Minimum Activation Cost Edge-Disjoint Paths in Graphs with Bounded Tree-Width

Transcription

1 Mnmum Actvaton Cost Edge-Dsont Paths n Graphs wth Bounded Tree-Wdth Hasna Mohsen Alqahtan (B) and Thomas Erlebach Department of Computer Scence, Unversty of Lecester, Lecester, UK {hmha1,terlebach}@lecester.ac.uk Abstract. In actvaton network problems we are gven a drected or undrected graph G = (V,E) wth a famly {f uv : (u, v) E} of monotone non-decreasng actvaton functons from D 2 to {0, 1}, where D s a constant-sze subset of the non-negatve real numbers, and the goal s to fnd actvaton values x v D for all v V of mnmum total cost v V xv such that the actvated set of edges satsfes some connectvty requrements. We propose an algorthm that optmally solves the mnmum actvaton cost of k edge-dsont st-paths (st-maedp) problem n O( V D tw+1 tw 3 (k +1) (tw+3)2(tw+3) (tw +3) 2(tw+3)+3 )tme for graphs wth treewdth bounded by a constant tw. 1 Introducton The actvaton network settng can be defned as follows. We are gven a drected or undrected graph G =(V,E) together wth a famly {f uv :(u, v) E} of monotone non-decreasng actvaton functons from D 2 to {0, 1}, where D s a constant-sze subset of the non-negatve real numbers, such that the actvaton of an edge depends on the chosen values from the doman D at ts endponts. An edge (u, v) E s actvated for chosen values x u and x v f f uv (x u,x v ) = 1, and the actvaton functon f uv s called monotone non-decreasng f f uv (x u,x v )=1 mples f uv (y u,y v ) = 1 for any y u x u, y v x v. The obectve of actvaton network problems s to fnd actvaton values x v D for all v V such that the total actvaton cost v V x v s mnmzed and the actvated set of edges satsfes some connectvty requrements. Actvaton problems generalze several problems studed n the network lterature such as power optmzaton, mnmum broadcast tree and nstallaton cost optmzaton. Several actvaton problems have been studed for arbtrary graphs n the recent research lterature. Unfortunately, many of these problems are computatonally hard. Obtanng a polynomal-tme approxmaton algorthm s a typcal approach to deal wth an NP-hard problem. Another mportant approach s studyng the problem on graph classes wth nce decomposton propertes such as bounded treewdth graphs to determne effcent algorthms. Pangrah [8] shows that the fundamental problem of fndng mnmum actvaton k edge-dsont st-paths (st-maedp) s NP-hard. It s an nterestng area of research to nvestgate ths problem for graphs wth treewdth bounded by a constant tw. In ths paper, we focus on developng a c Sprnger Internatonal Publshng Swtzerland 2016 Z. Lpták and W.F. Smyth (Eds.): IWOCA 2015, LNCS 9538, pp , DOI: /

2 14 H.M. Alqahtan and T. Erlebach polynomal-tme algorthm that optmally solves the st-maedp for graphs of bounded treewdth. (Note that k s not a constant but part of the nput.) Related Work. Actvaton network problems were frst ntroduced by Pangrah n [8] and varous of these problems have been nvestgated n [1,2,6 8]. The mnmum actvaton st-path (MAP) problem s optmally solvable n polynomaltme [8]. However, the mnmum spannng actvaton tree (MSpAT) problem s NP-hard to approxmate wthn a factor of o(log n) and there exsts a O(log n)- approxmaton algorthm for ths problem [8]. Actvaton network problems generalze several problems studed n the network lterature such as power optmzaton problems, whch are modelled by a graph G =(V,E) where each edge (u, v) ng s assgned a threshold power requrement θ uv. For undrected graphs, an edge (u, v) s actvated for chosen values x u and x v f each of these values s at least θ uv, and t s actvated f x u θ uv for the drected case. The mnmum power st-path (MPP) problem can be solved n polynomal tme for both drected and undrected graphs [5]. For drected graphs, the mnmum power k node-dsont st-paths problem s also optmally solvable n polynomal tme [4, 11]. However, the mnmum power k edge-dsont st-paths problem s unlkely to admt even a polylogarthmc approxmaton algorthm for both the drected and undrected varants [4]. There s a 2k-approxmaton algorthm for the st-maedp problem and a 2- approxmaton algorthm for the mnmum actvaton node-dsont st-paths (st- MANDP) problem [6]. In [1], we have studed the st-maedp and st-mandp problems when k = 2, denoted by st-ma2edp and st-ma2ndp respectvely. We proved that a ρ-approxmaton algorthm for the st-ma2ndp problem mples a ρ-approxmaton algorthm for the st-ma2edp problem. In the same paper, we obtaned a 1.5-approxmaton algorthm for the st-ma2ndp problem and hence for the st-ma2edp problem. The problems st-maedp and st-mandp for the restrcted verson of actvaton networks wth D = 2 and a sngle actvaton functon for all edges have also been studed n [1]. Under ths restrcton, the st-mandp problem s optmally solvable n polynomal tme for arbtrary k (except for one case of the actvaton functon, n whch we requre k = 2). The st-maedp problem, however, remans NP-hard [1, 8]. So far these problems for an arbtrary constant-sze D and fxed k 2 are not known to be NP-hard. Recently, n [2], we consdered the st-mandp and the problem of fndng mnmum actvaton cost node-dsont paths (MANDP) between k dsont termnal pars, (s 1,t 1 ),...,(s k,t k ), for graphs of bounded treewdth. We proposed algorthms that optmally solve the st-mandp problem n polynomal-tme and the MANDP problem n lnear-tme for graphs wth bounded treewdth [2]. There exsts a polynomal-tme algorthm that optmally solves the st-ma2edp problem for graphs of bounded treewdth [1, 2]. Other relevant work, applcatons and motvatons of actvaton network problems have been addressed n [1, 2, 6 8]. Our Results. We develop a polynomal-tme algorthm for the st-maedp problem for graphs wth treewdth bounded by tw. Our algorthm effcently combnes an edge-colorng scheme wth dynamc programmng over a nce treedecomposton (see Sect. 2). The edge-colorng scheme was ntroduced n [12] to

3 Mnmum Actvaton Cost Edge-Dsont Paths n Graphs 15 develop a polynomal-tme algorthm that solves the mnmum shared-edge kstpaths (MSEP) problem (fndng k paths between s and t wth mnmum number of shared edges among the paths) for graphs of bounded treewdth. The MSEP s a generalzaton of the problem of fndng k edge dsont st-paths. The rest of the paper s organzed as follows. In Sect. 2 we recall some defntons and results of the class of graphs wth bounded treewdth. Then n Sect. 3, we present a polynomal-tme algorthm that solves the st-maedp problem optmally for graphs wth treewdth bounded by tw. We conclude the paper by statng some open problems n Sect Prelmnares In ths paper we consder the class of graphs of bounded treewdth. A graph G =(V,E) has treewdth tw f t has a tree-decomposton of wdth tw [9]. The tree-decomposton concept s defned as follows. Defnton 1. Gven a graph G = (V,E), a tree T = (I,F) and a famly X = {X } I of subsets of V (called bags). The par (X, T ) s called a treedecomposton of G f t satsfes the followng condtons: V = I X. For every edge (v, w) E, there exsts an I wth v X and w X. For every vertex v V, the nodes I wth v X form a subtree of T. The wdth of (X, T ) s the number max I X 1. The treewdth tw of the graph G s the mnmum wdth among all possble tree-decompostons of the graph. Theorem 1 ([3]). For any fxed tw, there exsts a lnear-tme algorthm that checks whether a gven graph G =(V,E) has treewdth at most tw, andfso, outputs a tree-decomposton (X, T ) of G wth wdth at most tw. Defnton 2. A tree-decomposton (X, T ) s called a nce tree-decomposton, f T s a bnary tree rooted at some r I that satsfes the followng: Each node s ether a leaf, or has exactly one or two chldren. Let I be a leaf. Then X = {u, v} for some (u, v) E. For every edge (u, v) E, there s exactly one leaf I such that u, v X. (We say that the edge (u, v) s assocated wth that leaf I). Let Ibethe only chld of I, then ether X = X {v} or X = X \{v}. The node s called an ntroduce node or forget node, respectvely. Let, I be the two chld nodes of a node I, then X = X = X.The node s called a on node of T. Scheffler presented n [10] a specal tree-decomposton that follows the structure of a nce tree-decomposton as defned above but wth no restrcton on the sze of leaf bags (.e., t does not requre leaf bags to be of sze 2). We call that type of tree-decomposton a Scheffler-type nce tree-decomposton. Any gven tree-decomposton for a graph G =(V,E) wth treewdth at most tw can be

4 16 H.M. Alqahtan and T. Erlebach easly converted nto a Scheffler-type nce tree-decomposton of wdth tw and sze O( V ) n lnear tme, f tw s a fxed constant [10]. One can also transform a Scheffler-type nce tree-decomposton to have leaves wth bags of sze 2 n lnear tme. Therefore, a nce tree-decomposton wth leaf bags of sze 2 and wdth tw can be constructed from any gven tree-decomposton of the same treewdth. However, each leaf node of a Scheffler-type nce tree-decomposton may produce O(tw 3 ) new nodes n the constructon of leaves wth sze 2. The resultng tree-decomposton, therefore, has O( V tw 3 ) nodes. Throughout ths paper, we consder smple undrected graphs G =(V,E) gven as an nput wth a nce tree-decomposton of wdth at most tw. We defne X + to be the set of all vertces n X for all nodes I such that = or s a descendant of. We denote by G + a partal graph of G. For a leaf node, G + s the subgraph of G wth vertex set X and the edge of G that s assocated wth. For a non-leaf node, G + s the graph that s the unon of G + over all chldren of. Note that the graph G + r for the root r of the tree-decomposton s equal to G. 3 Mnmum Actvaton Cost k Edge-Dsont st-paths Gven are an actvaton network G =(V,E) and a par of source and destnaton vertces s, t V. In ths secton, we consder the st-maedp problem where the goal s to fnd actvaton values {x v : v V } of mnmum total cost v V x v such that the actvated set of edges contans k edge-dsont stpaths P st = P 1,...,P k. We present a polynomal-tme algorthm that solves the st-maedp problem optmally n the case of graphs of bounded tree-wdth usng dynamc programmng technques. The algorthm follows a bottom-up approach to compute a number of possble sub-solutons per nce tree-decomposton node I. It s easy to compute the sub-solutons for a leaf node I because the partal graph G + conssts of two vertces that are connected by an edge n G assocated wth. For a non-leaf node, we use the nformaton prevously computed for ts chldren. The algorthm also constructs a table tab to store the computed nformaton for each node I. We use an edge-colorng scheme to compute the sub-solutons per tree node. Let C = {0, 1,...,k} be a set of colors. We consder a colorng f : E(G + ) C for the graph G +. For each color c C, we defne G+ (f,c) to be the subgraph of G + nduced by the edges wth color c. Each color c C \{0} represents the edges used by P c. Denote by P (X) the set of all possble parttons of the set X. We defne C(X )=(Y 1,...,Y k )tobeacolor vector on X such that Y c P (X {s, t}) for all c C \{0}. P st (X ) denotes the set P (X {s, t}). A color vector C(X )=(Y 1,...,Y k )onx s called actve f G + has a colorng f such that every element of the partton Y c,foreachc C \{0}, s a set resultng from the ntersecton between X {s, t} and the vertex set of a connected component of G + (f,c) (see Fg. 1 for an example of an actve color vector). Y(X,f,c) denotes the partton Y c of X {s, t}. Ths color vector concept was ntroduced n [12] for developng a polynomal-tme algorthm that optmally

5 Mnmum Actvaton Cost Edge-Dsont Paths n Graphs 17 G + v 1 v 2 v 3 v 4 v 5 v 6 C(X )={Y 1, Y 2, Y 3 } X = {v 1,v 2,v 3,v 4,v 5,v 6 } and s, t X Y 1 = {{v 1,v 2,v 3 }, {v 4,v 5 }, {v 6 }} Y 2 = {{v 1 }, {v 2,v 3,v 4 }, {v 5 }, {v 6 }} Y 3 = {{v 2 }, {v 3 }, {v 1,v 4 }, {v 5,v 6 }} G + (f, 0) G + (f, 1) G + (f, 2) G + (f, 3) Fg. 1. An example of an actve color vector solves the MSEP problem for graphs wth bounded treewdth. One smple way to compute the sub-solutons s by storng a color vector and an actvatonvalue functon per each row of table tab. However, the number of color vectors per node can only be bounded by (tw +3) k(tw+3) because X tw +1, X {s, t} tw +3 and P st (X ) (tw +3) tw+3. Therefore, the number of color vectors s not always polynomally bounded. Hence the sze of the table tab s also not always polynomally bounded. Therefore we defne a mappng γ : P st (X ) {0, 1,...,k} to be a count on X to obtan a polynomal-tme algorthm to optmally solve the st-maedp problem. We say that the count γ on X s an actve count f G + has a colorng f wth a color vector C(X )= (Y 1,...,Y k ) such that γ (A) = {c C \{0} : A = Y c } for each A P st (X ). For any actve count γ, t s clear that A P γ st(x ) (A) =k. In Fg. 1, the counts would be 1 for all parttons A {Y 1, Y 2, Y 3 } P st (X ) and 0 otherwse. Consder a soluton P = P 1,...,P k for the st-maedp problem. Let P = P[G + ] be the nduced soluton n a partal graph G + (.e., the set of vertces and edges that are both n P and n G + ). Snce the nteracton between P n G + and the rest of the graph happens only n vertces of X, we can consder an actvatonvalue functon Λ : X D and a count γ : P st (X ) {0, 1,...,k} to represent P n G +. The dea of usng counts nstead of color vectors s based on [12]. 3.1 Processng the Tree Decomposton For a tree node I, the table tab has multple rows and each row represents a soluton of a unque combnaton of a count γ and a functon of actvaton values Λ.Letval(γ,Λ ) denote the mnmum cost value of an assgnment of actvaton values for G + whch satsfes the restrctons Λ and actvates an edge-colored subgraph of G + that satsfes the count γ. The value val(γ,λ ) s also stored n tab. We compute the sub-soluton tables startng at the leaves towards the root. Leaf. Let I be a leaf, X = {u, v}. Let (γ,λ )beanyrowoftab.we dstngush the followng cases and defne the value val(γ,λ ) for each case.

6 18 H.M. Alqahtan and T. Erlebach Each case corresponds to a possble sub-soluton n G +. Recall that G+ s a sngle edge. The sub-soluton s cost val(γ,λ )ssetto v X Λ (v) foneof the followng cases apples: γ (A ) = k 1 for A = {{u}, {v}, {s}, {t}} and γ (A ) = 1 for A = {{u, v}, {s}, {t}} and γ (A) = 0 for all A P st (X ) \{A,A }, where f uv (Λ (u),λ (v)) = 1 and s, t / X. Intutvely, ths means that the subsoluton s a path wth one edge not contanng s or t. γ (A )=k 1forA = {{u}, {v}} and γ (A )=1forA = {{u, v}} and γ (A) = 0 for all A P st (X ) \{A,A }, where f uv (Λ (u),λ (v)) = 1 and s, t X. Intutvely, ths means that the sub-soluton s a path wth one edge contanng s and t. γ (A )=k 1forA = {{u}, {v}, {s}} and γ (A )=1forA = {{u, v}, {s}} and γ (A) = 0 for all A P st (X ) \{A,A }, where f uv (Λ (u),λ (v)) = 1 and s/ X and t X. Intutvely, the sub-soluton s a path wth one edge and one endpont equal to t. (The roles of s and t can be exchanged.) γ (A )=k for A = {{u}, {v}, {s}, {t}} and γ (A) = 0 for all A P st (X ) \ {A }, where s, t / X. Intutvely, ths means that the sub-soluton has no edges. γ (A )=k for A = {{u}, {v}} and γ (A) = 0 for all A P st (X ) \{A }, where s, t X. Intutvely, ths means that the sub-soluton has no edges. γ (A )=k for A = {{u}, {v}, {s}} and γ (A) = 0 for all A P st (X ) \{A }, where s/ X and t X. Intutvely, ths means that the sub-soluton has no edges. (The roles of s and t can be exchanged.) In these cases we construct an edge-colored subgraph of G + plus actvatonvalues that may be part of a global soluton. If none of the above cases apples, val(γ,λ )=+. Introduce. Let I be an ntroduce node, and I ts only chld. We have X X, X = X +1andletv be the addtonal vertex n X. The vertex v s solated snce does not ntroduce edges n G +. For every row (γ,λ ) n tab, there are D rows n tab such that for all A P st (X ) and all u X \{v}, γ (A {{v}}) =γ (A) fv / {s, t} and γ (A) =γ (A) fv {s, t} and Λ (u) =Λ (u). The sub-soluton s cost val(γ,λ ) for these rows s set to val(γ,λ )+Λ (v). Forget. Let I be a forget node, and I ts only chld. We have X X, X = X +1 and let v be the dscarded vertex. Let A v be the partton A P st (X ) after removng the vertex v f v / {s, t} and A v equals the partton A f v {s, t}. Note that A v P st (X ) for all A P st (X ). For A P st (X ), we say that W (A) s the set of all parttons B P st (X ) such that B v = A (.e., W (A) ={B P st (X ):B v = A}). For each row (γ,λ ), we consder all (γ,λ ) such that for all u X and A P st (X ), Λ (u) =Λ (u) and γ (A) = B W (A) γ (B). The sub-soluton s cost val(γ,λ ) s the mnmum of val(γ,λ ) over all these (γ,λ ).

7 Mnmum Actvaton Cost Edge-Dsont Paths n Graphs 19 Jon. Let I be a on node, and, I ts two chldren. We have X = X = X. We call a mappng β : P st (X ) P st (X ) {0, 1,...,k} a par-count on X. We defne β to be an actve par-count f G + has a colorng f such that β (A,A )= {c C : A = Y(X,f,c)andA = Y(X,f,c)} for each A,A P st (X ) where f and f are the restrcton of f to E(G + ) and E(G + ), respectvely. Let val(β,λ ) denote the mnmum cost value of an assgnment of actvaton values for G + whch satsfes the restrctons Λ and actvates an edge-colored subgraph of G + that satsfes the par-count β.the algorthm computes all val(β,λ ) of all β on X from all pars of sub-solutons (γ,λ ) and (γ,λ ) such that both have the same actvaton-value functon (Λ (u) =Λ (u) for all u X ) and the par of actve counts γ and γ satsfy the followng condtons: C 1.γ (A )= A P st(x ) β (A,A) for all A P st (X ). C 2.γ (A )= A P st(x ) β (A, A ) for all A P st (X ). The value val(β,λ ) s set to be the summaton value of the par of sub-solutons (γ,λ ) and (γ,λ ) that satsfy the above condtons mnus the actvaton cost of X. To determne the par (γ,λ ) that corresponds to the par (β,λ ) of par-count and an actvaton-value functon, we construct a bpartte graph for each par of parttons wth a par-count greater than 0 as follows. For each par A,A P st (X ) where β (A,A ) 1, we construct a bpartte graph H (A,A ) β =(A A,E (A,A ) β ) wth partte sets A and A, where the vertces a A and a A are oned by an edge n E (A,A ) β ff a a. Assume that D 1,D 2,...,D b are the connected components of the graph H (A,A ) β.we defne U(A,A ) to be the famly of vertex sets { v D l v :1 l b}. Weset the value val(γ,λ ) to be the mnmum val(β,λ ) over all (β,λ ) such that for each A P st (X ): γ (A )= β (A,A ) where the summaton above s taken over all pars A,A A = U(A,A ). P st (X ) such that Extractng the Soluton at the Root. The algorthm checks all the pars (γ r,λ r )of the root bag X r such that for all A P st (X r ) where γ r (A) 1, there s a set n A contanng both s and t. Inthscase(γ r,λ r ) corresponds to a feasble soluton. The output of the algorthm s the mnmum cost value among all the feasble solutons obtaned at the root. For each row (γ,λ )ofbagx, we store the rows of X s chldren that were used n the calculaton of val(γ,λ ). Computng the optmum soluton s possble by traversng top-down n the decomposton tree to the leaves (traceback) to get the actvaton values, and then runnng a maxmum flow algorthm on the unt-capacty graph of the actvated edges to get the k edge-dsont st-paths.

8 20 H.M. Alqahtan and T. Erlebach 3.2 Analyss The followng lemmas analyse the runnng tme of the algorthm and show that we can effcently compute an optmal soluton for the st-maedp problem for graphs wth bounded treewdth. Let an nstance of the problem be gven by an actvaton network G =(V,E) wth treewdth bounded by tw and termnals s, t V.LetP OPT represent an optmal soluton for ths nstance. We use C(P ) to denote the actvaton cost of a sub-soluton P n a partal graph G +. Lemma 1. The st-maedp algorthm requres O( V D tw+1 (k +1) (tw+3)2(tw+3) (tw +3) 2(tw+3)+3 tw 3 ) tme. Proof. The runnng tme of the algorthm depends on the sze of the tables and the combnaton of tables durng the bottom-up traversal. For each set of vertces X, there are at most X X possble parttons. Therefore, for each node, P st (X ) ( X +2) ( X +2) (tw +3) (tw+3). That means there are at most (k +1) (tw+3)(tw+3) possble actve counts γ on X. The table tab n a processed bag X contans no more than (k +1) (tw+3)(tw+3) D tw+1 rows correspondng to the possble actve counts γ and the D possble actvaton values for each vertex of X. Consder all possble row combnatons wth equal actvaton functons for two tables for a on node. Snce there are at most (tw +3) 2(tw+3) possble pars of parttons, there are at most (k +1) (tw+3)2(tw+3) possble actve par-counts on X. For each par-count β, there s a par of actve counts γ and γ such that the condtons C 1 and C 2 are satsfed. Computng γ and γ from β takes O((tw +3) 2(tw+3) ) tme. Therefore, the computaton of the value val(β,λ) for each combnaton of par-count β and actvaton functon Λ needs O((tw + 3) 2(tw+3) ) tme. Thus we see that all pars (β,λ) andval(β,λ) can be computed n O((k +1) (tw+3)2(tw+3) (tw +3) 2(tw+3) D tw+1 ) tme. Snce A tw +3 for all A P st (X ), the bpartte graph H (A,A ) β defned n the on node contans no more than (tw +3) 2 edges and can be constructed n O((tw +3) 3 ) tme. Therefore, for each β, one can compute the actve count γ that satsfes γ (A )= β (A,A ) where the summaton s taken over all A,A P st (X ) such that A = U(A,A ) and update the value val(γ,λ )tobeequaltoval(β,λ ) f val(β,λ ) val(γ,λ )no((tw +3) 2(tw+3) (tw +3) 3 ) tme. Thus, we can compute all val(γ,λ) of all combnatons of actve count γ and actvaton-value functon Λ on X n tme O((k +1) (tw+3)2(tw+3) (tw +3) 2(tw+3)+3 D tw+1 ). Snce the tree-decomposton T has O( V tw 3 ) nodes, one can compute all the tables for all nodes n O( V D tw+1 tw 3 (k +1) (tw+3)2(tw+3) (tw +3) 2(tw+3)+3 ) tme. Lemma 2. For any processed bag X,letP OPT be the nduced soluton of P OPT n G + and (γ OPT ) be the correspondng count and actvaton values, then ) C(P OPT ) (1) Proof. We use nducton over the tree decomposton to prove that the value of (γ OPT ) s at most the actvaton cost of P OPT. The base case are the

9 Mnmum Actvaton Cost Edge-Dsont Paths n Graphs 21 leaf nodes where the hypothess clearly holds. Let us assume that the nducton hypothess holds for all descendants of bag X. We want to prove that the nducton hypothess holds also for X and that can be proven by showng that the hypothess holds for all dfferent types of the bag X. Introduce. Let us assume that s an ntroduce node and ts only chld and let v be the addtonal vertex n X. The statement (1) holds for X because t s a chld of the bag X. Let (γ OPT ) be the correspondng actve count and actvaton values of P OPT. The followng statement holds: ) C(P OPT ) (γ OPT ) and (γ OPT ) both agree on the actvaton values for all vertces n X \{v}. The addtonal vertex v s an solated vertex n the nduced graph G + γ OPT and the one vertex set {v} s n A for all A P st (X ) such that (v) D s the actvaton value of the vertex v, then: (A) 1. Snce Λ OPT )= C(P OPT )+Λ OPT (v) )+Λ OPT (v) = C(P OPT ) The statement (1) holds for an ntroduce node. Forget. Assume that s a forget node and ts only chld and let v be the dscarded vertex n X. The statement (1) holds for X because t s a chld of the bag X. Let (γ OPT ) be the correspondng actve count and actvaton values of P OPT. The followng statement holds: ) C(P OPT ) (γ OPT ) and (γ OPT ) both agree on actvaton values for all vertces n X \{v}. The dscarded vertex v s ether an solated vertex or part of a and that means the value ) s one of the values that has been consdered when calculatng ). Then: connected component n the nduced soluton P OPT ) Then the statement (1) holds for a forget node. ) C(P OPT )=C(P OPT ) Jon. Assume that s a on node and and ts chldren. Let (γ OPT ) be the correspondng actve count and actvaton values of the nduced soluton P OPT n G +. POPT s the unon of ts chldren sub-solutons P OPT and P OPT. Let (γ OPT ) and (γ OPT,ΛOPT ) be the correspondng actve count and actvaton values of P OPT and P OPT, respectvely. The statement (1) holds for X and X because they are chldren of the bag X. For all l {, } the followng statement holds: l l ) C(Pl OPT )

10 22 H.M. Alqahtan and T. Erlebach Let (β OPT of the nduced soluton P OPT (γ OPT ) be the correspondng actve par-count and actvaton values n G + whch satsfes condtons C 1 and C 2 for,λopt ). We know that POPT s a sub-soluton n G +, ) s one of the values that has been consdered when ). Then: ) and (γ OPT therefore, val(β OPT computng ) val(β OPT ) = C(P OPT = C(P OPT ) )+ )+C(P OPT ) Λ OPT (v) v X ) Λ OPT (v) v X Then the nducton hypothess holds for a on node. Lemma 3. For any processed bag X,anypar(γ,Λ ) where val(γ,λ )=c < corresponds to an edge-colorng plus actvaton-values P =(f,λ + ) where f : E(G + ) {0,...,k} and Λ+ : X + D wth the followng propertes: The actve count of P n X s γ. The actvaton values of P n X are Λ. The total actvaton cost n X + s c. Proof. We prove by nducton that for any bag X there exsts an edge-colorng and actvaton-values P wth the above propertes. The base case are the leaf nodes where the hypothess clearly holds. Let us assume that the nducton hypothess holds for all the descendants of bag X. We want to prove that the nducton hypothess holds also for X and that can be proved by showng that the hypothess holds for all dfferent types of the bag X. Introduce. Assume that s an ntroduce node and ts only chld and v the addtonal vertex n X. Let (γ,λ ) be some entry wth val(γ,λ )=c <.The nducton hypothess holds for X. Let (γ,λ ) be the correspondng actve count and actvaton functon that have been used for the calculaton of val(γ,λ ). Then (γ,λ ) corresponds to an edge-colorng and actvaton-values P that satsfes the above propertes. From the algorthm, (γ,λ ) and (γ,λ ) both agree on the actvaton values for all vertces n X \{v}. Moreover, the addtonal vertex v s an solated vertex n the nduced graph G + and the one vertex set {v} s n A for all A P st (X ) such that γ (A) 1. The value of (γ,λ )s equal to the value of (γ,λ ) added to the actvaton value of the vertex v. The unon of the solated vertex v and P s an edge-colorng wth actvaton-values that satsfes all propertes of the nducton hypothess. Thus, the nducton hypothess holds for an ntroduce node. Forget. Assume that s a forget node and ts only chld and v the dscarded vertex. Let (γ,λ ) be some entry wth val(γ,λ )=c <. The nducton hypothess holds for X. Let (γ,λ ) be the correspondng actve count and

11 Mnmum Actvaton Cost Edge-Dsont Paths n Graphs 23 actvaton functon that have been used for the calculaton of val(γ,λ ). Then (γ,λ ) corresponds to an edge-colorng and actvaton-values P that satsfes the above propertes. From the algorthm, (γ,λ ) has the mnmum value cost among all possble rows of X that can produce (γ,λ ) and both agree on actvaton values for all vertces n X \{v}. The dscarded vertex v s part of the edge-colorng P. Therefore (γ,λ ) corresponds to the edge-colorng and actvaton-values P andwecansetp = P. The nducton hypothess holds for a forget node. Jon. Assume that s a on node and and ts chldren. Let (γ,λ )be some entry wth val(γ,λ ) = c <. Snce val(γ,λ ) = c <, then there s a par of actve par-count and actvaton functon (β,λ ) such that val(γ,λ ) = val(β,λ ) and for each A P st (X ), γ (A ) = β (A,A ) where the summaton s taken over all pars A,A P st (X ) satsfyng A = U(A,A ). Let (γ,λ ) and (γ,λ ) be the pars that have been used for the calculaton of val(β,λ ) whch satsfy the condtons C 1 and C 2.We know that val(γ,λ ) < and val(γ,λ ) < and the nducton hypothess holds for X and X. Therefore, (γ,λ ) and (γ,λ ) correspond to edgecolorngs wth actvaton-values P =(f,λ + =(f )andp,λ + ), respectvely. For each par of parttons A and A where γ (A )=r, γ (A )=r and β (A,A )=r > 0, there are r l colors for partton A l n P l =(f l,λ l )for l {, }. Therefore, we choose r unused colors z 1,z 2,...,z r for A n P and r unused colors z 1,z 2,...,z G + (f,z w)andz w r for A n P and then recolor both zw n for all w {1,...,r}. n G + (f,z w ) wth a new color zw The colors z 1,z 2,...,z r n P and colors z 1,z 2,...,z r n P are now marked as used. Note that there are always enough unused colors because the condtons C 1 and C 2 are satsfed. After recolorng P and P and snce for each A P st (X ), γ (A )= β (A,A ) where the summaton s taken over all pars A,A P st (X ) satsfyng A = U(A,A ), t follows that the actve count of the unon P P n X s γ. Moreover, (γ,λ ), (γ,λ ) and (γ,λ ) all have the same actvaton-value functon (Λ (u) = Λ (u) = Λ (u) for all u X ). That means the actvaton values of the unon P P n X are Λ. The algorthm also computes the cost value of (γ,λ ) as follows: val(γ,λ )=val(β,λ )=val(γ,λ )+val(γ,λ ) v X Λ (v) Snce val(γ,λ )andval(γ,λ ) are the total actvaton costs n X + and X +, respectvely, then the summaton of these actvaton costs mnus the actvaton values for all u X s the total actvaton cost n X +.WecansetP to be the edge-colorng and actvaton values P P that satsfes the propertes of the lemma. The nducton hypothess holds for a on node. We obtan the followng theorem by combnng the above lemmas. Theorem 2. The st-maedp problem for graphs wth treewdth bounded by tw can be solved optmally n O( V D tw+1 tw 3 (k+1) (tw+3)2(tw+3) (tw+3) 2(tw+3)+3 ) tme.

12 24 H.M. Alqahtan and T. Erlebach Corollary 1. For any fxed k, the st-maedp problem for graphs of bounded treewdth can be solved optmally n lnear-tme FPT parameterzed by the treewdth tw. 4 Concluson To the best of our knowledge, the st-maedp problem for graphs wth treewdth bounded by tw consdered here has not been addressed before. We establshed an algorthm that solves the problem optmally n O( V D tw+1 tw 3 (k + 1) (tw+3)2(tw+3) (tw +3) 2(tw+3)+3 ) tme. Our algorthm also solves the st-maedp problem when k = 2 n lnear-tme and ths s an mprovement over the cubc algorthm obtaned n [2]. It would be nterestng f one can obtan a faster or even lnear-tme algorthm for the st-maedp problem n graphs wth treewdth bounded by a constant. References 1. Alqahtan, H.M., Erlebach, T.: Approxmaton algorthms for dsont st-paths wth mnmum actvaton cost. In: Spraks, P.G., Serna, M. (eds.) CIAC LNCS, vol. 7878, pp Sprnger, Hedelberg (2013) 2. Alqahtan, H.M., Erlebach, T.: Mnmum actvaton cost node-dsont paths n graphs wth bounded treewdth. In: Geffert, V., Preneel, B., Rovan, B., Štuller, J., Toa, A.M. (eds.) SOFSEM LNCS, vol. 8327, pp Sprnger, Hedelberg (2014) 3. Bodlaender, H.L.: A lnear tme algorthm for fndng tree-decompostons of small treewdth. In: STOC ACM, pp (1993) 4. Haaghay, M.T., Kortsarz, G., Mrrokn, V.S., Nutov, Z.: Power optmzaton for connectvty problems. In: Jünger, M., Kabel, V. (eds.) IPCO LNCS, vol. 3509, pp Sprnger, Hedelberg (2005) 5. Lando, Y., Nutov, Z.: On mnmum power connectvty problems. In: Arge, L., Hoffmann, M., Welzl, E. (eds.) ESA LNCS, vol. 4698, pp Sprnger, Hedelberg (2007) 6. Nutov, Z.: Survvable network actvaton problems. In: Fernández-Baca, D. (ed.) LATIN LNCS, vol. 7256, pp Sprnger, Hedelberg (2012) 7. Nutov, Z.: Approxmatng Stener networks wth node-weghts. SIAM J. Comput. 39(7), (2010) 8. Pangrah, D.: Survvable network desgn problems n wreless networks. In: 22nd Annual SIAM Symposum on Dscrete Algorthms, pp ACM (2011) 9. Robertson, N., Seymour, P.D.: Graph mnors. XIII. The dsont paths problem. J. Comb. Theory, Ser. B 63, (1995) 10. Scheffler, P.: A practcal lnear tme algorthm for dsont paths n graphs wth bound tree-wdth. Techncal report 396, Dept. Mathematcs, Technsche Unverstät Berln (1994) 11. Srnvas, A., Modano, E.: Fndng mnmum energy dsont paths n wreless adhoc networks. Wreless Netw. 11(4), (2005) 12. Ye, Z.Q., L, Y.M., Lu, H.Q., Zhou, X.: Fndng paths wth mnmum shared edges n graphs wth bounded treewdth. In: The 2013 World Congress n Computer Scence, Computer Engneerng, and Appled Computng, FCS 2013, pp (2013). ISBN:

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