Asymmetric Traveling Salesman Path and Directed Latency Problems

Transcription

1 Asymmetrc Travelng Salesman Path and Drected Latency Problems Zachary Frggstad Mohammad R. Salavatpour Zoya Svtkna February 28, 202 Abstract We study ntegralty gaps and approxmablty of three closely related problems on drected graphs wth edge lengths. Gven two specfed nodes s and t, two of these problems ask to fnd an s-t path n the graph vstng all other nodes. In the asymmetrc travelng salesman path problem (ATSPP), the objectve s to mnmze the total length of ths path. In the drected latency problem, the objectve s to mnmze the sum of the latences of the nodes, where the latency of a node v s the dstance from s to v along the path. The thrd problem that we study s the k-person ATSPP, n whch the goal s to fnd k paths of mnmum total cost from a source node s to a snk node t such that every pont s on at least one of these paths. All of these problems are NP-hard. The best known approxmaton algorthms for ATSPP had rato O(log n) [9, ] untl the very recent result that mproves t to O(log n/ log log n) [4, ]. However, the best known bound on any lnear programmng relaxaton for ATSPP was only O( n). For drected latency, the best prevously known approxmaton algorthm has a guarantee of O(n /2+ǫ ), for any constant ǫ > 0 [27]. We present a new algorthm for the ATSPP problem that has an approxmaton rato of O(log n), but whose analyss also bounds the ntegralty gap of the standard LP relaxaton of ATSPP by the same factor. Ths solves an open problem posed n [9]. We then pursue a deeper study of ths lnear program and ts varatons, whch leads to an O(log n)-approxmaton for the drected latency problem, a sgnfcant mprovement over prevously-known results. Our result for k-person ATSPP s an O(k 2 logn) approxmaton that bounds the ntegralty gap of an LP relaxaton by the same factor. We are not aware of any prevous work on ths problem. Introducton Let G = (V,E) be a complete drected graph on a set of n nodes and let d : E R + be a cost functon (also referred to as length or dstance) satsfyng the drected trangle nequalty: A prelmnary verson of ths paper appeared n the Proceedngs of 2st Annual ACM-SIAM Symposum on Dscrete Algorthms Department of Combnatorcs and Optmzaton, Unversty of Waterloo, Waterloo, Ontaro N2L 3G, Canada. Supported by NSERC and CORE scholarshps. Most of ths work was done whle the author was studyng at the Unversty of Alberta. Department of Computng Scence, Unversty of Alberta, Edmonton, Alberta T6G 2E8, Canada. Supported by NSERC and an Alberta Ingenuty New Faculty award. Google, Inc. Mountan Vew, Calforna, USA. Most of ths work was done whle the author was at the Unversty of Alberta and supported by Alberta Ingenuty.

2 d uw d uv + d vw for all u,v,w V. However, d s not necessarly symmetrc: t may be that d uv d vu for somenodesu,v V. IntheAsymmetrcTravelng Salesman Path Problem (ATSPP), gven such a graph G = (V,E) along wth two dstnct nodes s,t V, the goal s to fnd a Hamltonan path (a path contanng all nodes of V) s = v,v 2,...,v n = t wth mnmum total length n j= d v j v j+. Ths s a varant of the classcal Asymmetrc Travelng Salesman Problem (ATSP), where the goal s to fnd a mnmum-cost cycle contanng all the nodes. In k-person ATSPP we are also gven an nteger k and the goal s to fnd k paths from s to t such that every node les on at least one path and the sum of path lengths s mnmzed. Related to ATSPP s the drected latency problem. On the same nput as ATSPP, the goal s to fnd a Hamltonan path s = v,v 2,...,v n = t that mnmzes the sum of latences of the nodes. Here, the latency of a node v n the path s defned as j= d v j v j+. Ths objectve s qute natural as t can be thought of as the total or average watng tme of clents who are watng to be served by a reparman. There are possble varatons n the problem defnton, such as askng for a cycle nstead of a path, or specfyng only s but not t, but they easly reduce to the verson that we consder. Other names used n the lterature for ths problem are the delveryman problem [25] and the travelng reparman problem []. The assumpton of trangle nequalty can be elmnated from the problem formulatons that we consder f one does not requre that each node be vsted exactly once, and nstead allow walks that can pass through the same node multple tmes. The latency of a node v n ths case s defned as the dstance along the walk from s to the frst occurrence of v. The verson of a problem on a strongly connected drected graph wth non-negatve edge costs can be reduced to one satsfyng the drected trangle nequalty by assgnng new edge costs to be the lengths of the correspondng shortest paths n the orgnal graph (a.k.a. shortest path metrc completon).. Related work Both ATSPP and the drected latency problem are closely related to the classcal Travelng Salesman Problem (TSP), whch asks to fnd the cheapest Hamltonan cycle n a complete undrected graph wth nonnegatve edge costs [7,23] that satsfy the trangle nequalty d uv d uw +d vw. Though TSP s NP-hard, the well-known algorthm of Chrstofdes has an approxmaton rato of 3 /2 [0]. Later the analyss n [30, 32] showed that ths approxmaton algorthm bounds the ntegralty gap of a well-known lnear programmng relaxaton for TSP known as the Held-Karp LP. On the other hand, the ntegralty gap of ths LP relaxaton s known to be at least 4 /3. Furthermore, for all ǫ > 0, approxmatng TSP wthn a factor of 220 /29 ǫ s NP-hard for any constant ǫ > 0 [28]. Hoogeveen adapted Chrstofdes heurstc to the problem of fndng Hamltonan paths [8]. Specfcally, he obtans a 3 /2-approxmaton for the problems of fndng the cheapest Hamltonan path or fndng the cheapest Hamltonan path that starts at a gven node. When two dstnct nodes s,t are gven, hedescrbesan effcent algorthm that fndsahamltonan path wth endponts s and t whose cost s at most 5/3 tmes the mnmum-cost Hamltonan s-t path. Ths has been mproved very recently to a (+ 5)/2-approxmaton [2]. In contrast to TSP, no constant-factor approxmaton for ATSP s known. The current best approxmatonforatspstherecentresultofasadpouretal.[4], whchgves ano(logn/loglogn)- approxmaton algorthm. It also upper-bounds the ntegralty gap of the Held-Karp LP relaxaton Often the notaton uv s reserved for undrected edges, whereas (u,v) s used for drected edges. Apart from the ntroducton, all graphs n ths paper are drected, so we often use uv to refer to a drected edge as well to avod clutter. 2

3 for ATSP by the same factor. Prevous algorthms guarantee a soluton of cost wthn O(log n) factor of optmum [,3,20,2]. The algorthm of Freze et al. [3] s shown to upper-bound the Held-Karp ntegralty gap by logn n [3], and a dfferent proof that bounds the ntegralty gap of a slghtly weaker LP by O(logn) s obtaned n [26]. The best known lower bound on the Held-Karp ntegralty gap s essentally 2 [7], and tghtenng these bounds remans an mportant open problem. Fnally, ATSP s NP-hard to approxmate wthn 7 /6 ǫ [28]. Approxmaton algorthms for ATSPP, the path verson of ATSP, have only recently been studed. The frst one was an O( n) approxmaton algorthm by Lam and Newman [22], whch was subsequently mproved to O(log n) by Chekur and Pal [9]. Fege and Sngh [] mproved upon ths guarantee by a constant factor and also showed that the approxmablty of ATSP and ATSPP are wthn a constant factor of each other,.e. an α-approxmaton for one mples an O(α)- approxmaton for the other. Combned wth the result of [4], ths mples an O(logn/loglogn) approxmaton for ATSPP. However, none of these algorthms bound the ntegralty gap of any LP relaxaton for ATSPP. The ntegralty gap of an LP relaxaton was consdered by Nagarajan and Rav [27], who showed that t s at most O( n). To the best of our knowledge, the asymmetrc path verson of the k-person problem has not been studed prevously. However, some work has been done on ts symmetrc verson, where the goal s to fnd k rooted cycles of mnmum total cost (e.g., [4]). Both the drected latency problem and the undrectedlatency problem (when d uv = d vu for each u,v V) are NP-hard because an exact algorthm for ether of these could be used to effcently solve the drected or undrected Hamltonan Path problem, respectvely. The frst constant-factor approxmaton for undrected latency was developed by Blum et al. [5]. Ths was subsequently mproved n a seres of papers from 44 to 2.55 [5], then to 7.8 [3] and ultmately to 3.59 [8]. Blum et al. [5] also observed that there s some constant c > such that there s noc-approxmaton for undrected latency unless P = NP. Recently, Chakrabarty and Swamy [6] ntroduced an LP relaxaton for the undrected latency problem whose value can be effcently approxmated wthn a ( + ǫ)-factor for any constant ǫ > 0 and proved an upper bound of 0.78 on the ntegralty gap of ths relaxaton. For drected latency, Nagarajan and Rav [27] gave an O((ρ+logn)n ǫ ǫ 3 ) approxmaton algorthm for any logn < ǫ < that runs n tme no(/ǫ), where ρ s the ntegralty gapofanlprelaxaton foratspp.usngthero( n)upperboundonρ, theyobtanedaguarantee of O(n /2+ǫ ), whch was the best approxmaton rato known for ths problem before our present results..2 Our results In ths paper we study the ATSPP, the k-person ATSPP, and the drected latency problem. For all of these we use the followng lnear program, LP(α), wth dfferent values for the parameter α, 0 < α (there s a slght modfcaton for k-person ATSPP to be mentoned later). A natural LP relaxaton for ATSPP s LP(α) wth α =, where δ + ( ) denotes the set of outgong edges from a vertex or a set of vertces, and δ ( ) denotes the set of ncomng edges. A varable x e ndcates that edge e s ncluded n a soluton. In an nteger soluton (.e. x uv {0,} for all uv E), constrants () ensure that the number of arcs enterng u s equal to the number of arcs extng u for any u V \{s,t}. Constrants (2) say that one arc exts s and one arc enters t. Smlarly, constrants (3) say that no arcs enter s or ext t. Fnally, constrants (4) say that any nonempty subset of V \{s} must have at least one arc uv enterng t. Thus, nteger feasble solutons to the LP correspond to Euleran s-t walks that nclude all nodes and do not enter s or ext t. Such a walk 3

4 mn e Ed e x e (LP(α)) s.t. x(δ + (u)) = x(δ (u)) u V \{s,t} () x(δ + (s)) = x(δ (t)) = (2) x(δ (s)) = x(δ + (t)) = 0 (3) x(δ (Y)) α Y V,Y,s / Y (4) x e 0 e E can be transformed to a Hamltonan path from s to t of no greater cost by bypassng repeated nodes usng the trangle nequalty. Thus, a vald ATSPP LP relaxaton would follow by ncludng constrants x(δ + (u)) = for all u V \ {s,t}, whereas n LP(α) only constrants x(δ + (u)) α are mpled by constrants (4). However, for us ths weaker LP relaxaton wll more useful as we consder t for dfferent values of α. InSecton2wepresentouralgorthm foratsppandprovethatthentegralty gapoflp(α = ) s O(logn). Theorem. Algorthm s a polynomal tme algorthm that fnds a Hamltonan path from s to t whose cost s at most (2logn+) tmes 2 the optmum value of LP(α) wth α =. We note that, despte boundng the ntegralty gap, our algorthm s actually combnatoral and does not requre solvng the LP. Next, we study a generalzaton of the ATSPP, namely the k-person asymmetrc travelng salesman path problem. In Secton 3 we prove the followng theorem usng an algorthm that can be seen as a generalzaton of the algorthm used n the proof of Theorem.. The LP relaxaton we use for k-person ATSPP s the nearly dentcal to LP(α = ), except we use δ + (s) = δ (t) = k nstead. Theorem.2 There s a polynomal tme O(k 2 logn) approxmaton algorthm for the k-person ATSPP. Moreover, the ntegralty gap of the LP relaxaton for t s bounded by the same factor. Our algorthm for the drected latency problem requres a more detaled study of LP(α) for general values 0 < α. We strengthen the result of Theorem. by extendng t to any α ( 2,]. Ths captures the LP of [27], whch has α = 2 3, and s also used n our algorthm for the drected latency problem. We prove the followng theorem n Secton 4. Theorem.3 For any ratonal 2 < α, the Hamltonan path from s to t found by Algorthm has cost at most 6logn+3 2α tmes the value of LP(α). It s worth observng that ths theorem, together wth the results of [27], mples a polynomaltme O(n ǫ )-approxmaton for the drected latency problem for any constant ǫ > 0. Furthermore, choosng ǫ = 2 logn gves an O(log4 n)-approxmaton that runs n quas-polynomal tme n O(logn). 2 All logarthms n ths paper are base 2. 4

5 However, t seems dffcult to adapt ther approach to obtan an approxmaton algorthm that both runs n polynomal tme and guarantees a polylogarthmc bound on the approxmaton rato. Specfcally, to obtan a polylogarthmc approxmaton guarantee usng ther algorthm wth the mproved bound on the ntegralty gap of LP(α), we have to guess a polylogarthmc number of ntermedate vertces of the path. Though parts of our algorthm are motvated by steps n the drected latency algorthm n [27], we wll use a dfferent approach to obtan a polynomal-tme approxmaton. Next we consder LP(α) for α (0, 2 ]. If we allow α 2 then LP(α), as a relaxaton of ATSPP, has an unbounded ntegralty gap. However, we also prove the followng theorem n Secton 4. Theorem.4 For any postve nteger k, Algorthm 2 fnds, n polynomal tme, a collecton of at most k logn paths from s to t, of total cost at most k logn tmes the value of LP(α) wth α = k, such that each vertex of G appears on at least one of these paths. Gven these results concernng LP(α), we ntroduce and study a partcular LP relaxaton for the drected latency problem n Secton 5. We mprove upon the O(n /2+ǫ )-approxmaton of [27] substantally by provng the followng: Theorem.5 A soluton to the drected latency problem can be found n polynomal tme that has cost no more than O(logn) tmes the value of LP relaxaton (LatLP), whch s a lower bound on the nteger optmum. TheproofofTheorem.5relesheavly onmanyoftheconstrants nlprelaxaton (LatLP). So, for convenence, we present ths LP n Secton 5 rather than n ths ntroducton. We wsh to pont out that even though (LatLP) has exponentally many constrants, t can be solved n polynomal tme usng the ellpsod method. We also note that ths seems to be the frst tme that a bound s placed on the ntegralty gap of any LP relaxaton for the mnmum latency problem, even n the undrected case. As mentoned earler, there has been some recent work on LP relaxatons for undrected latency n [6] where they ntroduce a relaxaton and prove a constant upper bound on ts ntegralty gap..3 Prevous Approaches to Latency Problems and Our Technques Many of the known undrected latency approxmaton algorthms can be vewed as refnements of the followng basc approach. Suppose we know, for each k n, the cheapest path P k startng at s that vsts precsely k nodes (the endponts of these paths are not fxed). Then the total cost of all paths P k s a lower bound on the latency of the optmum path snce, for each k n the subpath of a mnmum latency path startng at s and vstng k nodes must have cost at least that of P k. Consder a subsequence = k < k 2 <... < k p = n of ndces (from,...,n) havng the property such that k + s the largest ndex such that the cost of P k+ s at most twce the cost of P k +. Construct the fnal path by concatenatng the P k s n ncreasng order of ndex where by concatenatng P k+ to P k we mean that after followng P k, we return to s before followng P k+. The cost of movng from the end of some P k back to s can be bounded by the cost of P k tself (n undrected graphs). Furthermore, snce the costs of the P k ncrease geometrcally one can argue that the total dstance travelled by the fnal path the moment that P k s completely traversed s wthn a constant factor of the length of P k ; let l denote ths dstance. Fnally, t s not too hard to see that the frst k 2 nodes on ths path (apart from s) have latency at most l, the next k 2 k 5

6 nodes have latency at most l 2, and so on; so the total latency of the path s p =2 (k k ) l, whch s wthn a constant factor of the total cost of all P k paths. Of course, fndng the paths P k s NP-hard, but a constant-factor approxmaton for ths problem can be used at the expense of a constant factor loss n the overall approxmaton rato for the undrected latency problem. It s also useful to revew the approach n [27] for drected latency. For a fxed value k, they guess the k nodes F := {v,...,v k } such that the length of the s-v subpath of the optmal soluton s a n/k n -fracton of the latency of the optmal soluton. They then present an LP that contans a varable y,v for each v V \F where y,v = ndcates that v should le between v and v + n the optmum path. They also use a v v + flow f for each and use cut constrants to ensure, for each v V \F, that at least y,v of ths v v + flow passes through v. Frst, they argue that for an approprate guess for the set F, the assocated LP relaxaton has value at most 2n k tmes the latency of the optmal path. They then round these flows usng ther bound of O( n) on the ntegralty gap for LP(α = 2 3 ) to obtan v v + paths where each node v appears on such a path. Concatenatng these paths yelds the fnal path whch, by the geometrc groupng of the latences of the nodes n F, yelds a path wth latency O(n (/2+/k) k 3 ) (the extra k 3 factor s lost n the detals of the analyss). Our approach for drected latency borrows some deas from both the known undrected algorthm and the algorthm n [27] mentoned above. However, there are some sgnfcant dfferences. Instead of computng an approxmate soluton to the cheapest path P k startng at s that vsts k nodes, we begn by consderng an s-v flow f v of value for each node v n the LP. In an ntegral soluton, ths can be thought of as an s-v path. We also consder orderng varables x uv where the dea s that x uv = means u appears before v. Then, for each dstnct u,v, the s-v flow must send at least x uv flow across any u-v cut whch captures the fact that an ntegral soluton must have the s-v path vstng u f u appears before v on the fnal s-t path. To round ths LP, we group the nodes v geometrcally accordng to the cost of ther f v flow. In each group, we would lke to use our ATSPP ntegralty gap bound to construct an s-v path from the flow f v for some v n the group that vsts all other nodes n the group. Followng these paths n ncreasng order of ther length then provdes an O(logn) approxmaton for the drected latency problem f we can fnd a good bound on the cost of travellng from the end of one path to the start of the next path. We note that ths s not a bg problem n the undrected latency problem snce we can travel from the end of a path back to s wth cost at most that of the path tself. In the drected settng, we bound ths cost ths by ntroducng a refnement of the x uv varables and addng certan constrants to ensure that the end of any path appears to a sgnfcant extent (accordng to the orderng varables) before a node that appears early n the next path. There are other techncal hurdles that we address n Secton 5 to make ths approach work. One such problem s that the s-v flow f v only guarantees the amount of flow sent over any u-v cut s x uv. It mght be that x uv s close to (or even equal to) /2 for many nodes u so the f v can only be vewed as a soluton to LP(α) for some α /2 whch s why we requre the ntegralty gap results for values α smaller than..4 Outlne of the Paper To summarze, n Secton 2 we bound the ntegralty gap of LP relaxaton (LP(α)) of ATSPP wth α = by O(logn). In Secton 3, the ATSPP algorthm s extended to k-person ATSPP and Theorem.2 s proven. We prove the supportng results n Theorems.3 and.4 n Secton 4. The drected latency algorthm s presented n Secton 5 where we prove Theorem.5. Secton 6 then 6

7 concludes ths paper. 2 Integralty gap of relaxaton LP(α) for ATSPP We show that LP relaxaton LP(α) of ATSPP wth α = has ntegralty gap O(logn). Let x be an optmal fractonal soluton, and let L be ts cost. We defne a path-cycle cover on a subset of vertces W V contanng s and t to be the unon of one s-t path and zero or more cycles, such that each v W occurs n exactly one of these subgraphs. The cost of a path-cycle cover s the total cost of ts edges. Our approach s an extenson of the algorthm by Freze et al. [3], analyzed by Wllamson [3] to bound the ntegralty gap for ATSP. That algorthm fnds a mnmum-cost cycle cover on the current set of vertces, chooses an arbtrary representatve vertex for each cycle, deletes other vertces of the cycles, and repeats untl the cycle cover contans only one cycle. The unon of all cycle covers over all teratons s a strongly connected Euleran graph whch, by the trangle nequalty, can be transformed nto a Hamltonan cycle of no greater cost. As ths s repeated at most logn tmes, and the cost of each cycle cover s at most the cost of the LP soluton, the upper bound of logn on the ntegralty gap s obtaned. In our algorthm for ATSPP, the analogue of a cycle cover s a path-cycle cover (also used n [22]), whose cost s at most the cost of the LP soluton (Lemma 2.3). At the end we combne the edges of O(logn) path-cycle covers to produce a Hamltonan path. However, the whole procedure s more nvolved than n the case of ATSP cycle. For example, we don t choose arbtrary representatve vertces, but use an amortzed analyss to ensure that each vertex only serves as a representatve a bounded number of tmes. Though t wll be convenent for our analyss to choose the representatves more carefully, we do not know f ths s necessary. In the proof of Lemma 2.3 below, we make use of the followng splttng-off theorem, as s also done n [26], where splttng off edges uv and vw refers to replacng these edges wth the edge uw (unless u = w, n whch case the two edges are just deleted). The drected connectvty λ(u,v) between two vertces u and v of a drected graph G s defned as the maxmum number of edge-dsjont drected paths from u to v n G. Theorem 2. (Frank [2, Theorem 4.3] and Jackson [9, Theorem 3]) Let G = (V,E) be a Euleran drected graph whch may contan multple edges but no loops. Then for any vw E there exsts an edge uv E such that splttng off uv and vw does not reduce λ(y,z) for any y,z V \{v}. We next consder the followng lnear program, whch dffers from LP(α) n that t s defned only on a subset of nodes W V and does not nclude constrants (4) for any but the sngleton sets. Let E W E be the set of edges wth both endponts n W. mn d e x e (LP(α,W)) e E W s.t. x(δ + (u)) = x(δ (u)) α u W \{s,t} (5) x(δ + (s)) = x(δ (t)) = (6) x(δ (s)) = x(δ + (t)) = 0 (7) x e 0 e E W 7

8 Lemma 2.2 For any ratonal α [0,] and any subset W V wth s,t W, there exsts a feasble soluton to LP(α,W) of cost no more than the cost of an optmal soluton to LP(α). Proof. AsallparametersofLP(α)areratonal, thasaratonal optmalsoluton. Let{x e : e E} be such a soluton, and let Q be a common denomnator of all x e s. We construct a drected graph H = (V,F) on the set of nodes V by ncludng Q x uv parallel edges from u to v, for any u,v V, as well as Q edges from t to s. The resultng graph H s Euleran, whch s ensured by constrants () of LP(α) for nodes other than s and t, and by constrants (2)-(3) and the extra Q edges for s and t. We now apply splttng off to nodes of V \W n H untl all of them are dsconnected from W. In partcular, whle there s an edge vw F such that v V \W, fnd an edge uv as guaranteed by Theorem 2., and modfy H by splttng off uv and vw. The graph remans Euleran after ths operaton, so the process can contnue untl nodes outsde of W have no more ncomng or outgong edges. Let x uv be defned as the number of edges from u to v n H dvded by Q, except for x ts, whch s set to zero. We argue that x s a feasble soluton to LP(α,W) whose cost s no more than the cost of x for LP(α). Let u be any node n W \ {s,t}. We have x(δ + (u)) = x(δ (u)) because H remaned Euleran. Consder the orgnal drected connectvty λ(s, u) n H. By Menger s theorem, λ(s,u) = mn{δ (Y) : u Y V \{s}}. But from constrants (4) and the constructon of H, ths s at least α Q. By the guarantee of Theorem 2., λ(s,u) does not decrease durng the splttng-off process, whch mples that the n-degree of u also remans at least α Q. So x (δ (u)) α, showng that x satsfes constrants (5) of LP(α,W). The only way that the n- or out-degree of a node w W can change durng a splttng-off process s f the edges wv and vw are splt off for some node v / W. However, nodes s and t do not partcpate n such 2-cycles wth nodes outsde of W, so ther degrees n H never change. Thus, constrants (6) and (7) are satsfed for x. The cost of x s no more than that of x because of trangle nequalty. If we assgn the costs d e to edges of H, we can see that splttng-off does not ncrease the total cost: whenever a new edge uw s ntroduced, t replaces two old edges uv and vw, whose cost, by the trangle nequalty, s the same or hgher as the cost of uw. Now, the orgnal cost of H, not countng the ts edges, s Q tmes the cost of x as a soluton to LP(α), and the cost of x as a soluton to LP(α,W) s at most the fnal cost of H (also not countng the ts edges) dvded by Q. A path-cycle cover of mnmum cost can be found by a combnatoral algorthm, usng a reducton to mnmum-cost perfect matchng, as explaned n [22]. We use the followng lemma to bound ts cost. Lemma 2.3 For any subset W V that ncludes s and t, there s a path-cycle cover of W of cost at most the cost of LP(α =,W). Proof. LP(,W) s equvalent to a crculaton problem on a network (whch can be seen by dentfyng s and t), and therefore s ntegral (see [29, p. 207] or the proof of Clam n [26]). In partcular, t has an nteger optmal soluton. By Lemma 2.2, the cost of ths soluton s at most L. In prncple, ths nteger soluton can have x(δ + (u)) > for some nodes u. In ths case we fnd a path-cycle cover of no greater cost as follows. Consder the Euleran graph formed by addngatsedgetothenteger soluton tolp(,w)andfndaeulertourforeach ofts components. Shortcut these tours over extra copes of vertces that appear more than once, whch, by the trangle nequalty, does not ncrease the cost. Note that the edge ts s not nvolved n such a shortcut snce 8

9 the n and out degree of both s and t s exactly n the Euleran graph. Fnally, removng the edge ts produces a path-cycle cover of W of cost at most L. We next present our algorthm for approxmatng ATSPP, Algorthm. For convenence of presentaton, we equate an nteger flow to a drected mult-graph that has an edge uv for each unt of flow assgned to the edge uv. In the same way, we regard an nteger crculaton as equvalent to a drected Euleran multgraph. Addng two such graphs (flows, crculatons) that are defned on the same set of nodes means takng dsjont unon of ther edges. Algorthm Asymmetrc Travelng Salesman Path : Let a set W V; nteger labels l v 0 for all v V; flow F and crculaton H 2: for 2 log 2 n + teratons do 3: Fnd an nteger mnmum-cost s-t path-cycle cover F on W 4: F F +F F s acyclc before ths operaton 5: Fnd a path-cycle decomposton of F, wth cycles C...C k and paths P...P h, such that P s acyclc 6: for each strongly connected component A of j C j do A s a crculaton 7: For each vertex u A, let d u be the n-degree of u n A 8: Fnd a representatve node r A mnmzng l r +d r 9: F F A subtract flows 0: for each w A, w r, and for each path P : f w P then modfy F by shortcuttng P over w 2: Remove all nodes n A, except r, from W they don t partcpate n F anymore 3: H H +A add crculatons 4: l r l r +d r 5: end for 6: end for 7: Let P be an s-t path consstng of nodes n W n the order found by topologcally sortng F F s an acyclc flow on the nodes W 8: for every connected component X of H of sze X > do 9: Fnd a Euler tour of X, shortcut over nodes that appear more than once 20: Incorporate the resultng cycle nto P usng a shared node 2: end for 22: return P P s a Hamltonan s-t path The basc dea of our algorthm s adapted from the frst part of the O( n)-approxmaton of Lam and Newman [22]: fnd an nteger path-cycle cover, select a representatve node for each cycle, delete the other cycle nodes, and repeat untl we can construct a relatvely cheap Hamltonan path on the remanng nodes. Then, the deleted nodes can be added to ths path usng edges of the deleted cycles n a manner smlar to Freze et al. for ATSP [3]. Our approach s a bt more nvolved n that a representatve s selected for a subgraph more general than a smple cycle. The algorthm mantans several structures that help to keep track of ts progress. The set of nodes W, whch s ntally equal to V, contans the nodes that have not been deleted yet. Crculaton H contans the edges of the subgraphs that have been removed, and whch wll be used n the last stage to reconnect the deleted nodes to the fnal path. Flow F conssts of leftover acyclc path-cycle edges from prevous teratons. In each teraton, a new path-cycle cover s added 9

10 to F, and then the cyclc parts of F are removed and transferred to H. Components that can be removed from F are Euleran subgraphs. In order to be able to reconnect them at the end, a representatve node, whch s not deleted from F wth the rest of ts component A, s chosen for each one. The labels l v are used to load-balance the number of tmes that vertces are used as representatves. At each teraton of the man for-loop, we fnd an nteger path-cycle cover F over W (as guaranteed by Lemma 2.3) and add t to F. Now the flow (mult-graph) F mght have some cycles. We fnd a path-cycle decomposton of F, say C,...,C k are the cycles and P,...,P h are the paths n ths decomposton. By fndng and removng all the cycles frst, and then fndng the path decomposton, we can ensure that the unon of paths of ths decomposton s acyclc. Our goal s to keep one representatve for each strongly connected component A of j C j and delete the rest of those nodes from W. Thus, when we eventually fnd an s-t path that goes through the remanng nodes of W, and n partcular the representatve node of A, we can expand ths path to vst all the nodes of A as well, and hence obtan an s-t path spannng all of V. By load-balancng usng labels, we ensure that after 2 log n + teratons, each survvng vertex has partcpated n the acyclc part of the path-cycle covers at least logn + tmes. Usng ths fact and a technque of [27], we show that by the end of the man loop, F contans enough edges to form a spannng s-t path over all the survvng vertces of W. Then we expand ths s-t path at the representatve nodes by addng the subpaths obtaned from the cyclc part, H, of the unon of path-cycle covers. Lemma 2.4 Durng the course of the algorthm, for each v V, l v logn. The dea of the proof s, as the algorthm proceeds, to mantan a forest on the set of nodes V, such that the number of nodes n a subtree rooted at any v V s at least 2 lv. The lemma then follows because the number of nodes n a subtree cannot exceed n. We frst prove an auxlary clam. Clam 2.5 In each component A found by Algorthm on lne 6, there are two dstnct nodes x and y such that d x = d y =. Proof. Let F be the value of F at the start of the current teraton of the outsde loop,.e. before F s added to t on lne 4. F s acyclc, because durng the course of the loop, all cycles of F are subtracted from t. So A s a unon of cycles, formed from the sum of an acyclc flow F and a path-cycle cover F, whch sends exactly one unt of flow through each vertex. Consder a topologcal orderng of nodes based on the flow F, and let x and y be the frst and last nodes of A, respectvely, n ths orderng. As A always contans at least two nodes, x and y are dstnct. Snce x and y partcpate n some cycle(s) n A, ther n-degrees are at least. We now clam that the n-degree of x n A s at most. Indeed, snce all other nodes of A are later than x n the topologcal orderng, t cannot have any flow comng from them n F. So the only ncomng flow to x can be n F. But snce F sends a flow of exactly one unt through each vertex, the n-degree of x n A s at most one. A symmetrcal argument can be made for y, showng that ts out-degree n A s at most one. But snce A s a unon of cycles, every node s n-degree s equal to ts out-degree, and the n-degree of y s also at most. Proof of Lemma 2.4. As the algorthm proceeds, let us construct a forest on the set of nodes V. Intally, each node s the root of ts own tree. We mantan the nvarant that W s the set of tree roots n ths forest. For each component A that the algorthm consders, and the representatve 0

11 node r found on lne 8, we attach the nodes of A, except r, as chldren of r. Note that the nvarant s mantaned, as these nodes are removed from W on lne 2. The set of nodes of each component A found on lne 6 s always a subset of W, and thus our constructon ndeed produces a forest. We show by nducton on the steps of the algorthm that f a node has label l, then ts subtree contans at least 2 l nodes. Thus, snce there are n nodes total, and the labels are all nteger values, no label can exceed log 2 n. At the begnnng of the algorthm, all labels are 0, and all trees have one node each, so the base case holds. Now consder some teraton n whch the label of vertex r A s ncreased from l r to l r +d r. By Clam 2.5, there are nodes x,y A (possbly one of them equal to r) wth d x = d y =. Snce r mnmzes l v + d v among all vertces v A, we have that l x +d x l r +d r and l y +d y l r +d r, and thus l x l r +d r and l y l r +d r. Thus, by the nducton hypothess, the trees rooted at x and y each have at least 2 lr+dr nodes. Because we update the forest n such a way that r s new tree contans all the nodes of trees prevously rooted at x and y, ths tree now has at least 2 2 lr+dr = 2 lr+dr nodes. Lemma 2.6 At the end of the algorthm s man loop, the flow n F passng through any node v W s equal to 2 logn + l v, and thus (by Lemma 2.4) s at least logn +. Proof. There are 2 logn + teratons, each of whch adds one unt of flow through each vertex v W. We now clam that, by the end of the algorthm, for a vertex v W, the amount of flow passng through t that has been removed from F s equal to ts label, l v. Flow s removed from v only f v becomes part of some component A. Now, f t s ever part of A, but not chosen as a representatve on lne 8, then t s removed from W. Thus, we are only concerned about vertces that are chosen as representatves every tme that they are part of A. Such a vertex has flow d v gong through t n A, whch s the amount subtracted from F. But snce ths s also the amount by whch ts label ncreases, the lemma follows. Lemma 2.7 When Algorthm reaches lne 7, F contans a spannng s-t path on the set of nodes W. Proof. Recall that F s acyclc at ths pont n the algorthm, and let P be an orderng of nodes n W obtaned by a topologcal sort of F. We show that P s actually a path n F,.e. that there s an edge between each par (u,v) of consecutve nodes of P. Ths s smlar to an argument used n [27]. Suppose that we fnd a flow decomposton of F nto s-t paths. There are at precsely 2 logn + such paths, and, by Lemma 2.6, each vertex of W partcpates n at least logn +, or more than half, of them. Ths means that any two consecutve vertces n P, such as u and v, must share a path, say p, n ths decomposton. Because v appears later than u n the topologcal order, v must come after u n p. Moreover, we clam that n p, v s the mmedate successor of u. If not, suppose that there s a node w that appears between u and v n p. But ths means that n the topologcal orderng, w wll appear after u and before v, whch contradcts the fact that they are consecutve n P. So u and v are neghbors on p, whch means that F contans an edge uv. Lemma 2.8 Algorthm fnds a Hamltonan s-t path n the graph G whose cost s at most the sum of costs of the 2 logn + path-cycle covers computed by the algorthm. Proof. As the algorthm proceeds, the total cost of edges n F and H never exceeds the cost of all the path-cycle covers found so far. The edges of the path-cycle covers are ether moved from F to H, whch does not change the total cost, or a shortcuttng operaton s performed, whch does

12 not ncrease the cost due to the trangle nequalty. We bound the cost of the fnal path by the cost of F and H. By Lemma 2.7, the path P found on lne 7 s a subgraph of F, and thus costs no more than F does. On the other hand, the Euler tours found on lne 9 cost no more than the edges of H. We now show that the algorthm ndeed produces a Hamltonan path and bound the cost of connectng the components of H to P. Now we descrbe how the cycles obtaned n Step 9 are ncorporated nto path P n lne 20. At the end of the man loop, all nodes of V are part of ether W or H or both. For every component X of the second for-loop, assume that C s the cycle obtaned over nodes of X on lne 9. We clam that C shares exactly one node wth W (and thus wth P). Note that every component A added to H contans only nodes that are n W at that tme. Moreover, when ths s done, all but one nodes of A are expelled from W (on lne 2). So when several components of H are connected by the addton of A, the nvarant s mantaned that there s exactly one node per component that s shared wth W. Now, suppose that v s the vertex shared by the cycle C and the path P. On lne 20, we ncorporate the cycle nto the path by followng the path up to v, then traversng the cycle from v to the predecessor of v, then connectng t to the successor of v on the path. So when all the components of H are ncorporated nto the path P, all nodes of V become part of the path. By trangle nequalty, the resultng longer path costs no more than the sum of costs of the old path and the cycles. Theorem. now easly follows from Lemmas 2.2, 2.3 and Algorthm for k-person ATSPP In ths secton we consder the k-person asymmetrc travelng salesman path problem. The LP relaxaton we present for ths problem s smlar to LP(α), but wth x(δ + (s)) = x(δ (t)) = k for constrant (2) and x(δ + (S)) for constrant (4). Smlar to path-cycle covers, we defne a k-pathcycle cover on a subset W of V contanng s and t to be the unon of k s-t paths and zero or more cycles such that each v W {s,t} occurs n exactly one of these subgraphs and nether s nor t appears on any cycle. Lke a path-cycle cover, the mnmum-cost k-path-cycle cover can be found by a combnatoral algorthm by creatng k copes each of s and t and usng the matchng algorthm descrbed n [22]. Arguments smlar to those of Lemmas 2.2 and 2.3 show that a k-path-cycle cover on any subset W V s a lower bound on the value of the LP relaxaton for the k-person ATSPP. Our algorthm constructs a soluton that uses each edge of O(k log n) k-path-cycle covers at most k tmes, provng a bound of O(k 2 logn) on the approxmaton rato and the ntegralty gap. The algorthm starts by runnng lnes -6 of Algorthm, except wth T = (k +) logn + teratons of the loop and fndng mnmum-cost k-path-cycle covers nstead of the path-cycle covers on lne 3. Then t fnds k s-t paths n the resultng acyclc graph F, satsfyng condtons of Lemma 3.2 below. The algorthm concludes by ncorporatng each component of the crculaton H nto one of the obtaned paths, smlar to lnes 8-2 of Algorthm. Essentally the same proof as for Lemma 2.4 shows that the labels l v do not exceed logn n our k-person ATSPP algorthm. Lemma 2.6 also generalzes wth a nearly dentcal proof to the followng. Lemma 3. At the end of the algorthm s man loop, the flow n F passng through any node v W s equal to T l v, and thus s at least k logn +. 2

13 Proof. The proof s nearly dentcal to that of Lemma 2.6, except that there are now T = (k +) logn + teratons of the man loop. Lemma 3.2 After lnes -6 of Algorthm are executed wth T teratons of the loop on lne 2 and fndng mnmum-cost k-path-cycle covers on lne 3, there exst k s-t paths n the resultng acyclc graph F, such that each edge of F belongs to at most k of them, and every node of W s contaned n at least one path. Moreover, these paths can be found n polynomal tme. Proof. We note that the graph F can support kt unts of flow from s to t. Ths s because n each of the T teratons, k s-t paths are added to the graph, whereas the removal of cycles does not decrease the amount of flow supported. So F can be decomposed nto a set P of kt edge-dsjont paths from s to t. Moreover, each node of F partcpates n at least k logn + of these paths by Lemma 3.. Let K = (W,F ) be the drected graph on W wth uv F f there s a path from u to v usng only edges n F. Note that K s an acyclc graph where for any u,v,w W such that uv F and vw F, t must also be that uw F. Let K be the undrected graph obtaned by removng the orentaton on each edge n K. Then K s a comparablty graph, whch s perfect [6]. We clam that the nodes of K can be parttoned nto k clques. Snce K s a perfect graph, the mnmum number of clques requred to cover the nodes of K s equal to the sze of the largest ndependent set. So, we only have to show that there s no ndependent set of sze k+. Suppose, for the sake of contradcton, that I s an ndependent set of sze k +. By the constructon of K, no two nodes n I can le on a common path n the path decomposton P of F. Snce each node n W les on at least k logn + of these paths, the number of paths n P s at least (k + )(k logn + ) > k((k + ) logn + ) = kt. Ths contradcts P = kt and we conclude that K can be covered wth k clques. Note that the mnmum clque cover of a comparablty graph can be found n polynomal tme (eg. [6]). Say k k s the sze of the mnmum clque cover. To transform these clques nto paths, we frst add both s and t to each of the k clques n the clque cover. If k < k, then we add k k copes of the clque {s,t}. Say the resultng clques are C,...,C k. Order each C topologcally (accordng to F) to get an s-t path P n K that ncludes each node n C. Fnally, let P denote the path obtaned by replacng each edge of P by ts correspondng path n F. Note that for each P, the paths n F correspondng to dfferent edges n P are edge-dsjont snce F s acyclc. Overall, we have that each edge of each path P s also an edge n F and each edge n F s used at most once by any partcular path P. Thus, the unon of the k paths P,...,P k uses only edges n F and each edge n F s used by at most k paths. Proof of Theorem.2. Let L be the cost of a lnear programmng relaxaton for the problem. Theedges of F as well as theedges usedto connect theeulerancomponents of H tothe pathscome from the unon of T k-path-cycle covers on subsets of V, and thus cost at most T L = O(klogn) L. However, the algorthm may use each edge of F up to k tmes n the paths of Lemma 3.2, whch makes the total cost of the produced soluton at most O(k 2 logn) L. 3

14 4 Analyss of relaxed ATSPP LP 4. Case α (,]: Proof of Theorem.3 2 The case of α = follows from Theorem.. Assume that α ( 2,) s ratonal and consder LP(α) wth cost L. We show that Algorthm fnds a Hamltonan s-t path of cost at most 6logn+3 2α L, thus boundng the ntegralty gap of ths LP for ATSPP. By Lemma 2.2, for any subset W V, the cost of the optmal soluton to LP(α,W) s at most L. As we prove n Lemma 4. below, ths mples that the cost of the optmal soluton to LP(,W) 3 s at most 2α L. So, by Lemma 2.3, the cost of a mnmum path-cycle cover of any subset W s also bounded by ths value. An applcaton of Lemma 2.8 concludes the proof by showng that 3 Algorthm fnds a Hamltonan s-t path of cost at most (2logn+) 2α L. Lemma 4. For any ratonal value of α ( 2,), the cost of LP(,W) s upper-bounded by 3 2α tmes the cost of LP(α,W). Proof. Let x be a ratonal optmal soluton to LP(α,W), and let ˆx = x/α be a scaled flow vector. If we vew ˆx as a soluton to LP(,W), we see that constrants (5) and (7) are satsfed, but constrants (6) are volated, as ˆx(δ + (s)) = ˆx(δ (t)) = /α >. The rest of the proof shows how to transform ˆx nto a feasble soluton for LP(,W). We can thnkof ˆx as aflow F of /α untsfrom sto t. Fnd aflow decomposton of F nto paths and cycles, each carryng the same amount of flow (say σ), so that the unon of the paths s acyclc. Ths s possble for suffcently small σ, as all quanttes nvolved are ratonal. Let F = F p + F c, where F p s the sum of flows on the paths n our decomposton, and F c s the sum of flows on the cycles. Observe that we have a total of /(ασ) paths. Let γ be a quantty satsfyng 2α < γ <. For any node u such that the amount of F p flow gong through u s less than γ, shortcut any flow decomposton paths that contan u, so that there s no more F p flow gong through u. Let U W be the set of vertces stll partcpatng n the F p flow. Then each vertex n U has at least γ unts of F p flow gong through t (and so partcpates n at least γ/σ such paths), and each vertex n W \U has at least γ unts of F c flow gong through t. We fnd a topologcal orderng of vertces n U accordng to F p (whch s acyclc), and let P be an s-t path that contans the nodes of U n ths topologcal order. Clam 4.2 The cost of P s at most 2γ /α tmes the cost of F p. Proof. The argument for ths s smlar to the one n the proof of Lemma 2.7. Out of the /α unts of flow gong from s to t n F p, each vertex u U carres at least γ unts, whch s more than half of the total amount (as γ > 2α ). Consder any two consecutve vertces u,v on P, n ths order. Snce we have a total of ασ paths n F p, out of whch at least γ/σ contan u and at least γ/σ contan v, ths means that at least 2γ σ ασ paths must contan both u and v. On these paths, u s followed mmedately by v, as any other node w between u and v would contradct u and v beng consecutve n the topologcal orderng. Snce each such path has a flow of σ, the cost of P s at most /(2γ α ) tmes the cost of F p. We now defne x as a flow equal to one unt of s-t flow on the path P plus γ tmes the flow F c. We clam that x s a feasble soluton to LP(,W): there s exactly one unt of flow from s to t (as F c conssts of cycles not contanng s or t); there s flow conservaton at all nodes except s and 4

15 2 3 D Fgure : Large ntegralty gap example for LP(α) wth α = /2. Here, D s an arbtrarly large nteger. t; each vertex n U (and thus n P) has at least one unt of flow gong through t; and each vertex n W \U has at least one unt of flow gong through t (as t had at least γ unts of F c flow). The cost of ths soluton s at most 2γ /α cost(f p)+ γ cost(f c) max ( 2γ /α, γ ) α L. If we set γ = 3 + 3α, whch satsfes 2α < γ <, we see that the cost of x s at most 3 2α L. 4.2 Case α : Proof of Theorem.4 2 Consder LP(α) wth α = k for some nteger k 2. As a relaxaton for the ATSPP problem, ths LP has unbounded ntegralty gap. For example, let D be an arbtrarly large value and consder the shortest path metrc obtaned from the graph n Fgure. One can verfy that the followng assgnment of x-values to the arcs s feasble for LP (LP(α)) wth α = /2. Assgn a value of /2 to arcs (,2), (3,2), (3,6), (,4), (5,4), and (5,6) and a value of to arcs (2,3) and (4,5). Every other arc s assgned a value of 0. Ths assgnment s feasble for the lnear program and has objectve functon value 5. On the other hand, any Hamltonan path from to 6 has cost at least D. Let L be the cost of the optmal soluton to LP(α = k ). We present Algorthm 2 that fnds k logn paths from s to t, contanng all the vertces, wth total cost of at most klogn L. It s smlar to our prevous algorthms, except the paths and ther ntermedate nodes are also removed n each teraton and the representatve for a cycle s chosen arbtrarly. Lemma 4.3 For any subset W V that ncludes s and t, there s a k-path-cycle cover of W wth total cost at most kl. Proof. By Lemma 2.2, there s a soluton to LP( k,w) of cost at most L. Now, f we multply each x e n ths soluton by k, we get a feasble soluton to a modfcaton of LP(,W) n whch constrants (6) are replaced wth x(δ + (s)) = x(δ (t)) = k. Note that ths s the same LP that we consdered for the k-person ATSPP. The cost of ths soluton s no more than kl. As n the proof of Lemma 2.3, ths LP also has an nteger optmum, whch, possbly after shortcuttng, s exactly a k-path-cycle cover. 5

16 Algorthm 2 : Intalze 0, W 0 V. Let k be an nteger parameter. 2: whle W > 2 do stops when W = {s,t} 3: Fnd a mnmum-cost k-path-cycle cover F of W, wth paths P and cycles C 4: For each cycle C C, choose a representatve vertex v C C 5: Let W + {s,t} {v C : C C }; + 6: end whle 7: Let T be the number of teratons and F T j=0 F j be the unon of all k-path-cycle covers 8: Add kt t-s arcs to F to produce an Euleran graph and fnd a Euler tour H on t 9: Delete the t-s arcs from H to produce kt s-t walks 0: Shortcut these walks over repeated nodes and return the resultng paths Lemma 4.4 The unon of k-path-cycle covers F found by Algorthm 2 contans a v-t path for each v V. Proof. We use a smple backward nducton to show that for every, T j= F j contans a v-t path for each v W. The base case s = T, where the last k-path-cycle cover found does not have any cycles (as otherwse there would be a node v s,t that remans n W + after ths teraton). In ths case all the vertces of W partcpate n s-t paths. For the nducton step, for 0 < T, assume that T j=+ F j contans a v-t path for every v W +. Every vertex v W \ W + partcpates ether n an s-t path n F, n whch case t s connected to t by ths path, or n a cycle, say C. In ths case, the representatve v C s n W +, and v s connected to t by frst followng the cycle C to v C and then followng the v C -t path n T j=+. Proof of Theorem.4. We observe that W + \{s,t} W \{s,t} /2 for each, and so T s at most logn. Ths s because each cycle n C contans at least two vertces, so for each v C that s ncluded n W + \{s,t} there s at least one vertex n W \{s,t} that s not. Each k-path-cycle cover adds an equal number of ncomng and outgong arcs to each vertex except for s and t, to whch t adds k outgong and ncomng arcs respectvely. Thus, addng kt t-s arcs to F produces a graph wth equal n and out degree at each node. By Lemma 4.4 and the smple fact that a weakly connected graph wth equal n and out degree at each node, the addton of these t-s arcs yelds an Euleran graph. By Lemma 4.3, the cost of any F s at most kl, so the cost of F and the fnal soluton s at most klt kllogn. 5 Approxmaton algorthm for Drected Latency 5. Lnear programmng relaxaton We ntroduce LP relaxaton (LatLP) for the drected latency problem. In LatLP, a varable x uw ndcates that node u appears before node w on the path. Smlarly, x uvw for three dstnct nodes u, v, w ndcates that they appear n ths order on the path. We do not know f these x uvw varables are necessary to obtan an LP relaxaton wth ntegralty gap O(log n), but they wll be convenent n the analyss of our roundng algorthm. The basc dea why we use these varables s that we wll generate many paths endng at dfferent nodes and we want to somehow transform these paths nto a sngle path. Roughly speakng, we do ths by appendng some nodes of one path to the end of another. The cost of the edge used when concatenatng subpaths n ths manner can be bounded 6