On Approximating Restricted Cycle Covers

Transcription

1 On Approximting Restricted Cycle Covers Bodo Mnthey Universität zu Lüeck, Institut für Theoretische Informtik Rtzeurger Allee 160, Lüeck, Germny Astrct. A cycle cover of grph is set of cycles such tht every vertex is prt of exctly one cycle. An L-cycle cover is cycle cover in which the length of every cycle is in the set L. A specil cse of L-cycle covers re k-cycle covers for k N, where the length of ech cycle must e t lest k. The weight of cycle cover of n edge-weighted grph is the sum of the weights of its edges. We come close to settling the complexity nd pproximility of computing L-cycle covers. On the one hnd, we show tht for lmost ll L, computing L-cycle covers of mximum weight in directed nd undirected grphs is APX-hrd nd NP-hrd. Most of our hrdness results hold even if the edge weights re restricted to zero nd one. On the other hnd, we show tht the prolem of computing L-cycle covers of mximum weight cn e pproximted with fctor 2.5 for undirected grphs nd with fctor 3 in the cse of directed grphs. Finlly, we show tht 4-cycle covers of mximum weight in grphs with edge weights zero nd one cn e computed in polynomil time. As y-product, we show tht the prolem of computing minimum vertex covers in λ-regulr grphs is APX-complete for every λ 3. 1 Introduction The trvelling slesmn prolem (TSP) is perhps the est-known comintoril optimistion prolem. An instnce of the TSP is complete grph with edge weights, nd the im is to find minimum or mximum weight cycle tht visits every vertex exctly once. Such cycle is clled Hmiltonin cycle. Since the TSP is NP-hrd [10, ND22+23], we cnnot hope to lwys find n optiml cycle efficiently. For prcticl purposes, however, it is often sufficient to otin cycle tht is close to optiml. In such cses, we require pproximtion lgorithms, i.e. polynomil-time lgorithms tht compute such ner-optiml cycles. The prolem of computing cycle covers is relxtion of the TSP: A cycle cover of grph is spnning sugrph such tht every vertex is prt of exctly one simple cycle. Thus, solution to the TSP is cycle cover consisting of single cycle. In nlogy to the TSP, the weight of cycle cover in n edge-weighted grph is the sum of the weights of its edges. A full version of this work is ville t Supported y DFG reserch grnt RE 672/3. 3rd Workshop on Approximtion nd Online Algo. (WAOA 2005) c Springer

2 In contrst to the TSP, cycle covers of mximum weight cn e computed efficiently. This fct is exploited in pproximtion lgorithms for the TSP; the computtion of cycle covers forms the sis for the currently est known pproximtion lgorithms for mny vritions of the TSP. These lgorithms usully strt y computing n initil cycle cover nd then join cycles to otin Hmiltonin cycle. Short cycles in cycle cover limit the pproximtion rtios chieved y such lgorithms. In generl, the longer the cycles in the initil cover re, the etter the pproximtion rtio. Thus, we re interested in computing cycle covers without short cycles. Moreover, there re pproximtion lgorithms tht ehve prticulrly well if the cycle covers tht re computed do not contin cycles of odd length [6]. Finlly, some so-clled vehicle routing prolems (cf. e.g. Hssin nd Ruinstein [12]) require covering vertices with cycles of ounded length. Therefore, we consider restricted cycle covers, where cycles of certin lengths re ruled out priori: Let L N, then n L-cycle cover is cycle cover in which the length of ech cycle is in L. To fthom the possiility of designing pproximtion lgorithms sed on computing cycle covers, we im to chrcterise the sets L for which L-cycle covers of mximum weight cn e computed efficiently. 1.1 Preliminries A cycle cover of grph G = (V, E) is sugrph of G tht consists solely of cycles such tht ll vertices in V re prt of exctly one cycle. The length of cycle is the numer of edges it consists of. We re concerned with simple grphs, i.e. the grphs do not contin multiple edges or loops. Thus, the shortest cycles of undirected nd directed grphs hve length three nd two, respectively. An L-cycle cover is cycle cover in which the length of every cycle is in the set L N. For undirected grphs, we hve L U = {3, 4, 5,...}, while L D = {2, 3, 4,...} in cse of directed grphs. A k-cycle cover is {k, k + 1,...}-cycle cover. Let L = U \ L in the cse of undirected grphs nd L = D \ L in the cse of directed grphs (this will e cler from the context). Given n edge weight function w : E N, the weight w(c) of suset C E of the edges of G is w(c) = e C w(e). This prticulrly defines the weight of cycle cover since we view cycle covers s sets of edges. Let U V e ny suset of the vertices of G. The internl edges of U re ll edges of G tht hve oth vertices in U. We denote y w U (C) the sum of the weights of ll internl edges of U in C. The externl edges t U re ll edges of G with exctly one vertex in U. For L U, the set L-UCC contins ll undirected grphs tht hve n L-cycle cover s spnning sugrph. Mx-L-UCC is the following optimistion prolem: Given complete undirected grph with edge weights zero nd one, find n L-cycle cover of mximum weight. We cn lso consider the grph s eing not complete nd without edge weights. Then we try to find n L-cycle cover with minimum numer of nonedges ( non-edges correspond to weight zero edges, edges to weight one edges). Thus, Mx-L-UCC cn e viewed s generlistion of L-UCC.

3 Mx-W-L-UCC is the prolem of finding mximum-weight L-cycle covers in grphs with ritrry non-negtive edge weights. For k 3, k-ucc, Mx-k-UCC, nd Mx-W-k-UCC re defined like L-UCC, Mx-L-UCC nd Mx-W-L-UCC except tht k-cycle covers insted of L-cycle covers re sought. L-DCC, Mx-L-DCC, Mx-W-L-DCC, k-dcc, Mx-k-DCC, nd Mx-W-k-DCC re defined for directed grphs like L-UCC, Mx-L-UCC, Mx-W-L-UCC, k-ucc, Mx-k-UCC, nd Mx-W-k-UCC for undirected grphs except tht L D nd k 2. An instnce of Min-Vertex-Cover is n undirected grph H = (X, F ). A vertex cover of H is suset X X such tht t lest one vertex of every edge in F is in X. The im is to find vertex cover of minimum crdinlity. Min-Vertex-Cover(λ) is Min-Vertex-Cover restricted to λ-regulr grphs, i.e. to simple grphs in which every vertex is incident to exctly λ edges. Alredy Min-Vertex-Cover(3) is APX-complete [2]. We refer to Ausiello et l. [3] for survey on NP optimistion prolems. 1.2 Existing Results Undirected Grphs. U-UCC, Mx-U-UCC, nd Mx-W-U-UCC cn e solved in polynomil time vi reduction to the clssicl perfect mtching prolem, which cn e solved in polynomil time [1, Chp. 12]. Hrtvigsen presented polynomil-time lgorithm for computing mximum-crdinlity tringle-free two-mtching [11] (see lso Sect. 5). His lgorithm cn e used to decide 4-UCC in polynomil time. Furthermore, it cn e used to pproximte Mx-4-UCC within n dditive error of one ccording to Bläser [4]. Mx-W-k-UCC dmits simple fctor 3/2 pproximtion for ll k: Compute mximum weight cycle cover, rek the lightest edge of ech cycle, nd join the cycles to otin Hmiltonin cycle, which is sufficiently long if the grph contins t lest k vertices. Unfortuntely, this lgorithm cnnot e generlised to work for Mx-W-L-UCC with ritrry L. For the prolem of computing k- cycle covers of minimum weight in grphs with edge weights one nd two, there exists fctor 7/6 pproximtion lgorithm for ll k [8]. Cornuéjols nd Pulleylnk presented proof due to Ppdimitriou tht 6- UCC is NP-complete [9]. Vornerger showed tht Mx-W-5-UCC is NP-hrd [14]. For k 7, Mx-k-UCC nd Mx-W-k-UCC re APX-complete [5]. Hell et l. [13] proved tht L-UCC is NP-hrd for L {3, 4}. For most L, L-UCC, Mx-L-UCC, nd Mx-W-L-UCC re not even recursive since there re uncountly mny L. Thus for most L, L-UCC is not in NP nd Mx-L-UCC nd Mx-W-L-UCC re not in NPO. This does not mtter for hrdness results ut my cuse prolems when one wnts to design pproximtion lgorithms tht se on computing L-cycle covers. However, our pproximtion lgorithms work for ritrry L, independently of the complexity of L. Directed Grphs. D-DCC, Mx-D-DCC, nd Mx-W-D-DCC cn e solved in polynomil time y reduction to the mximum weight perfect mtching prolem

4 L-UCC Mx-L-UCC Mx-W-L-UCC L = in P in PO in PO L = {3} in P in PO L = {4} L = {3, 4} APX-complete else NP-hrd APX-hrd APX-hrd () Undirected cycle covers. L-DCC Mx-L-DCC Mx-W-L-DCC L = {2} L = D in P in PO in PO else NP-hrd APX-hrd APX-hrd () Directed cycle covers. Tle 1. The complexity of computing L-cycle covers. in iprtite grphs [1, Chp. 12]. But lredy 3-DCC is NP-complete [10, GT13]. Mx-k-DCC nd Mx-W-k-DCC re APX-complete for ll k 3 [5]. Similr to the fctor 3/2 pproximtion lgorithm for undirected cycle covers, Mx-W-k-DCC hs simple fctor 2 pproximtion lgorithm for ll k: Compute mximum weight cycle cover, rek the lightest edge of every cycle, nd join the cycles to otin Hmiltonin cycle. Agin, this lgorithm cnnot e generlised to work for Mx-W-L-DCC with ritrry L. There re fctor 4/3 pproximtion lgorithm for Mx-W-3-DCC [7] nd fctor 3/2 pproximtion lgorithm for Mx-k-DCC for k 3 [5]. As in the cse of cycle covers in undirected grphs, for most L, L-DCC, Mx-L-DCC, nd Mx-W-L-DCC re not recursive. 1.3 New Results We come close to settling the complexity nd pproximility of restricted cycle covers. Only the complexity of the five prolems 5-UCC, {4}-UCC Mx-5-UCC, Mx-{4}-UCC, nd Mx-W-4-UCC remins open. Tle 1 shows n overview on the complexity of computing restricted cycle covers. Hrdness Results. We prove tht Mx-L-UCC is APX-hrd for ll L with L {3, 4} (Sect. 3). We lso prove tht Mx-W-L-UCC is APX-hrd if L {3} (this follows from the results of Sect. 3 nd the APX-completeness of Mx-W-5-UCC nd Mx-W-{4}-UCC shown in Sect. 2). The hrdness results for Mx-W-L- UCC hold even if we llow only the edge weights zero, one, nd two. We show dichotomy for cycle covers of directed grphs: For ll L with L {2} nd L D, L-DCC is NP-hrd (Theorem 6) nd Mx-L-DCC nd Mx-W-L-DCC re APX-hrd (Theorem 5), while it is known tht ll three prolems re solvle in polynomil time if L = {2} or L = D. To show the hrdness of directed cycle covers, we show tht certin kinds of grphs, clled L-clmps, exist for non-empty L D if nd only if L D (Theorem 4). This grph-theoreticl result might e of independent interest.

5 As y-product, we prove tht Min-Vertex-Cover(λ) is APX-complete for ll λ 3 (Sect. 6). We need this result for the APX-hrdness proofs in Sect. 3. Algorithms. We present polynomil-time fctor 2.5 pproximtion lgorithm Mx-W-L-UCC nd fctor 3 pproximtion lgorithm for Mx-W-L-DCC (Sect. 4). Both lgorithms work for ritrry L. Finlly, we prove tht Mx-4-UCC is solvle in polynomil time (Sect. 5). 2 A Generic Reduction for L-Cycle Covers In this section, we present generic reduction from Min-Vertex-Cover(3) to Mx- L-UCC or Mx-W-L-UCC. To instntite the reduction for certin L, we use smll grph, which we cll gdget, the specific structure of which depends on L. Such gdget together with the generic reduction is n L-reduction from Min- Vertex-Cover(3) to Mx-L-UCC or Mx-W-L-UCC. The mere im is to prove the APX-hrdness of Mx-W-{4}-UCC nd Mx-W-5-UCC. 2.1 The Generic Reduction Let H = (X, F ) e cuic grph with vertex set X nd edge set F s n instnce of Min-Vertex-Cover(3). Let n = X nd m = F = 3n/2. We construct n undirected complete grph G with edge weight function w s generic instnce of Mx-L-UCC or Mx-W-L-UCC. For ech edge = {x, y} F, we construct sugrph F of G clled the gdget of. We define F s set of vertices, thus w F (C) for suset C of the edges of G is well defined. This gdget contins four distinguished vertices x in, x out, y in, nd y out. These four vertices re used to connect F to the rest of the grph. Wht such gdget looks like depends on L. If ll edges in such gdget hve weight zero or one, we otin n instnce of Mx-L-UCC since ll edges etween different gdgets will hve weight zero or one. Otherwise, we hve n instnce of Mx-W-L-UCC. Figure 2 shows n exmple of such gdget. Let,, c F e the three edges incident to vertex x X (the order is ritrry). Then we ssign weight one to the edges connecting x out to x in nd x out to x in c nd weight zero to the edge connecting x out c to x in. We cll the three edges {x out, x in }, {xout, x in c }, nd {x out c, x in } the junctions of x. We sy tht {x out, x in } nd {xout c, x in } re the junctions of x t F. Figure 1 shows n exmple. We cll n edge illegl if it connects two different gdgets ut is not junction. Thus, n illegl edge is n externl edge t two different gdgets. All illegl edges hve weight zero, i.e. there re no edges of weight one tht connect two different gdgets except for the junctions. The weights of the internl edges of the gdgets depend on the gdget, which in turn depends on L. The following terms re defined for ritrry susets C of the edges of G, nd so in prticulr for L-cycle covers. We sy tht C leglly connects F if C contins no illegl edges incident to F,

6 y x y c y () Vertex x nd its edges. x in y in F x out y out x in y in F x out y out x in c y in c F c x out c y out c () The gdgets F, F, nd F c nd their connections vi the three junctions of x. The dshed edge hs weight zero. Other weight zero edges nd the junctions of y, y, nd y re not shown. Fig. 1. The construction for vertex x X incident to,, c F. C contins exctly two or four junctions t F, nd if C contins exctly two junctions t F, then these elong to the sme vertex x. We cll C legl if C leglly connects ll gdgets. If C is legl, then for ll x X, either ll junctions of x re in C or no junction of x is in C. Furthermore, from legl set C we otin vertex cover X = {x the junctions of x re in C}. Let us now define the requirements the gdgets must fulfil. In the following, let C e n ritrry L-cycle cover of G nd = {x, y} F e n ritrry edge of H. R0: There exists fixed numer s N, which we cll the gdget prmeter, tht depends only on the gdget. The role of the gdget prmeter will ecome cler in the susequent requirements. R1: w F (C) s 1. R2: If C contins 2α externl edges t F, then w F (C) s α. R3: If C contins exctly one junction of x t F nd exctly one junction of y t F, then w F (C) s 2. (In this cse, C does not leglly connect F.) R4: Let C e n ritrry suset of the edges of G tht leglly connects F. Assume tht there re 2α junctions (α {1, 2}) t F in C. Then there exists C with the following properties: C differs from C only in F s internl edges nd w F (C ) = s α. Thus, given C, C cn e otined y loclly modifying C within F. We cll the process of otining C from C rerrnging C in F. R5: Let C e legl suset of the edges of G. Then there exists suset C of edges otined y rerrnging ll gdgets s descried in R4 such tht C is n L-cycle cover. The requirements ssert tht connecting the gdgets leglly is never worse thn connecting them illeglly. This yields the min result of this section. Lemm 1. Assume tht gdget s descried exists for L U. Then the reduction presented is n L-reduction from Min-Vertex-Cover(3) to Mx-W-L-UCC. If the gdget contins only edges of weight zero or one, then the reduction is n L-reduction from Min-Vertex-Cover(3) to Mx-L-UCC.

7 x in x out y in y out Fig. 2. The edge gdget F for n edge = {x, y} tht is used to prove the APXcompleteness of Mx-W-5-UCC. Bold edges re internl edges of weight two, solid edges re internl edges of weight one, internl edges of weight zero re not shown. The dshed nd dotted edges re the junctions of x nd y, respectively, t F. () x X. () y X. (c) x, y X. Fig. 3. Trversls of the gdget for Mx-W-5-UCC tht chieve mximum weight. 2.2 Mx-W-5-UCC nd Mx-W-{4}-UCC The gdget for Mx-W-5-UCC is shown in Fig. 2. Let G e the grph constructed vi the reduction presented in Sect. 2.1 with the gdget of this section. Let C e n ritrry L-cycle cover of G nd = {x, y} F. By proving tht it fulfils ll requirements, we otin the following result. Theorem 1. Mx-W-5-UCC is APX-hrd, even if the edge weights re restricted to e zero, one, or two. Although the sttus of Mx-5-UCC is still open, llowing only one dditionl edge weight of two lredy yields n APX-complete prolem. The generic reduction together with the gdget used for Mx-W-5-UCC works lso for Mx-W-{4}-UCC. The gdget only requires tht cycles of length four re foridden since otherwise R1 is not stisfied. Thus, ll requirements re fulfilled for Mx-W-{4}-UCC in exctly the sme wy s for Mx-W-5-UCC. Theorem 2. Mx-W-{4}-UCC is APX-hrd, even if the edge weights re restricted to e zero, one, or two. 3 A Uniform Reduction for L-Cycle Covers 3.1 Clmps We now define so-clled clmps, which were introduced y Hell et l. [13]. Clmps re crucil for the hrdness proof presented in this section. Let K = (V, E) e n undirected grph, let u, v V e two vertices of K, nd let L U. We denote y K u nd K v the grphs otined from K y deleting u nd v, respectively, nd their incident edges. Moreover, K u v denotes the

8 v } {{ } Λ 3 vertices u Fig. 4. An L-clmp for finite L with mx(l) = Λ. grph otined from K y deleting oth u nd v. Finlly, for k N, K k is the following grph: Let y 1,..., y k e vertices with y i / V, dd edges {u, y 1 }, {y i, y i+1 } for 1 i k 1, nd {y k, v}. For k = 0, we directly connect u to v. The grph K is clled n L-clmp if the following properties hold: Both K u nd K v contin n L-cycle cover. Neither K nor K u v nor K k for ny k N contins n L-cycle cover. We cll u nd v the connectors of the L-clmp K. Lemm 2 (Hell et l. [13]). Let L U e non-empty. Then there exists n L-clmp if nd only if L {3, 4}. Figure 4 shows n exmple of n L-clmp for finite L. If there exists n L-clmp for some L, then we cn ssume tht the connectors u nd v oth hve degree two since we cn remove ll edges tht re not used in the L-cycle covers of K v nd K u. For our purpose, consider ny non-empty set L {3, 4, 5,...} with L {3, 4}. We fix one L-clmp K with connectors u, v V ritrrily nd refer to it in the following s the L-clmp, lthough there exists more thn one L-clmp. Let σ e the numer of vertices of K. We re concerned with edge-weighted grphs. Therefore, we trnsfer the notion of clmps to grphs with edge weights zero nd one in the ovious wy: Let G e n undirected complete grph with vertex set V nd edge weights zero nd one nd let K e n L-clmp. Let U V. We sy tht U is n L-clmp with connectors u, v U if the sugrph of G induced y U restricted to the edges of weight one is isomorphic to K with u nd v mpped to connectors of K. 3.2 The Reduction Let L U e non-empty with L {3, 4}. Thus, L-clmps exist nd we fix one s in the previous section. Let σ e the numer of vertices in the L-clmp. Let λ = min(l). (This choice is ritrry. We could choose ny numer in L.) We will reduce Min-Vertex-Cover(λ) to Mx-L-UCC. Min-Vertex-Cover(λ) is APX-complete since λ 3 (see Sect. 6). Let H = (X, F ) e n instnce of Min-Vertex-Cover(λ) with n = X vertices nd m = F = λn/2 edges. Our instnce G for Mx-L-UCC consists of λ sugrphs G 1,..., G λ, ech contining 2σm vertices. We strt y descriing G 1.

9 X 1 Y 1 x 1 z 1 p 1 y 1 q 1 t 1 Fig. 5. The edge gdget for = {x, y} consisting of two L-clmps. The vertex z 1 is the only vertex tht elongs to oth clmps X 1 nd Y 1. Then we stte the differences etween G 1 nd G 2,..., G λ nd sy to which edges etween these grphs we ssign weight one. Let = {x, y} F e ny edge of H. We construct n edge gdget F for tht consists of two L-clmps X 1 nd Y 1 nd one dditionl vertex t 1 s shown in Fig. 5. The connectors of X 1 re x 1 nd z 1 while the connectors of Y 1 re y 1 nd z, 1 i.e. X 1 nd Y 1 shre the connector z. 1 Let p 1 nd q 1 e the two unique vertices in Y 1 tht shre weight one edge with z. 1 (The choice of Y 1 is ritrry, we could choose the corresponding vertices in X 1 s well.) We ssign weight one to oth {p 1, t 1 } nd {q, 1 t 1 }. Thus, the vertex t 1 cn lso serve s connector for Y 1. Now let x X e ny vertex of H nd let 1,..., λ F e the λ edges tht re incident to x. We connect the vertices x 1 1,..., x 1 λ to form pth y ssigning weight one to the edges {x 1 η, x 1 η+1 } for η {1,..., λ 1}. Together with edge {x 1 λ, x 1 1 }, these edges form cycle of length λ L, ut note tht w({x 1 λ, x 1 1 }) = 0. These λ edges re clled the junctions of x. The junctions t F for some = {x, y} F re the junctions of x nd y tht re incident to F. Overll, the grph G 1 consists of 2σm vertices since every edge gdget consists of 2σ vertices. The grphs G 2,..., G λ re lmost exct copies of G 1. The grph G ξ, ξ {2,..., λ} hs clmps X ξ nd Y ξ nd vertices x ξ, y, ξ z, ξ t ξ, p ξ, q ξ for ech edge = {x, y} F, just s ove. The edge weights re lso identicl with the single exception tht the edge {x ξ λ, x ξ 1 } lso hs weight one. Note tht we only use the term gdget for the sugrphs of G 1 defined ove lthough lmost the sme sugrphs occur in G 2,..., G λ s well. Similrly, the term junction refers only to n edge in G 1 s defined ove. Finlly, we descrie how to connect G 1,..., G λ with ech other. For every edge F, there re λ vertices t 1,..., t λ. These re connected to form cycle consisting solely of weight one edges, i.e. we ssign weight one to ll edges {t ξ, t ξ+1 } for ξ {1,..., λ 1} nd to {t λ, t 1 }. Figure 6 shows n exmple of the whole construction from the viewpoint of single vertex. As in the previous section, we cll edges tht re not junctions ut connect two different gdgets illegl. Edges with oth vertices in the sme gdget re gin clled internl edges. In ddition to junctions, illegl edges, nd internl edges, we hve fourth kind of edges: The t-edges of F for F re the two edges {t 1, t 2 } nd {t 1, t λ }. The t-edges re not illegl. All other edges connecting G 1 to G ξ for ξ 1 re illegl.

10 F F F c x 1 t 1 y 1 x 1 t 1 y 1 x 1 c t 1 c y 1 c x 2 t 2 y 2 x 2 t 2 y 2 x 2 c t 2 c y 2 c x 3 t 3 y 3 x 3 t 3 y 3 x 3 c t 3 c y 3 c Fig. 6. The construction for vertex x X incident to edges,, c F for λ = 3 (Fig. 1() on pge 6 shows the corresponding grph). The drk grey res re the edge gdgets F, F, nd F c. Their copies in G 2 nd G 3 re light grey. The cycles connecting the t-vertices re dotted. The cycles connecting the x-vertices re solid, except for the edge {x 1 c, x 1 }, which hs weight zero nd is dshed. The vertices z, 1..., zc 3 re not shown for legiility. Let C e ny suset of the edges of the grph G thus constructed, nd let = {x, y} F e n ritrry edge of H. We sy tht C leglly connects F if the following properties re fulfilled: C contins no illegl edges incident to F nd exctly two or four junctions t F. If C contins exctly two junctions t F, then these elong to the sme vertex nd there re two t-edges t F in C. If C contins four junctions t F, then these re the only externl edges in C incident to F. In prticulr, C does not contin t-edges t F. We cll C legl if C leglly connects ll gdgets. We cn prove tht the construction descried ove is n L-reduction from Min-Vertex-Cover(λ) to Mx-L-UCC for ll L with L {3, 4}. Theorem 3. For ll L U with L {3, 4}, Mx-L-UCC is APX-hrd. 3.3 Clmps in Directed Grphs The im of this section is to introduce directed L-clmps. Let K = (V, E) e directed grph nd u, v V. Agin, K u, K v, nd K u v denote the grphs otined y deleting u, v, nd oth u nd v, respectively. For k N, Ku k denotes the following grph: Let y 1,..., y k / V e new vertices nd dd edges (u, y 1 ), (y 1, y 2 ),..., (y k, v). For k = 0, we dd the edge (u, v). The grph Kv k is similrly defined, except tht we now strt t v, i.e. we dd the edges (v, y 1 ), (y 1, y 2 ),..., (y k, u). Kv 0 is K with the dditionl edge (v, u). Now we cn define clmps for directed grphs: Let L D. A directed grph K = (V, E) with u, v V is directed L-clmp with connectors u nd v if the following properties hold:

11 Both K u nd K v contin n L-cycle cover. Neither K nor K u v nor K k u nor K k v for ny k N contins n L-cycle cover. Theorem 4. Let L D e non-empty. Then there exists directed L-clmp if nd only if L D. 3.4 L-Cycle Covers in Directed Grphs From the hrdness results in the previous sections nd the work y Hell et l. [13], we otin the NP-hrdness nd APX-hrdness of L-DCC nd Mx- L-DCC, respectively, for ll L with 2 / L nd L {2, 3, 4}: We use the sme reduction s for undirected cycle covers nd replce every undirected edge {u, v} y pir of directed edges (u, v) nd (v, u). However, this does not work if 2 L nd lso leves open the cses when L {2, 3, 4}. We will show tht L = {2} nd L = D re the only cses in which directed L-cycle covers cn e computed efficiently y proving the NP-hrdness of L-DCC nd the APX-hrdness of Mx- L-DCC for ll other L. Thus, we settle the complexity for directed grphs. The APX-hrdness of the directed cycle cover prolem is otined y proof similr to the proof for undirected cycle covers. All we need is λ L with λ 3 nd the existence of n L-clmp. Theorem 5. Let L D e non-empty set. If L {2} nd L D, then Mx-L-DCC nd Mx-W-L-DCC re APX-hrd. We cn lso prove tht for ll L / {{2}, D}, L-DCC is NP-hrd. Theorem 6. Let L D e non-empty set. If L {2} nd L D, then L-DCC is NP-hrd. Let L / {{2}, D}. L-DCC is in NP nd therefore NP-complete if nd only if the lnguge {1 λ λ L} is in NP. 4 Approximtion Algorithms The gol of this section is to devise pproximtion lgorithms for Mx-W-L- UCC nd Mx-W-L-DCC tht work for ritrry L. The ctch is tht for most L it is impossile to decide whether some cycle length is in L or not. One possiility would e to restrict ourselves to sets L such tht {1 λ λ L} is in P. For such L, Mx-W-L-UCC nd Mx-W-L-DCC re NP optimistion prolems. Another possiility for circumventing the prolem is to include the permitted cycle lengths in the input. However, it turns out tht such restrictions re not necessry since we cn restrict ourselves to finite sets L. A necessry nd sufficient condition for complete grph with n vertices to hve n L-cycle cover is tht there exist (not necessrily distinct) lengths λ 1,..., λ k L for some k N with k i=1 λ i = n. We cll such n n L- dmissile nd define L = {n n is L-dmissile}.

12 Input: n undirected grph G = (V, U(V )) with V = n; n edge weight function w : U(V ) N Output: n L-cycle cover C px of G if n is L-dmissile, otherwise 1. If n / L, then return. 2. Compute cycle cover C init of mximum weight. 3. Compute suset P C init of mximum weight such tht (V, P ) consists of n/5 pths of length two nd n 3 n/5 isolted vertices. 4. Join the pths to otin n L-cycle cover C px, return C px. Fig. 7. A fctor 2.5 pproximtion lgorithm for Mx-W-L-UCC. Lemm 3. For ll L N, there exists finite set L L with L = L. Insted of computing L -cycle covers in the following, we ssume without loss of generlity tht L is lredy finite set. The min ide of the two pproximtion lgorithms is s follows: We strt y computing cycle cover C init of mximum weight. Then we tke suset S of the edges of C init tht weighs s much s possile under the restriction tht there exists n L-cycle cover tht includes ll edges of S. We dd edges to otin n L-cycle cover C px S. Let C e n L-cycle cover of mximum weight, nd ssume tht we cn gurntee ρ w(s) w(c init ) for some ρ 1. Then w(c ) w(c init ) ρ w(s) ρ w(c px ). Thus, we hve computed fctor ρ pproximtion to n L-cycle cover of mximum weight. 4.1 Approximting Undirected Cycle Covers The input of our lgorithm for undirected grphs is n undirected complete grph G = (V, U(V )) with V = n nd n edge weight function w : U(V ) N. The min ide of the pproximtion lgorithm is s follows: Every cycle cover cn e decomposed into n/5 vertex-disjoint pths of length two nd n 3 n/5 isolted vertices. Conversely, every collection P of n/5 pths of length two together with n 3 n/5 isolted vertices cn e extended to form n L-cycle cover, provided tht n is L-dmissile. Theorem 7. For every fixed L, the lgorithm shown in Fig. 7 is fctor 2.5 pproximtion lgorithm for Mx-W-L-UCC with running time O(n 3 ). 4.2 Approximting Directed Cycle Covers Now we present n pproximtion lgorithm for Mx-W-L-DCC tht chieves n pproximtion rtio of 3. The input consists of directed complete grph G = (V, D(V )) with V = n nd n edge weight function w : D(V ) N. Given cycle cover C, we cn otin mtching M C consisting of n/3 edges such tht w(m) w(c)/3. Conversely, if n is L-dmissile, then every mtching of crdinlity n/3 cn e extended to form n L-cycle cover.

13 Input: directed grph G = (V, D(V )) with V = n; n edge weight function w : D(V ) N Output: n L-cycle cover C px of G if n is L-dmissile, otherwise 1. If n / L, then return. 2. Compute mximum weight mtching M init of G of crdinlity n/3. 3. Join the edges in M init to otin n L-cycle cover C px, return C px. Fig. 8. A fctor 3 pproximtion lgorithm for Mx-W-L-DCC. Insted of computing n initil cycle cover, the lgorithm shown in Fig. 8 directly computes mtching of crdinlity n/3. Theorem 8. For every fixed L, the lgorithm shown in Fig. 8 is fctor 3 pproximtion lgorithm for Mx-W-L-UCC with running time O(n 3 ). 5 Solving Mx-4-UCC in Polynomil Time The im of this section is to show tht Mx-4-UCC cn e solved deterministiclly in polynomil time. To do this, we exploit Hrtvigsen s lgorithm for computing mximum-crdinlity tringle-free two-mtching. A two-mtching of n undirected grph G is spnning sugrph in which every vertex of G hs degree t most two. Thus, two-mtchings consist of disjoint simple cycles nd pths. A two-mtching is relxtion of cycle cover (or two-fctor): In cycle cover, every vertex hs degree exctly two. A tringlefree two-mtching is two-mtching in which ech cycle hs length of t lest four. The pths cn hve ritrry lengths. A tringle-free two-mtching of mximum weight in grphs with edge weights zero nd one cn e computed deterministiclly in time O(n 3 ), where n is the numer of vertices [11, Chp. 3]. We wnt to solve Mx-4-UCC, i.e. ll cycles must hve length of t lest four nd no pths re llowed. Therefore, let M e mximum weight tringlefree two-mtching of grph G of n vertices. If M does not contin ny pths, then M is lredy 4-cycle cover of mximum weight. Let l e the numer of vertices of G tht lie on pths in M. If l 4, then we connect these pths to get cycle of length l. No weight is lost in this wy, nd the result is mximum weight 4-cycle cover. We run into troule if l {1, 2, 3}. Let Y = {y 1,..., y l } e the set of vertices tht lie on pths in M. Let l e the numer of edges of weight one in M tht connect two vertices of Y. Then 0 l l 1 nd w(m) = n l + l n 1. An ovious wy to otin cycle cover from M is to rek one edge of one cycle nd connect the vertices of Y to this cycle. Unfortuntely, reking n edge might cuse loss of weight one. This yields the forementioned pproximtion within n dditive error of one. We cn prove the following with more creful nlysis: Either we cn void the loss of weight one, or indeed mximum weight 4-cycle cover hs only weight w(m) 1. This yields the following result.

14 Theorem 9. Mx-4-UCC cn e solved deterministiclly in time O(n 3 ). 6 Vertex Cover in Regulr Grphs We cn prove tht Min-Vertex-Cover(λ) is APX-complete for every λ 3. Previously, this ws only known for cuic, i.e. three-regulr, grphs [2]. We need the APX-hrdness of Min-Vertex-Cover(λ) for ll λ 3 in Sect. 3. Theorem 10. For every λ N, λ 3, Min-Vertex-Cover(λ) is APX-complete. References 1. Rvindr K. Ahuj, Thoms L. Mgnnti, nd Jmes B. Orlin. Network Flows: Theory, Algorithms, nd Applictions. Prentice-Hll, Pol Alimonti nd Viggo Knn. Some APX-completeness results for cuic grphs. Theoret. Comput. Sci., 237(1 2): , Giorgio Ausiello, Pierluigi Crescenzi, Giorgio Gmosi, Viggo Knn, Alerto Mrchetti-Spccmel, nd Mrco Protsi. Complexity nd Approximtion: Comintoril Optimiztion Prolems nd Their Approximility Properties. Springer, Mrkus Bläser. Approximtionslgorithmen für Grphüerdeckungsproleme. Hilittionsschrift, Institut für Theoretische Informtik, Universität zu Lüeck, Lüeck, Germny, Mrkus Bläser nd Bodo Mnthey. Approximting mximum weight cycle covers in directed grphs with weights zero nd one. Algorithmic, 42(2): , Mrkus Bläser, Bodo Mnthey, nd Jiří Sgll. An improved pproximtion lgorithm for the symmetric TSP with strengthened tringle inequlity. J. Discrete Algorithms, to pper. 7. Mrkus Bläser, L. Shnkr Rm, nd Mxim I. Sviridenko. Improved pproximtion lgorithms for metric mximum ATSP nd mximum 3-cycle cover prolems. In Proc. of the 9th Workshop on Algorithms nd Dt Structures (WADS), vol of Lecture Notes in Comput. Sci., pp Springer, Mrkus Bläser nd Bodo Sieert. Computing cycle covers without short cycles. In Proc. of the 9th Ann. Europen Symp. on Algorithms (ESA), vol of Lecture Notes in Comput. Sci., pp Springer, Bodo Sieert is the irth nme of Bodo Mnthey. 9. Gérrd P. Cornuéjols nd Willim R. Pulleylnk. A mtching prolem with side conditions. Discrete Mth., 29(2): , Michel R. Grey nd Dvid S. Johnson. Computers nd Intrctility: A Guide to the Theory of NP-Completeness. W. H. Freemn nd Compny, Dvid Hrtvigsen. An Extension of Mtching Theory. PhD thesis, Deprtment of Mthemtics, Crnegie Mellon University, Refel Hssin nd Shlomi Ruinstein. On the complexity of the k-customer vehicle routing prolem. Oper. Res. Lett., 33:1, Pvol Hell, Dvid G. Kirkptrick, Jn Krtochvíl, nd Igor Kríz. On restricted two-fctors. SIAM J. Discrete Mth., 1(4): , Oliver Vornerger. Esy nd hrd cycle covers. Technicl report, Universität/Gesmthochschule Pderorn, 1980.