workers and the tasks are grouped and the addtonal constrants are of the form At least one worker from group A must be assgned a task n group B. An assgnment that satsfes such addtonal constrants may be favorable n settngs where the workers are assgned to projects and the projects demand at least one worker wth addtonal qualfcatons. Over the course of the followng subsectons, we prove the followng theorem. heorem 1. Conjonng Bpartte Matchng and Euleran Extenson are FP -equvalent wth respect to the parameters jon set sze and number of connected components n the nput graph. he proof of heorem 1 conssts of four reductons, one of whch s a parameterzed urng reducton. he other three reductons are polynomal-tme polynomal-parameter many-one reductons. It s easy to see that the equvalence of EE and RP gven by Dorn et al. [4] also holds for the parameters number of components and number of components n the graph nduced by the requred arcs. hus, we obtan the followng from heorem 1. heorem 2. Conjonng Bpartte Matchng and Rural Postman are FP -equvalent wth respect to the parameters number of components n the graph nduced by the requred arcs and jon set sze. 3.1 From Euleran Extenson to Matchng In ths secton we sketch a reducton from EE CA to CBM. By Lemma 1 ths reducton leads to the followng theorem. heorem 3. Euleran Extenson s PPP m -reducble to Conjonng Bpartte Matchng wth respect to the parameters number of components and jon set sze. Outlne of the Reducton. he basc dea of our reducton s to use vertces of postve balance and negatve balance n an nstance of EE CA as the two cells of the graph bpartton n a desgnated nstance of CBM. Edges between vertces n the new nstances

7 (a) EE CA nstance (b) Long-hnt gadget n CBM nstance Fgure 1: Example of the long-hnt gadget. In (a) an EE A-nstance s shown, consstng of a graph wth three connected components and an advce that contans a sngle path hnt h (dashed lnes). In (b) a part of an nstance of CBM s shown, comprsng the cells that correspond to the ntal and termnal vertces of h and a gadget to model h. he gadget conssts of new vertces put nto a new cell whch s connected by two jons (dashed and dotted lnes) to the cells correspondng to the ntal and termnal vertces of h. represent shortest paths between these vertces that consst of allowed extenson arcs n the orgnal nstance. Every connected component n the orgnal nstance s represented by a cell n the partton n the matchng nstance and hnts are bascally modeled by jons. Descrpton of the Reducton. For the descrpton of the reducton, we need the followng defnton. Defnton 1. Let C 1,..., C c be the connected components of a drected multgraph G, and let H be a cycle-free advce for G. For every h H defne connect(h) {, j}, where C, C j are the components correspondng to the ntal and termnal vertces of h. Frst, consder an EE CA-nstance (G, ω max, ω, H) such that H s a cycle-free mnmal connectng advce that contans only hnts of length one. We wll deal wth longer hnts later. We create an nstance I CBM of CBM by frst defnng B 0 = (I G + I G, E 0) as a bpartte graph. Here, the set E 0 conssts of all edges {u, v} such that u I G +, v I G, and ω(u, v) <. 6 Second, we derve a vertex partton {V 1,..., V c} of B 0 by ntersectng the connected components of G wth (I G + I G ). he vertex partton obvously models the connected components n the nput graph, and the need for connectng them accordng to the advce H s modeled by an approprate jon-set J 0, defned as {connect(h) : h H}. Fnally, we make sure that matchngs also correspond to Euleran extensons weght-wse, by defnng the weght functon ω ({u, v}) for every u I G +, v I G as ω(u, v) wth ω max = ω max. By Observaton 3 we may assume that every hnt n H of length one s realzed by a sngle arc. Snce the advce connects all connected components, by the same observaton, we may assume that all other trals n a vald Euleran extenson have length one. Fnally, by Observaton 1, we may assume that every vertex has at most one ncdent ncomng or outgong arc n the extenson and, hence, we get an ntutve correspondence between conjonng matchngs and Euleran extensons. o model hnts of length at least two, we utlze gadgets smlar to the one shown n Fgure 1. he gadget comprses two vertces (u v and u v) for every par (u, v) of vertces wth one vertex n the component the hnt starts and one n the component the hnt ends. he vertces u v and u v are adjacent and each of these two vertces s connected wth 6 hs serves the purpose of modelng the structure of allowed arcs n the matchng nstance.

8 one vertex of the par t represents. he edge {u v, u} s weghted wth the cost t takes to connect u, v wth a path that realzes h. hs cost s computed usng Observaton 5. he edges {u v, u v} and {u v, v} have weght 0. Intutvely these three edges n the gadget represent one concrete realzaton of h. If u v and u v are matched, ths means that ths specfc path does not occur n a desgnated Euleran extenson. However, by addng the vertces of the gadget as cell to the vertex partton and by extendng the jon set to the gadget, we enforce that there s at least one outgong edge that s matched. If a perfect matchng matches u v wth u, then t also matches u v wth v and vce versa. hs ntroduces an edge to the matchng that has weght correspondng to a path that realzes h. 3.2 From Matchng to Euleran Extenson wth Advce We reduce Conjonng Bpartte Matchng to Euleran Extenson wth Advce: heorem 4. Conjonng Bpartte Matchng s PPP m -reducble to Euleran Extenson wth Advce wth respect to the parameters jon set sze and connected components n the nput graph. o reduce CBM to EEA we frst observe that for every nstance of CBM there s an equvalent nstance such that every cell n the nput vertex partton contans equal numbers of vertces from both cells of the graph bpartton. hs observaton enables us to model cells as connected components and vertces n the bpartte graph as unbalanced vertces n the desgnated nstance of EEA. Lemma 2. For every nstance of CBM there s an equvalent nstance comprsng the bpartte graph G = (V 1 V 2, E), the vertex partton P = {C 1,..., C k } and the jon set J, such that () for every 1 k t holds that V 1 C = V 2 C, and () the graph (P, {{C, C j } : {, j} J}) s connected. hs equvalent nstance contans at most one cell more than the orgnal nstance. Descrpton of the Reducton. o reduce nstances of CBM that conform to Lemma 2 to nstances of EEA we use the smple dea of modelng every cell as connected component, vertces n V 1 as vertces wth balance 1, vertces n V 2 as vertces wth balance 1, and jons as hnts. Constructon 1. Let the bpartte graph B = (V 1 V 2, E), the weght functon ω: E [0, ω max ], the vertex partton P = {C 1,..., C k } and the jon set J consttute an nstance I CBM of CBM that corresponds to Lemma 2. Let v 1 1, v2 1,..., v1 n/2, v2 n/2 be a sequence of all vertces chosen alternatngly from V 1 and V 2. Let the graph G = (V, A) (V 1 V 2, A 1 A 2 ) where the arc sets A 1 and A 2 are as follows: A 1 {(v 1, v2 ) : 1 n/2}. For every 1 j k let C j = {v 1,..., v jk }, and let A j 2 {(v, v +1 ) : 1 j k 1} {(v jk, v 1 }

9 and defne A 2 k j=1 A j 2. Defne a new weght functon ω for every par of vertces (u, v) V V by ω ω({u, v}), u V 2, v V 1, {u, v} E (u, v), otherwse. Fnally, derve an advce H for G by addng a length-one hnt h to H for every jon {o, p} J such that h conssts of the edge that connects vertces n G that correspond to the connected components C o, and C p. he graph G, the weght functon ω, the maxmum weght ω max and the advce H consttute an nstance of EEA. heorem 4 follows, snce Constructon 1 s a PPP m -reducton. 3.3 Removng Advce For heorem 1, t remans to show, that the advce n an nstance of EEA created by Constructon 1 can be removed. hat s, t remans to show the followng theorem. heorem 5. Euleran Extenson wth Advce s PPP m -reducble to Euleran Extenson wth respect to the parameter number of components n the nput graph. he basc deas for provng heorem 5 are as follows. Frst, we remove every cycle-hnt usng Observaton 4. We use the fact that every Euleran extenson has to connect all connected components of the nput graph. hus, for each hnt h, we ntroduce a new connected component C h. Let the components C s, C t correspond to the endponts of hnt h. o enforce that hnt h s realzed, we use the weght functon to allow an arc from every vertex wth balance 1 n C s to a number of dstnct vertces n C h. hs number s the number of vertces wth balance 1 n C t. hat s, for every par of unbalanced vertces n C s C t, we have an assocated vertex n C h. hen, for every nner vertex v on h, we copy C h and connect t to the component correspondng to v. From one copy to another, usng the weght functon, we allow only arcs that start and end n vertces correspondng to the same par of unbalanced vertces n C s C t. hs enforces that every hnt s realzed and connects every component t vsts. Usng the weght functon and Observaton 5 we can ensure that the arcs correspondng to a realzaton of a hnt have the weght of an optmal realzaton wth the same endponts. Usng ths constructon, heorem 5 can be proven whch concludes the proof of heorem 1. 4 Conjonng Bpartte Matchng: Propertes and Specal Cases hs secton nvestgates the propertes of CBM ntroduced n Secton 3. As dscussed before, CBM mght eventually help us derve a fxed-parameter algorthm for EE wth respect to the parameter number of connected components. Secton 4.1 frst shows that also CBM s NP-complete. Secton 4.2 then establshes tractablty of the problem on restrcted graph classes and translates ths tractablty result nto the world of EE and RP.

10 4.1 NP-Hardness NP-Hardness for Conjonng Bpartte Matchng (CBM) does not follow from the parameterzed equvalence to Euleran Extenson (EE) we gave n Secton 3, snce the reducton from EE we gave s a parameterzed urng reducton. o show that CBM s NP-hard, we polynomal-tme many-one reduce from the well-known 3SA, where a Boolean formula φ n 3-conjunctve normal form (3-CNF) s gven and t s asked whether there s an assgnment to the varables of φ that satsfes φ. Heren, a formula φ n 3-CNF s a conjuncton of dsjunctons of three lterals each, where each lteral s ether x or x and x s a varable of φ. In the followng, we represent each clause as three-element-set γ X {+, }, where (x, +) γ means that x s contaned n the clause represented by γ and (x, ) γ means that x s contaned n the clause represented by γ. Constructon 2. Let φ be a Boolean formula n 3-CNF wth the varables X {x 1,..., x n } and the clauses γ 1,..., γ m X {+, }. We translate φ nto an nstance of CBM that s a yes-nstance f and only f φ s satsfable. o ths end, for every varable x, ntroduce a cycle wth 4m edges consstng of the vertex set V {v j : 1 j 4m} and the edge set E {e k {v k, vk+1 } V } {e 4m {v 1, v4m }}. Let G ( n =1 V, n =1 E ), and let ω(e) 0, e E for any 1 n, and defne ω max 1. o construct an nstance of CBM, t remans to fnd a sutable partton of the vertces of G and a jon set. Inductvely defne the vertex partton P m of V(G) and the jon set J m as follows: Let J 0 =, and let P 0. For every clause γ j ntroduce the cell C j {v 4 j 1 : (x, +) γ j (x, ) γ j } {v 4 j 2 : (x, +) γ j } {v 4 j : (x, ) γ j }. Defne P j P j 1 {C j } and J j J j 1 {{0, j}}. Fnally, defne C 0 V(G) \ ( m j=1 C j ). he graph G, the weght functon ω, the vertex partton P m {C 0 } and the jon set J m consttute an nstance of CBM. Usng ths constructon, we can prove the followng theorem. heorem 6. CBM s NP-complete, even n the unweghted case and when the nput graph G = (V W, E) has maxmum degree two, and for every cell C n the gven vertex partton of G t holds that C V = C W. Proof. CBM s contaned n NP, because a perfect conjonng matchng of weght at most ω max s a certfcate for a yes-nstance. We prove that Constructon 2 s a polynomal-tme many-one reducton from 3SA to CBM. Notce that n nstances created by Constructon 2 any matchng has weght lower than ω max and, thus, the soundness of the reducton mples that CBM s hard even wthout the addtonal weght constrant. Also, snce the cells n the nstances of CBM are dsjont unons of edges, every cell n the partton P m contans the same number of vertces from each cell of the graph bpartton. It s easy to check that Constructon 2 s polynomal-tme computable. For the correctness we frst need the followng defnton: For every varable x X let M true M false {e k E : k odd} and E \ M true = {e k E : k even}.

11 Observe that all perfect matchngs n G are of the form n =1 M ν(x ), where ν s an assgnment of truth values to varables n X. We show that the matchng n =1 M ν(x ) s a conjonng matchng for G wth respect to the jon set J m f and only f ν s satsfes φ. For ths, t suffces to show that for every varable x X t holds that { j : (x, +) γ j } = { j : M true satsfes the jon {0, j}}, and (1) { j : (x, ) γ j } = { j : M false satsfes the jon {0, j}}. (2) We only show that (1) holds; (2) can be proven analogously. Assume that (x, +) γ j. By Constructon 2 v 4 j 2 C j, v 4 j 3 C 0 and thus, snce {v 4 j 2, v 4 j 3 } = e 4 j 3 M true the matchng M true satsfes the jon {0, j}. Now assume that (x, +) γ j, that s, ether (1) (x, ±) γ j or (2) (x, ) γ j. If (x, ±) γ j, then V and C j are dsjont and, thus, no matchng n G[V ] can satsfy the jon {0, j}. If (x, ) γ j, then the only edges n E that can satsfy the jon {0, j} are e 4 j 2 and e 4 j. Both edges are not n M true and, thus, ths matchng cannot satsfy the jon {0, j}., 4.2 ractablty on Restrcted Graph Classes hs secton presents data reducton rules and employs them to sketch an algorthm for CBM on a restrcted graph class, leadng to the followng theorem: heorem 7. Conjonng Bpartte Matchng can be solved n O(2 j( j+1) n + n 3 ) tme, where j s the sze of the jon set, provded that n the bpartte nput graph G = (V 1 V 2, E) each vertex n V 1 has maxmum degree two. In ths secton, let (G, ω max, ω, P = {C 1,..., C c }, J) be an nstance of CBM, where n G s as n heorem 7. he followng lemma plays a central role n the proof of heorem 7. It mples that, n a yes-nstance, every component of G conssts of an even-length cycle wth a collecton of parwse vertex-dsjont paths ncdent to t. Lemma 3. If G has a perfect matchng, then every connected component of G contans at most one cycle as subgraph. Proof. We show that f G contans a connected component that contans two cycles c 1, c 2 as subgraphs, then G does not have a perfect matchng. Frst assume that c 1, c 2 are vertex-dsjont. hen, there s a path p from a vertex v V(c 1 ) to a vertex w V(c 2 ) such that p s vertex-dsjont from c 1 and c 2 except for v, w. It s clear that both v, w V 2 because they have degree three. Consder the vertces V cp 1 (V(c 1 ) V(p) V(c 2 )) V 1 and the set V cp 2 (V(c 1 ) V(p) V(c 2 )) V 2. he set V cp 2 s the set of neghbors of vertces n V cp 1, because they have degree two and thus have neghbors only wthn p, c 1, and c 2. It s V cp 1 = ( E(c 1) + E(p) + E(c 2 ) )/2 snce nether of these paths and cycles overlap n a vertex n V 1. However, t s V cp 2 = Vcp 1 1 because c 1 and p overlap n v and c 2 and p overlap n w. hs s a volaton of Hall s condton and thus G does not have a perfect matchng.

12 he case where c 1 and c 2 share vertces can be proven analogously. (Observe that then there s a subpath of c 2 that s vertex-dsjont from c 1 and contans an even number of edges.) We now present four polynomal-tme executable data reducton rules for CBM. he correctness of the frst three rules s easy to verfy, whle the correctness of the fourth one s more techncal and omtted. We note that all rules can be appled exhaustvely n O(n 3 ) tme. Reducton Rule 1 removes paths ncdent to the cycles of a graph G n a yes-nstance. As a sde-result, Reducton Rule 1 solves CBM n lnear tme on forests. Reducton Rule 1. If there s an edge {v, w} E(G) such that deg(v) = 1, then remove both v and w from G, and remove all jons {, j} from J, wth v C, w C j. Decrease ω max by ω({v, w}). If exhaustvely applyng Reducton Rule 1 to G does not transform G such that each connected component s a cycle, whch s checkable n lnear tme, then, by Lemma 3, G does not have a perfect matchng and we can return NO. Hence, n the followng, assume that each connected component of G s a cycle. Reducton Rule 2 now deletes connected components that cannot satsfy jons. Reducton Rule 2. If there s a connected component D of G such that t contans no edge that could satsfy any jon n J, then compute a mnmum-weght perfect matchng M n G[D], remove D from G and decrease ω max by ω(m). After exhaustvely applyng Reducton Rule 2, we may assume that each connected component of G contans an edge that could satsfy a jon. We next present a data reducton rule that removes jons that are always satsfed. o ths end, we need the followng defnton. Defnton 2. For each connected component D (that s, each cycle) n G, denote by M 1 (D) a mnmum-weght perfect matchng of D wth respect to ω and denote by M 2 (D) E(D) \ M 1 (D) the other perfect matchng of D. 7 Furthermore, denote σ 1 (D) { j J : e M 1 (D) : e satsfes j}, σ 2 (D) { j J : e M 2 (D) : e satsfes j}, and the sgnature σ(d) of D as {σ 1 (D), σ 2 (D)}. Reducton Rule 3. Let D be a connected component of G. If there s a jon j σ 1 (D) σ 2 (D), then remove j from J. A fnal data reducton rule removes connected components that satsfy the same jons. Reducton Rule 4. Let S = {D 1,..., D j } be a maxmal set of connected components of G such that σ(d 1 ) =... = σ(d j ) and j 2. Let M1 = j k=1 M 1(D k ), let D l S such that ω(m 2 (D l )) ω(m 1 (D l )) s mnmum, and let M1 = M 1 \ M 1(D l ). 7 Note that n bpartte graphs every cycle s of even length.

13 () If the matchng M 1 s conjonng for the jon set σ 1(D 1 ) σ 2 (D 1 ), then remove each component n S from G, remove each jon n σ 1 (D 1 ) σ 2 (D 1 ) from the jon set J, and reduce ω max by ω(m 1 ). () If the matchng M 1 s not conjonng for the jon set σ 1(D 1 ) σ 2 (D 1 ), then remove each component n S \ {D l } from G, remove any jon n σ 1 (D 1 ) from the jon set J, and reduce ω max by ω(m 1 ). In ether case, update the partton P accordngly. Observaton 6. If Reducton Rule 4 s not applcable to G, then G contans at most one connected component for each of the 2 J +1 possble sgnatures. Now, heorem 7 follows: Explotng Observaton 6, a search-tree algorthm solvng CBM can n O(n) tme choose a jon j J and choose one of the at most 2 J +1 connected components of the graph that can satsfy j and match the component accordngly. hen, the algorthm can recurse on how the remanng J 1 jons are satsfed. Analyzng the pre-mages that lead to tractable nstances of CBM under the reductons we gave n Secton 3, heorem 7 can be translated to a tractablty result for EE. A smlar tractablty result can also be shown for Rural Postman. Due to ts length, we only state t for EE here. Corollary 1. Let the graph G and the weght functon ω consttute an nstance I EE of EE. Let c be the number of connected components n G. () If every path or cycle n the set of allowed arcs w.r.t. ω has length at most one, () f G contans only vertces wth balance between 1 and 1, () f every vertex n I G + (every vertex n I G ) has only outgong allowed arcs (ncomng allowed arcs), and (v) f n every connected component C of G, ether all vertces n I G + C or n I G C have at most two ncdent allowed arcs, then t s decdable n O(2 c(c+log(2c4 )) (n 4 + m)) tme whether I EE s a yes-nstance. 5 Concluson Clearly, the most mportant remanng open queston s to determne whether Rural Postman s fxed-parameter tractable wth respect to the number of connected components of the graph nduced by the requred arcs. hs queston also extends to the presumably harder undrected case. he newly ntroduced Conjonng Bpartte Matchng (CBM) problem mght also be useful n spottng new, computatonally feasble specal cases of Rural Postman and Euleran Extenson. he development of polynomal-tme approxmaton algorthms for CBM or the nvestgaton of other (structural) parameterzatons for CBM seem worthwhle challenges as well. Fnally, we remark that prevous work [10, 4] also left open a number of nterestng open problems referrng to varants of Euleran Extenson. Due to the practcal relevance of the consdered problems, our work s also meant to further stmulate more research on these challengng combnatoral problems.

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