As a generalization of paths, the notion of paths of bandwidth w is introduced. We show that, for a given constant w >= 1, the corresponding search problem for such a path of length k in a given graph is NP-complete and fixed-parameter tractable in the parameter k, like this is known for the special case w=1, the LONGEST PATH problem. We state the FPT algorithm in terms of a guess and check protocol which uses witnesses of size polynomial in the parameter

Borchert, Bernd

Reinhardt, Klaus

Publikationsserver der Universität Tübingen

Searching Paths of Constant BandwidthBernd BorchertKlaus ReinhardtWSI-2004-10Universita¨t Tu¨bingenWilhelm-Schickard-Institut fu¨r InformatikArbeitsbereich Theoretische Informatik/Formale SprachenSand 13D-72076 Tu¨bingenborchert@informatik.uni-tuebingen.dec© WSI 2004ISSN 0946-3852Searching Paths of Constant BandwidthBernd Borchert Klaus ReinhardtUniversita¨t Tu¨bingen, Germany{borchert,reinhard}@informatik.uni-tuebingen.deAbstractAs a generalization of paths, the notion of paths of bandwidth w is introduced. We showthat, for a given constant w ≥ 1, the corresponding search problem for such a path of length kin a given graph is NP-complete and fixed-parameter tractable in the parameter k, like this isknown for the special case w = 1, the LONGEST PATH problem. We state the FPT algorithmin terms of a guess and check protocol which uses witnesses of size polynomial in the parameter.1 IntroductionA path within a graph is one of the most elementary notions of graph theory and its applications.The LONGEST PATH is the computational problem which asks for a given graph G and an integerk whether there is a path of length k in G which is simple, i.e. all vertices are different from eachother. The LONGEST PATH is NP-complete [GJ97]. Moreover, the LONGEST PATH problem isfixed-parameter tractable in the parameter k. This was shown by Monien [Mo85] and improved withrespect to running time by Alon, Yuster, Zwick [AYZ95], using randomization techniques.In this paper we generalize the notion of a path: a path of bandwidth w, or short w-path, in a graphG is a sequence (v1, . . . , vn) of vertices such that for all vi, vj with 1 ≤ j− i ≤ w the pair (vi, vj) is anedge in G, see Fig. 1 for an example of a 2-path. 1-paths are paths in the usual sense. It will be easyto show that for every w ≥ 1 the corresponding computational problem BANDWIDTH-w-PATH,which asks for a given graph G and an integers k whether there exists a simple w-path of length kin G, is NP-complete.The BANDWIDTH-w-PATH problem for every w is fixed-parameter tractable in the parameterk, this will be shown according to the characterization of FPT ∩ NP by Cai, Chen, Downey &Fellows [CCDF95] via an “FPT guess and check protocol” using witnesses of size only dependenton the parameter. The runtime obtained for our guess and check protocol, for the case w = 1,which is the LONGEST PATH problem, and seen as a deterministic exhaustiv search algorithm, isworse than the algorithms of Monien [Mo85] and Alon, Yuster, Zwick [AYZ95]. On the other hand,our algorithm is more easily stated and can immediately be applied to the BANDWIDTH-w-PATHproblem. Moreover, the algorithms of [Mo85, AYZ95] do not seem to give better FPT guess andcheck protocols.1v1v2v3 v7v5v4 v6v1 v2 v3 v4 v5 v6 v7Figure 1: Two drawings of the same 2-path of length 5, vertex-disjoint and deterministic2 Paths of constant bandwidthLet G be a digraph and let w, k ≥ 1. A path of bandwidth w and length k in G is a sequence of k +wvertices (v1, ..., vk+w) such that the pair (vi, vi+j) is an edge of G for every i with 1 ≤ i ≤ k andevery j with 1 ≤ j ≤ w. A path of bandwidth w and length k will also be called w-path of length kor, even shorter, (w, k)-path. A 1-path of length k is a path of length k in the usual sense. (For apath of length k some authors count the number of vertices while others count the number of edges– what is one less. In this paper we count the number of edges.) In Figures 1, 2, and 3 some 2-pathsand 3-paths are shown. Note that a (w, 1)-path is a (w+1)-clique: every two nodes are connected byan edge. A (w, k)-path can actually be seen as a sequence of k (w + 1)-cliques with two subsequentcliques “glued” together by their common w elements.A (w, k)-path (v1, ..., vk+w) is vertex-disjoint if all vi are different from each other, it is simple if all kw-tupels (v1, ..., vw), (v2, ..., vw+1), . . . , (vk, ..., vk+w) are different from each other. A vertex-disjoint(k, n)-path is simple, but not vice versa for k ≥ 2, see Figure 3. A vertex-disjoint (w, k)-path, as agraph on its own, is the graph with k + w vertices having bandwidth w and a maximal set of edges,that is why we choose the name “bandwidth” for the number w (see [PT99, GJ97] for the definitionof bandwidth of a graph).Though the notion of w-paths within a graph G is a rather natural generalization of paths theauthors could not find references for it in the literature. The closest concept found is the w-ray fromProskurowski & Telle [PT99], corresponding to a vertex-disjoint w-path (as a graph on its own).v1v2v3 v5v4 v6v7v8Figure 2: A 3-path of length 5, vertex-disjoint and deterministic2v1 v3 v5v4v7v9v11 = v10v2 v6 = v8Figure 3: A 2-path of length 10, deterministic and simple but not vertex-disjointA (w, k)-path (v1, ..., vk+w) is deterministic in G if for every 1 ≤ i ≤ k vi+w is the only vertex in thegraph G having the property that all edges (vi, vi+w), . . . , (vi+w−1, vi+w) are edges of the graph. Forexample, a deterministic 1-path has the property that every vertex of it – besides the last one – hasexactly one outgoing edge in G.For w < k, a (w, k)-path (v1, ..., vk+w) is a cycle of bandwidth w and length k, short w-cycle of lengthk or (w, k)-cycle, if (vk+1, . . . , vk+w) = (v1, . . . , vw). The cycle is vertex-disjoint if v1, ..., vk aredifferent from each other, it is simple if (v1, ..., vk+w−1) is a simple w-path, see Fig. 4 for an example.For undirected graphs the definitions can be transfered literally.For a fixed w let BANDWIDTH-w-PATH be the set of pairs 〈G, k〉 such that the digraph G containsa simple (w, k)-path. BANDWIDTH-1-PATH = LONGEST-PATH. Let BANDWIDTH-PATH bethe double-parameterized problem consisting of the triples 〈G, w, k〉 such that the digraph G containsa simple (w, k)-path.Some variations of these problems: Let the prefixes UNDIRECTED- and DISJOINT- in front ofthese problem names indicate that the input graph is undirected, or, independently, that the path tobe found has to be not only simple but vertex-disjoint, respectively. Let CYCLE instead of PATHin a problem name denote that the path to be found has to be a cycle. Call these further 7 problemsthe variations of the BANDWIDTH-w-PATH, resp. BANDWIDTH-PATH, problem.Figure 4: A 2-cycle of length 8, deterministic and vertex-disjoint3Proposition 1 (a) BANDWIDTH-PATH is NP-complete, likewise its variations.(b) For every w ≥ 1 the problem BANDWIDTH-w-PATH is NP-complete, likewise its variations.Proof. Obviously all problems are in NP. BANDWIDTH-PATH is NP-complete because LONGESTPATH is a subproblem. In order to show NP-completeness of BANDWIDTH-w-PATH we reduceLONGEST PATH to it. Let some directed graph G be given. Let the graph φ(G) consist of w copiesG1, . . . , Gw of G, and let an edge from u in Gi to v in Gj only exist if i < j and in G there is asimple path of length j − i from u to v. It holds: G has a simple path of length k iff φ(G) has asimple w-path of length k. q.e.d.We mention that for fixed w the problem of searching for a deterministic simple w-path of a givenlength k can be done in PTIME by a straightforward marking algorithm.3 Fixed-Parameter TractabilityThe following notion is from Downey & Fellows [DF92] though it can already be found – withoutgiving it a name – in Monien [Mo85][p. 240, the two paragraphs before and after Th. 1, resp.].Definition 1 (fixed-parameter tractability [Mo85, DF92]) A computational problem consist-ing of pairs 〈x, k〉 is fixed-parameter tractable in the parameter k if there is a deciding algorithm forit having run-time f(k) · |x|c for some recursive function f and some constant c.We use the following characterization of FPT ∩ NP by Cai, Chen, Downey & Fellows [CCDF95]:Theorem 1 (Cai et al. [CCDF95]) A language L ∈ NP consisting of pairs 〈x, k〉 is fixed-parametertractable in the parameter k iff there exists a recursive function s(k) and a PTIME computable lan-guage C such that 〈x, k〉 ∈ L ⇐⇒ ∃y ≤ s(k) : 〈x, k, y〉 ∈ C.We call the function s the witness size function, and the language C the witness checker, and we saythat these two together form an FPT guess and check protocol for L.Theorem 2 For every w ≥ 1 the problem BANDWIDTH-w-PATH is fixed parameter tractable inthe parameter k, likewise its variations. More specifically, there exists an FPT guess and checkprotocol for it with a witness size function s(k) =(k2)· log k and a witness checker having runtimeO(w · k2 · |E|w · |V |w).Proof. We first consider the case w = 1, i.e. the LONGEST PATH problem. Afterwards we will seethat the algorithm is generalizable to the BANDWIDTH-w-PATH problem for w > 1. We state anFPT guess and check protocol for LONGEST PATH with the witness size function s(k) =(k2)· log kand a witness checker with runtime O(k2 · |E| · |V |).Let a digraph G with n vertices be given. We want to find out whether the graph contains a simplepath p = (v1, . . . , vk+1) of length k. We will work with witnesses. The intention of a witness isto tell the algorithm in the moment when it is trying to build an initial segment (v1, . . . , vi) of thesimple path of length k which are the future vertices vi+1, . . . , vk+1 of the simple path – so that thealgorithm does not pick one of these future vertices as a part of the initial segment. Unfortunately,4aaa aaaaaaa1 1 02aaaa========002,1 2,2 2,3 2,4 23,1 3,2 2,3 34,1 4,2 45,150021Figure 5: Witness table for a simple path of length 4we cannot use the tuple (v1, . . . , vk+1) as a witness, because that way we would have nk+1 potentialwitnesses, so that we would need at least (k + 1) log(n) bits to encode them, a number growing in n.But for the FPT guess and check protocol we need some witness size function s(k) only dependenton k.We choose the following kind of witnesses. A witness for such a simple path of length k consistsof k(k + 1)/2 =(k+12)numbers ai,j ∈ {0, 1, . . . , k}, for 2 ≤ i ≤ k + 1 and j ∈ {1, . . . , k − i + 2}.The witness can be visualized as a half-matrix a, see Figure 5. Let ai for 2 ≤ i ≤ k + 1 be thetupel (ai,1, . . . , ai,k−i+2). We can restrict the witnesses to have these properties: ai contains onlynumbers ≤ i− 1 and at least one 0. There is some redundancy, for example ak+1,1 will always be 0.Nevertheless, the order of magnitude of the witness size function s(k) does not seem to be improvableby these “little savings”.For every witness a the main algorithm C does the following: In every of the k steps i = 2, 3, . . . , k+1it computes for every vertex v a value fa,i(v), defined further below, which is either a vertex or hasthe value nil (standing for “not existing”), and stores this function for use in the following steps.The following pseudo code shows the main structure of the algorithm.Main algorithm CInput: graph G, number k ≤ |G|, and a witness afor every vertex v set fa,1(v) := v;for i = 2, . . . , k + 1 dofor every vertex v in G docompute fa,i(v) and store it;if i = k + 1 and fa,i(v) 6= nil ACCEPT and STOP;REJECT and STOP;The computation of the value fa,i(v) – which is either nil or a vertex – is described in the pseudocode below. Assume w.l.o.g. that for each vertex there is a list of incoming edges (ending withthe nil list element) in which the edges appear according to the order on the vertices. As a useful5abbreviation let fda,i(v) for a vertex v and d with 1 ≤ d ≤ i + 1 be defined viaf1a,i(v) := v, f2a,i(v) := fa,i(v), and fd+1a,i (v) := fda,i−1(fa,i(v))with this value being nil in case fa,i(v) or fda,i−1(fa,i(v)) equals nil. Intuitively, fda,i(v) follows –starting in v – for growing d = 1, . . . , i + 1 the “backward path” given by the fa,i−d-functions, seeFigure 6. The upper index d numbers the vertices of this path, and the witness elements ai,j ≥ 0will refer to this numbering.By easy induction on i, the following invariant will be guaranteed for every witness a, every i with2 ≤ i ≤ k + 1, and every vertex v:(Inv1) If fa,i(v) 6= nil then the “backward path” (fia,i(v), . . . , f2a,i(v), f1a,i(v)) is a simple path oflength i− 1.Computing fa,i(v)Input: i, a, and v. Already computed: fa,1, . . . , fa,i−1.set F := {v};set j := 1;if there are no incoming edges for v set fa,i(v) := nil and STOP;set e = (u, v) to be the first edge incoming to v;while e 6= nil doif fa,i−1(u) 6= nil and none of the vertices f1a,i−1(u), f2a,i−1(u), . . . , fia,i−1(u) is in F doset c := ai,j ;if c = 0set fa,i(v) := u and STOP;otherwiseset F := F ∪ {f ca,i(u)};set j := j + 1;set e = (u, v) := next edge going into v;set fa,i(v) := nil and STOP;Verification of the main algorithm C: If the algorithm accepts then it has found for this witness a avertex v such that fa,k+1(v) 6= nil. By invariant (Inv1), case i = k + 1, the backward path startingin v is a simple path of length k.On the other hand assume that there is a simple path of length k in G. Let s = (s1, . . . , sk+1) be thelexicographically smallest among them (largest weight on sk+1, unlike, for example, with decimalnumbers). With the knowledge of this path and its vertices we will construct a witness b such thatthe main algorithm will accept for witness b.6(v)f (v)23f (v)4v =  fu =  f(v)1a,4a,4a,4a,4Figure 6: A “backward path”, starting in vConstructing bInput: s1, . . . , sk+1.for every vertex v set fb1,1(v) = v;for i = 2 to k + 1 doset e = (u, si) := first edge going into si;set F = {si};set j := 1;repeatwhile fbi−1,i−1(u) = nil or some of the vertices f1bi−1,i−1(u), . . . , f ibi−1,i−1(u) is in Fset e = (u, si) := next edge going into si;if there is a c ∈ {1, . . . , i} such that f cbi−1,i−1(u) ∈ {si+2, . . . , sk+1}set bi,j := c for the smallest such c;set F := F ∪ {f cbi−1,i−1(u)};set j := j + 1;until there is no such c;bi,j := 0compute fbi,i(v) for all vertices v;The crucial invariant kept by this construction is the following:(Inv2) For every i with 2 ≤ i ≤ k + 1 it holds: fb,i(si) = si−1.The invariant holds via induction on i: the construction of bi prevents fb,i(si) from choosing oneof the vertices si+1, . . . , sk+1 which will be needed in the future but which would be – without7the witness – unknown at step i. Because there are at most k − i + 1 such vertices the repeatloop will always terminate and, moreover, the part bi of the witness has sufficient size. For every2 ≤ i ≤ k + 1 it is guaranteed that the computation of fb,i(si) will terminate, i.e. will be not-nil,because at least (si−1, si) is a suitable edge, and this will be the first suitable edge which fb,i(si)will find, i.e. fb,i(si) = si−1, because otherwise s = (s1, . . . , sk+1) would not be lexicographicallyminimal.Invariant (Inv2) implies for i = k+1 that the back path (f k+1b,k+1(sk+1), . . . , f2b,k+1(sk+1), f1b,k+1(sk+1))at sk+1 equals s = (s1, . . . , sk+1), i.e. the main algorithm C will accept the input graph for this witnessb via a non-nil value of fb,k+1 at vertex sk+1. This finishes the correctness proof for the FPT guessand check protocol.The running time of all fai(v) for a fixed i is O(k · |E|) (we ignore some log(k) factors for thecomparison algorithms). Therefore, the main algorithm C has runtime O(k2 · |V | · |E|). Representingall witnesses can be done with(k2)· log k bits, i.e. the witness size function can be chosen this way(note that the diagonal of the half matrix does not need to be stored – it can be assumed to consistof 0’s). This finishes the proof that an FPT guess and check protocol exists for LONGEST PATH.Cases w > 1. We first do a graph transformation. From the given graph G construct the fol-lowing graph G′: Consider all w-tuples (v1, . . . , vw) of vertices of G. Make such a tuple a vertexof G′ if the tuple represents a directed w-clique in G, i.e. (vi, vj) is an edge in G for 1 ≤ i <j ≤ w. The edges in G′ are defined to consist of the pairs of such w-cliques of the special form((v1, . . . , vw), (v2, . . . , vw, vw+1)) such that also (v1, vw+1) is an edge in G. We have the property:G contains a simple w-path of length k iff G′ contains a 1-path of length k. The witness checkerconsists therefore of this graph transformation and subsequently the checking algorithm C for w = 1running on G′. In total the checking takes O(w · |V |w · |E|w) time, the first w stems from a slightlyhigher comparison time for tuples. The witnesses size function does not change.Variants: For the vertex disjoint case with w > 1 it is not enough to do the graph transformation,one has to go inside the checking algorithm C and maintain the vertex lists appropriately. q.e.d.It should be mentioned that, when given k as a constant, the problem whether a given graph hasa (w, k)-path does not seem to be fixed-parameter tractable in the parameter w because the W[1]-complete CLIQUE problem is obviously reducible to it, see for example [CCDF95] for the definitionof W[1].4 Conclusions and Open QuestionsWe introduced for every w ≥ 1 the NP-complete problem BANDWIDTH-w-PATH and showed thatit is fixed-parameter tractable in the length parameter k by presenting an FPT guess and checkprotocol for it, according to the characterization of Cai et al. [CCDF95].As an open problem we suggest to study whether the witness size function, especially for the caseLONGEST PATH, can be improved from the quasi-quadratic function(k2)log k to some quasi-linearfunction, for example by the methods of Monien [Mo85] or Alon, Yuster & Zwick [AYZ95].8References[AYZ95] N. Alon, R. Yuster, U. Zwick: Color-Coding, J. ACM 42(4): 844-856 (1995).[CCDF95] L. Cai, J. Chen, R. G. Downey, M. R. Fellows: On the Structure of ParameterizedProblems in NP, Inf. Comput. 123(1): 38-49 (1995).[DF92] R. G. Downey, M. R. Fellows: Fixed-Parameter Intractability, Structure in Com-plexity Theory Conference 1992: 36-49.[GJ97] M. R. Garey, D. S. Johnson: Computers and Intractability, Freeman, Ney York, 1979.[Mo85] B. Monien: How to find long paths efficiently, Annals of Discrete Mathematics 25, 239-254 (1985).[PT99] A. Proskurowski, J. A. Telle: Classes of graphs with restricted interval models,Discrete Mathematics & Theoretical Computer Science 3(4), 167-176 (1999).9

Searching paths of constant bandwidth

https://publikationen.uni-tuebingen.de/xmlui/bitstream/10900/48673/1/pdf/Borchert_Reinhardt-2004-bandwidth.pdf

Searching paths of constant bandwidth

Abstract

Similar works

Full text

Available Versions

Publikationsserver der Universität Tübingen