Operations Research Letters. Simpler analysis of LP extreme points for traveling salesman and survivable network design problems

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1 Operatons Research Letters 38 (010) Contents lsts avalable at ScenceDrect Operatons Research Letters journal homepage: Smpler analyss of LP extreme ponts for travelng salesman and survvable network desgn problems Vswanath Nagarajan a,, R. Rav b, Moht Sngh c a IBM T.J. Watson Research Center, Yorktown Heghts, NY 10598, USA b Tepper School of Busness, CMU, Pttsburgh, PA, USA c McGll Unversty, Montreal, QC, Canada a r t c l e n f o a b s t r a c t Artcle hstory: Receved 10 September 009 Accepted 5 January 010 Avalable onlne 13 February 010 Keywords: Lnear programmng relaxatons Travelng salesman Survvable network desgn Approxmaton algorthms We consder the Survvable Network Desgn Problem (SNDP) and the Symmetrc Travelng Salesman Problem (STSP). We gve smpler proofs of the exstence of a 1 -edge and 1-edge n any extreme pont of the natural LP relaxatons for the SNDP and STSP, respectvely. We formulate a common generalzaton of both problems and show our results by a new countng argument. We also obtan a smpler proof of the exstence of a 1 -edge n any extreme pont of the set-par LP relaxaton for the element connectvty Survvable Network Desgn Problem (SNDP elt ). 010 Elsever B.V. All rghts reserved. 1. Introducton We consder two well-studed combnatoral optmzaton problems, the Survvable Network Desgn Problem (SNDP) and the Symmetrc Travelng Salesman Problem (STSP). Gven an undrected graph G = (V, E) and connectvty requrements r uv for all undrected pars u, v V of vertces, a Stener network s a subgraph of G n whch there are at least r uv edge-dsjont paths between u and v for all pars u, v V. The Survvable Network Desgn Problem s a general network desgn problem where we are gven an edge-weghted graph G = (V, E) and connectvty requrements { r uv (u, v) ( V )}, and the task s to fnd a mnmum-cost Stener network. A Hamltonan cycle n graph G = (V, E) s a connected subgraph of G that has degree at every vertex of V. In the Symmetrc Travelng Salesman problem (STSP), we are gven an edge-weghted undrected graph G = (V, E), and the goal s to compute a mnmum-cost Hamltonan cycle. Lnear programmng methods have been successfully used n solvng both these problems n practce [1,8]. Strong theoretcal results have also been obtaned by analyzng lnear programmng (LP) relaxatons for these problems [7,8]. We present a common generalzaton of these problems and ts natural LP relaxaton. Correspondng author. E-mal address: vswanath@us.bm.com (V. Nagarajan). Usng ths LP and a new countng argument, we prove the followng results n Secton. Theorem 1.1. Gven any extreme pont x of the LP relaxaton (LP sndp ) for Survvable Network Desgn, there exsts an edge e such that x e 1. Theorem 1.. Gven any extreme pont x of the LP relaxaton (LP stsp ) for the Symmetrc Travelng Salesman, there exsts an edge e such that x e = 1. Theorem 1.1 was orgnally proved by Jan [9], and Theorem 1. by Boyd and Pulleyblank [3]. In fact [3] showed that any extreme pont of the LP relaxaton to the STSP has at least three 1-edges. We also consder the element connectvty Survvable Network Desgn Problem (SNDP elt ) n Secton 3. Ths s a well-known generalzaton of the usual (edge-connectvty) SNDP, where the nput s an edge-weghted undrected graph G = (V, E), a set U V of termnals, and connectvty requrements r uv for all undrected pars u, v U of termnals. The vertces V \U and edges E of the graph are called elements. The goal n SNDP elt s to fnd the mnmum-cost subgraph that contans at least r uv element-dsjont paths between u and v for every u, v U. Usng the new countng argument, we provde a shorter proof of the followng theorem for ts natural LP relaxaton consdered n Flescher et al. [5]. Theorem 1.3. Gven any extreme pont x of the LP relaxaton (LP elt ) for element connectvty Survvable Network Desgn, there exsts an edge e such that x e /$ see front matter 010 Elsever B.V. All rghts reserved. do: /j.orl

2 V. Nagarajan et al. / Operatons Research Letters 38 (010) Ths result s orgnally due to Flescher et al. [5], where they used t to obtan a -approxmaton algorthm for SNDP elt. Recently, Chuzhoy and Khanna [4] gave a very elegant reducton from the (even more general) vertex-connectvty SNDP to the elementconnectvty SNDP; usng ths -approxmaton for SNDP elt, they obtaned an O(k 3 log n)-approxmaton algorthm for the vertexconnectvty SNDP (here k s the maxmum requrement and n s the number of vertces). Our proofs are based on a new countng argument that nvolves dstrbutng fractonal tokens. Ths dea was used earler n Bansal et al. [] for degree-bounded network desgn problems n drected graphs, and also appears mplct n the proofs of Gabow et al. [6] for the k-edge connected subgraph problem. Notaton. For any subset F E of edges, the characterstc vector χ(f) {0, 1} E (also denoted χ F ) contans a 1 correspondng to each edge e F, and a 0 otherwse. For any assgnment x : E R + of non-negatve real values to the edges and any subset F E, x(f) denotes the sum e F x e.. The STSP and the edge-connectvty SNDP Gven a subset S V, let δ(s) = {(u, v) E u S, v S} denote the set of edges wth exactly one end-pont n S. We also denote δ({v}) by δ(v). Then, the classcal LP relaxaton (LP stsp ) for the STSP has the followng constrants: x(δ(s)) S V (cut constrants) x(δ(v)) = v V (degree constrants) We now consder the LP relaxaton (LP sndp ) for the SNDP. A functon f from subsets of V to the ntegers s called weakly supermodular f f (V) = f ( ) = 0 and, for all S, T V, one of the followng holds. f (S) + f (T) f (S T) + f (S T), f (S) + f (T) f (S \ T) + f (T \ S). It s easy to see that the functon f defned by f (S) = max u S,v S r uv for each subset S V s weakly supermodular. It can be verfed that the above functon encodes the connectvty requrements {r u,v }. We state the LP relaxaton [9] for any network desgn problem wth weakly supermodular connectvty requrement (whch contans the SNDP as a specal case). x(δ(s)) f (S) S V (cut constrants) Now we present the LP relaxaton of a generalzaton of both the SNDP and the STSP. The nput conssts of an undrected graph G = (V, E) wth edge-costs c : E R +, a weakly supermodular functon f : V Z, and a desgnated subset W V of vertces. The LP correspondng to ths s as follows. (LP) mnmze c e x e e E subject to x(δ(s)) f (S) S V x(δ(v)) = f (v) v W Note that the frst set of constrants above enforces the connectvty requrements f, the second set of constrants enforces the degree constrants on W, and the last set of constrants ensures that only a subgraph s chosen. Gven graph G, edge-costs c and connectvty requrements { r u,v (u, v) ( V or )}, the LP relaxaton (LP sndp ) of ths SNDP nstance s obtaned by settng, n (LP), f (S) = max u S,v S r uv for each subset S V and W =. For an nstance of the STSP gven by graph G and edge-costs c, the correspondng LP relaxaton (LP stsp ) s obtaned by settng f (S) = for each S V, f ( ) = f (V) = 0, and W = V. We prove the followng theorem, whch mples Theorems 1.1 and 1.. Theorem.1. Let x be a basc feasble soluton to (LP) where f s weakly supermodular. A. There exsts an edge e E such that x e 1. B. Moreover, f f (S) s even for each subset S V, then there exsts an edge e E such that x e = 1. The frst part of Theorem.1 was at the heart of the teratve - approxmaton algorthm for the SNDP [9]. Before the proof of Theorem.1, we state some propertes of tght constrants of extreme ponts. Two sets X, Y are ntersectng f X Y, X Y and Y X are nonempty. A famly of sets s lamnar f no two sets are ntersectng. The proof of the followng lemma s mmedate from the uncrossng lemma n Jan [9]. Lemma. ([9]). Let x be a basc feasble soluton to (LP) wth f beng weakly supermodular, such that 0 < x e < 1 for all edges e E. Then, there exsts a lamnar famly L of subsets such that 1. x s the unque soluton to {x(δ(s)) = f (S), S L};. the vectors χ δ(s) for S L are lnearly ndependent; and 3. E = L. Proof. Lemma 4.3 n [9] proves ths lemma when W = ; that proof s based on standard uncrossng arguments. In the general case, there are addtonal equaltes for sngleton vertexsets correspondng to W. Let (LP ) denote the polytope gven by just the frst and thrd sets of constrants n (LP),.e., wthout equalty constrants on W. Note that the polytope (LP) s a face of polytope (LP ). Hence any extreme pont n (LP) s also an extreme pont n (LP ), for whch the lemma from [9] apples. We now prove Theorem.1. Let x be any basc feasble soluton to (LP). Proof of Theorem.1 (A). We frst prove that x e 1 for some edge e E. Suppose for the sake of contradcton that x e < 1 for each e E. If x e = 0 for some e E, we can remove edge e from the graph G and varable x e from (LP). The resdual soluton x remans a basc feasble soluton to the modfed (LP). Thus we assume wthout loss of generalty that x e > 0 for all e E, and so Lemma. apples. We wll show a contradcton to Lemma. by means of a new countng argument. The countng argument proceeds as follows. We assgn one token to each edge n E, and then reassgn the tokens such that we can collect strctly more than one token per set n the lamnar famly L: ths would mply E > L, whch s the desred contradcton. For any sets S, R L, we say that S s the parent of R (or equvalently, that R s a chld of S) f S s the smallest set n L contanng R. Each edge e = (u, v) E s gven a unt token, whch t reassgns as follows. 1. (Rule 1) Let S be the smallest set n L contanng u, and R be the smallest set n L contanng v. Then e assgns x e tokens to each of S and R.. (Rule ) Let T be the smallest set n L contanng both u and v. Then e assgns 1 x e tokens to T.

3 158 V. Nagarajan et al. / Operatons Research Letters 38 (010) Fg. 1. Example for the expresson x(δ(s)) k x(δ(r )) wth k = chldren. The dashed edges cancel out n the expresson. Edge-sets A, B, C are shown wth ther respectve coeffcents. We now show that each set n L receves at least one token. Let S L have k chldren R 1,..., R k n L (f S does not have any chldren then k = 0). We have the followng tght nequaltes for the extreme pont x. x(δ(s)) = f (S) and x(δ(r )) = f (R ) 1 k. Subtractng, we obtan x(δ(s)) x(δ(r )) = f (S) x(a) x(b) x(c) = f (S) where f (R ) f (R ) A = {e : e ( R ) = 0, e S = 1} B = {e : e ( R ) = 1, e S = } C = {e : e ( R ) =, e S = }. Observe that A B C : otherwse, we have the dependence χ δ(s) = k χ δ(r ). Also, S receves x e tokens for each edge e A (by Rule 1), 1 x e tokens for each edge e B (by Rules 1 & ), and 1 x e tokens for each edge e C (by Rule ). Hence, the total number of tokens receved by S s exactly x e + (1 x e ) + (1 x e ) = x(a) + B x(b) + C x(c) = B + C + f (S) f (R ). (1) Observe that, for every edge e E, x e, 1 x e, 1 x e > 0 snce 0 < x e < 1 ; combned wth the fact that A B C, the number of tokens assgned to S s strctly postve (usng the frst expresson n Eq. (1)). On the other hand, the last expresson n (1) mples that the number of tokens assgned to S s ntegral. Thus every S L gets at least one token n ths assgnment (Fg. 1). Now we show that there are some unassgned tokens, thereby showng the strct nequalty L < E. Let R be a maxmumcardnalty set n L; note that none of the sets n L \ {R} contans R and R V snce f (V) = 0. Consder any edge e δ(r) : the token by Rule for edge e s unassgned as there s no set such that T e =. Ths gves us the desred contradcton, and proves the frst part of Theorem.1. Proof of Theorem.1 (B). We now consder the case when f (S) s even for each S V, and show that, for any basc feasble soluton x to (LP), there s always an edge e E wth x e = 1. The proof follows the same approach as above but wth a scaled token assgnment. For the sake of contradcton, we assume that x e < 1 for each e E. As before, we can assume wthout loss of generalty that x e > 0. Agan, we wll show a contradcton to Lemma. by showng that L < E. The countng argument proceeds as follows. We assgn one token to each edge e = (u, v) E, whch t redstrbutes as follows. 1. (Rule 1 ) Let S be the smallest set n L contanng u, and R be the smallest set n L contanng v. Then e assgns x e tokens to each of S and R.. (Rule ) Let T be the smallest set n L contanng both u and v. Then e assgns 1 x e tokens to T. We now show that each set n L receves at least one token. As before, let S L have chldren R 1,..., R k (k 0). We have the followng tght nequaltes. x(δ(s)) = f (S) and x(δ(r )) = f (R ) 1 k. Dvdng by two and subtractng, we obtan [ 1 x(δ(s)) ] [ x(δ(r )) = 1 f (S) ] f (R ) [ x(a) x(b) x(c) = 1 f (S) ] f (R ) where the edge sets A, B, C are exactly as n the earler case. Observe that A B C : else there s a dependence n the constrants for S and ts chldren. Also, S receves x e tokens for each edge e A (Rule 1 ), 1 x e tokens for each edge e B (Rules 1 & ), and 1 x e tokens for each edge e C (Rule ). Hence, the total number of tokens receved by S s x e + ( 1 x ) e + (1 x e ) = x(a) x(b) + B + C x(c) f (S) f (R ) = B + C +. Followng the same reasonng as before, ths quantty s a postve nteger (here f s an even-valued functon, so the number of tokens s stll ntegral). Thus every set S L receves at least one token n ths assgnment. Fnally, note that some tokens correspondng to the maxmal sets n L are unassgned. Ths shows the strct nequalty L < E, and gves us the desred contradcton. Ths proves the second part of Theorem The element-connectvty SNDP In ths secton, we consder the element-connectvty survvable network desgn problem (SNDP elt ). In ths problem, we are gven an undrected graph G = (V, E) wth edge-costs c : E R +, a set U V of termnals, and connectvty requrements r uv for all undrected pars u, v U of termnals. Vertces n V \ U are called non-termnals. The edges and non-termnals of the graph are called elements. The goal n SNDP elt s to fnd the mnmum-cost subgraph that contans at least r uv element-dsjont paths between u and v for every u, v U. Flescher et al. [5] used teratve roundng to obtan a -approxmaton algorthm for ths problem. They [5] showed that SNDP elt can be formulated as a sutable nteger program (defned formally below), such that any extreme pont soluton to ts LP-relaxaton contans an edge wth soluton value at least half. We gve a short proof of ths result usng a new countng argument generalzng the results n the prevous secton.

4 V. Nagarajan et al. / Operatons Research Letters 38 (010) A set-par s an ordered tuple (S, S ) where S, S V. Let F denote some famly of set-pars. A two-set functon f : F Z + s called weakly two-supermodular f, for any (S, S ) and (T, T ) F, at least one of the followng holds. 1. (S T, S T ) and (S T, S T ) F, and we have f (S T, S T ) + f (S T, S T ) f (S, S ) + f (T, T ).. (S T, S T) and (S T, S T) F, and we have f (S T, S T) + f (S T, S T) f (S, S ) + f (T, T ). For any set-par (S, S ), let E(S, S ) = {e = (u, v) E u S, v S } denote the edges wth one end-pont n S and the other n S. For any assgnment x : E R + and set-par (S, S ), we abbrevate x(e(s, S )) by just x(s, S ). The LP-relaxaton for SNDP elt consdered n [5] s the followng. (LP elt ) mnmze c e x e e E subject to x(s, S ) f (S, S ) (S, S ) F 0 x e 1 e E, where F = { (S, S ) S S =, U S S }, and f (S, S ) = max{r uv u S U, v S U} V S S for any (S, S ) F. Note that f s a weakly two-supermodular functon on set-pars F. We wll prove the followng that mmedately mples Theorem 1.3. Theorem 3.1. Let x be a basc feasble soluton to (LP elt ), where f : F Z + s weakly two-supermodular; then there exsts an e E such that x e 1. As mentoned earler, ths theorem was proved earler n Flescher et al. [5], and s the man ngredent n the -approxmaton algorthm for the element-connectvty SNDP. We frst ntroduce some defntons from [5] that are requred for the proof. Defne a partal order on set-pars where (S, S ) (T, T ) ff S T and T S ; n ths case we say that (S, S ) s smaller than (T, T ). We also say that (S, S ) and (T, T ) are comparable f ether (S, S ) (T, T ) or (T, T ) (S, S ); otherwse they are ncomparable. Set-pars (S, S ) and (T, T ) are sad to par-cross ff none of the followng holds. C1. S T and T S ;.e., (S, S ) (T, T ). C. S T and T S;.e., (S, S ) (T, T ). C3. S T and T S. A collecton of set-pars s par-lamnar f no two of them par-cross. The followng result appears as Corollary 4.6 and Lemma 4.7 n Flescher et al. [5]. Lemma 3. ([5]). Let x be a basc feasble soluton to (LP elt ), such that 0 < x e < 1 for all edges e E. Then, there exsts a par-lamnar famly L of set-pars such that 1. x s the unque soluton to {x(s, S ) = f (S, S ), (S, S ) L};. the vectors χ E(S,S ) for (S, S ) L are lnearly ndependent; 3. the poset nduced by on L s a forest;.e., for any (X, X ), (Y, Y ), (Z, Z ) L wth (X, X ) (Y, Y ) and (X, X ) (Z, Z ), the set-pars (Y, Y ) and (Z, Z ) are comparable; and 4. E = L. We also refer to set-pars n L as nodes. For any node (S, S ) L, ts parent s the smallest node (T, T ) L \ {(S, S )} that s larger than (S, S ) (.e., satsfyng (T, T ) (S, S )); n ths case (S, S ) s called a chld of (T, T ). If there s no node n L \ {(S, S )} that s larger than (S, S ), then node (S, S ) s called a maxmal node. Node (R, R ) L s a descendent of (S, S ) L ff (R, R ) (S, S ). Proof of Theorem 3.1. Suppose for a contradcton that the clam does not hold, and let x be an extreme pont soluton wth x e < 1 for all e E. If x e = 0 for some e E, we can remove edge e from the graph G and varable x e from (LP elt ). The resdual soluton x remans a basc feasble soluton to the modfed (LP elt ). Thus we assume wthout loss of generalty that x e > 0 for all e E, and so Lemma 3. apples. We wll derve a contradcton usng a countng argument smlar to the one n the prevous secton. Each edge e = (, j) E s assgned one unt of token, whch t dstrbutes to nodes n L as follows. 1. Rule I: assgn x e tokens to the smallest node (S, S ) L such that ether S or {, j} S =.. Rule II: assgn x e tokens to the smallest node (T, T ) L such that ether j T or {, j} T =. 3. Rule III: assgn 1 x e tokens to the smallest node (R, R ) L such that {, j} R =. Note that both x e and 1 x e are strctly postve for any edge e. Addtonally, by Lemma 4.8 n Flescher et al. [5], t follows that each of Rules I, II and III assgns tokens to at most one node. Hence each edge n E dstrbutes a total of at most one token. We now show that each node of L receves a total of at least one token. Consder any node (S, S ) L wth chldren {(R, R )}k ; f (S, S ) s a leaf then k = 0. For each [k], we have (A) R S, snce (S, S ) (R, R ); and (B) R R j for all j [k] \ {}, snce (R, R ) and (R j, R j ) are ncomparable, and they satsfy condton (C3). Addtonally, the {R } k are dsjont subsets of S. Defne the followng edge-sets: H = k E(R, R \ S ) C = {e H : e ( R ) = } B = {e H : e ( R ) = 1} k D = E(R, S ) A = E ( S \ ( R ), S ). k Thus we can wrte x(e(r, R )) = x(c) + x(b) + x(d), and x(e(s, S )) = x(d)+x(a). Recall that the tght LP constrants mply that x(e(s, S )) = f (S, S ) and x(e(r, R )) = f (R, R ) 1 k. Subtractng, we obtan (snce the f -values are all ntegral) x(e(s, S )) x(e(r, R )) = f (S, S ) x(a) x(b) x(c) Z. f (R, R ) Z Addng B + C (an nteger) to the above expresson, we obtan x e + (1 x e ) + (1 x e ) Z. Note that A B C ; otherwse, χ(e(s, S )) = k χ (E(R, R )), contradctng the lnear ndependence n Lemma 3.. Snce 0 < x e < 1 for all e E, the left-hand sde above s strctly postve, and x e + (1 x e ) + (1 x e ) 1. () We now show that the tokens assgned to (S, S ) total to at least the left-hand sde n Inequalty ().

5 160 V. Nagarajan et al. / Operatons Research Letters 38 (010) Edge e = (u, v) A. Let u S \ ( R ) and v S. We clam that the token assgned by Rule I goes to (S, S ). Clearly, (S, S ) s the smallest set-par wth u S. For any descendant (T, T ) of (S, S ), we must have T S v; thus we cannot have u, v T. Hence (S, S ) receves x e tokens from e. Edge e = (u, v) C. Let u R and v R j for, j [k], j. We clam that the token assgned by Rule III goes to (S, S ). Clearly u, v S. Furthermore, for any chld (R l, R l ) of (S, S ) we have R l R u or R l R j v. Hence (S, S ) receves 1 x e tokens from e. Edge e = (u, v) B. Let u R and v R \ S for some [k]. We frst clam that the token assgned by Rule III goes to (S, S ). Clearly u, v S. We show that {u, v} R l for every chld (R l, R l ) of (S, S ). 1. Suppose l = ; then v R.. Suppose l [k] \ {}; then u R R l. That s, (S, S ) receves the token by Rule III. We next clam that the token assgned by Rule II also goes to (S, S ). Note that v R, so no descendant (T, T ) of (S, S ) can have v T. As seen above, (S, S ) s the smallest node wth u, v S ;.e., (S, S ) receves the token by Rule II. Hence (S, S ) receves n total 1 x e tokens from e. Thus each node of L receves at least a unt token. We now show that there s some postve amount of unused tokens. Let (P, P ) L be any maxmal node n L. Note that there s at least one maxmal node (P, P ) L and E(P, P ). We clam that the token of any edge (u, v) E(P, P ) gven by Rule III s unused. Let u P and v P. For any descendent (T, T ) of (P, P ), we have T P v; so T {u, v}. Any node (Q, Q ) L that s not a descendent of (P, P ) s ncomparable to (P, P ), and we have Q P u. Thus {u, v} S for all (S, S ) L,.e., the Rule III token of edge (u, v) s unassgned. Thus there s a postve amount of unused tokens. However, ths mples that E > L, whch contradcts Lemma 3.. Ths completes the proof of Theorem 3.1. Acknowledgement Ths research was supported by NSF Grants CCF and CCF References [1] Davd L. Applegate, Robert E. Bxby, Vasek Chvátal, Wllam J. Cook, The Travelng Salesman Problem: A Computatonal Study, Prnceton Unversty Press, 006. [] N. Bansal, R. Khandekar, V. Nagarajan, Addtve guarantees for degree-bounded drected network desgn, SIAM J. Comput. 39 (4) (009) [3] S.C. Boyd, W.R. Pulleyblank, Optmzng over the subtour polytope of the travellng salesman problem, Math. Program. 49 () (1990) [4] J. Chuzhoy, S. Khanna, An O(k 3 log n)-approxmaton algorthm for vertexconnectvty survvable network desgn, n: Proceedngs of FOCS, 009. [5] L. Flescher, K. Jan, D.P. Wllamson, Iteratve roundng -approxmaton algorthms for mnmum-cost vertex connectvty problems, J. Comput. System Sc. 7 (5) (006) [6] H.N. Gabow, M.X. Goemans, E. Tardos, D.P. Wllamson, Approxmatng the smallest k-edge connected spannng subgraph by LP-roundng, Networks 53 (4) (009) [7] M. Goemans, Worst-case comparson of vald nequaltes for the TSP, Math. Program. 69 (1 3) (005) [8] M. Grötschel, C.L. Monma, M. Stoer, Desgn of Survvable Networks, n: M.O. Ball, et al. (Eds.), Handbooks n OR & MS, vol. 7, 1995, pp [9] K. Jan, A factor -approxmaton algorthm for the generalzed Stener network problem, Combnatorca 1 (1) (001)