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On minimal Eulerian graphs
V~~urne12, ~urnbe~4
Received 19 ~~v~rn~~ 1980; revisedversion received20 February1981
Suppose that we are @vena graph (; 2 [If, E) and we are asked whether there exists a cycle s ’ of G “-~on~dered as a subset C c E of the edger?t f4CT-such that (V, E - C) is connected. In the special case in which G is Eulerian (i.e., connected and witi. all or, in another version, all but two - vertices with c ttcstl degree) this problem cafl be restated as follows: given as Eule~~ graph, is it rn~~? That is, does it have an Eulerian proper subgraph? This problem has two important applications in rna~ernati~~ pr~~~~g. It has been obeyed sevI eral times that the special case of the traveling salesman problem (TSP) in which the distances satisfy the triarrngle irzequtlft,v(ATSP) has certain positive algorithmic propertie:a([I ,7] ; for an overviewof the TSP see the forcing hook 23)) Now, for the goners TSP there is an elemt polyto~l fo~nulation with many applicationsto the development of algorithms and the modeling cl’ltrcal search heuristics (see, e.g., [2,6])..We~on~de~beachtour of It pivots as a O-l vector tif the (~)-d~e~ion~ space, and take the canvex polytope which is the convex huil of these i(n - I)! points. It turns out that this polytope coma pletely raptures the TSP, and its pro~~ies are some* times sources of ~go~~~ ideas, or of further evil. dence about the difficulty of the TSP (see, eg., [2,4]). Is there such a polytopal formulation of the ATSP? It turns out that, under tha usual gumption that all
* Reslraarch supportedby NSF Grant ~~S79~896~
distances are positive, an indirect $1.ch formulation does exist, Consider the poly (Iape 111 in (~)~d~en~ sional mace ‘whoseve~ices aret not the tours~but the ~~i#i~l Eileen
alpha cmn nodes. Fcx WI
distance matrix D define its clogureD to be *.1c whose (i, jrh entry is the length of the shortest p&m from i to j under D, 6 by its very ~on~t~~*ion~a~~~~ fies the t~~~e inequity Fu~he~~r~ an ~pt~um tour of n points under 5 directly corresponds to a rn~~~ Eulerian graph under D (simply replace earth edge (i, j) by thr+~ortest path from i to j under D)* Thus thd p~~~~tope n does raptured in some ~en~clr, the ATSP. ?“he p_Dblsmof testing an Eulerian graph for minimality ii, therefore the problem of recognizingthe set sf vertices of lL Wenow iurn to the aced ~~l~~at~~ of tots Eule~~~~graphs for comity’ Consider the versionof the TSP: we wish to visit n cities, the f’irst kr times, the second kz, etc. This problem can be solved by dyn~ic prosing in time pr~~~ti~~ to n Cki e fI(ki + 1) [8]. By using a different techa nique, however, we can solve it in time e(n) *log Zki, thus rendering the ~omple~ty practically ~de~nd~n~ of the ki’s. Here e(e)is an e~ponenti~y hong fugue Zion [4,5]. This te~~que entails flog a tr~s~~aiioa problem for the optimal Eulerian graph correa Jpondirtgto each given degree sequence. The computae tion time would be reduce ~,~bst~ti~y if we could repeat this only for the og lal Eule*!an ~~ph%thst are rni~~~, Hence the nt 1 for an a ~~o~t~ that l
203
Volume 12, number4
INFOlRMATlON PRWESSZNG LETTERS
tests Eulerian graphs for minimality. Very ~~~rn~y, this problem is ~~ornplet~ (Theorem 1). The same result holds for the case of Eulerian d&mph, which is even more important with respect to the second application above. As a consequencc ll is to our knowledge the only natural polytope, *ACset of vertices of which is hard to recognize. Theorem 1. Testing whether a given Eulerian graph is not rninhal is NP4omplete. Proof.An Ederiaa graph G = (V, E) is not minimal iff (by def~tion) there is an Eulerian graph (V, E’), with E’ c E, Take E - E’; it is a non-empty graph with even degrees, and it is therefore the union of several simple cycles. Let C be one of these cycles. It follows that G is not minimal i$f there is a simple cycle C c E such that (V, E - C) is connected. This problem is certainly in MA To prove it NP=complete, we shall reduce the satisftibility problem to it. Suppose that we are given a Boolean fo.rmula F with variables xl, .... x, and clauses C1, l*., C,; flout loss of generality we may assume Sii;sithere are n clauses of the form (X*Vxi) for i -‘ 73 ..,, n in F. We shall construct an Eulerian graph G(& E) with the property that there is a simple cycle C of G such that (V, E - C) is connected iff F is storable. For each variable xi, G has two nodes ai and bi. They are, all 2n of them, connected to a node c via disjoint paths of length 2 (or double edges, as we shall call them, see Fig. 1). Double edges play an important role in our ~onst~ction. They cannot par~cipate in the cycle C, because their removal would isolate the
Figa I. 204
13 August 1981
Fig. 2.
~dpo~t. They do however enforce ~oMecti~ty and Eulerianness. We &O add the edges [bigai+r], i = 1, ..,, n - 1, as well as [bn,.al] - see Fig. 1. For each clause Ci with, say, ji literals we have ji nodes, namely dil, .... di$ They are CYC~C~Y connected via double edges (see Fig. 2). We call ‘thisset of nodes the Ci-component of G; the %‘s,hi’s and c make up ‘he central component. Notice that so far we have only taken into account the ‘syntax’ of F, that is, the integers n, m, and the ji’s. The precise structure of the clauses enters into play next. Suppose that the literal xi appears pl th in the qr th clause, ,.., prst in the 4rth clause. Ihen we add to E the path P5 = [q, dQaP1,.... dqrPr, bi]. This path will be taken by C jffxi is to be false_Also, suppose that the literal xi appears s1th in the t sth clause, ... . Sk* in the tk et clause. We add to E the path pjr = [Q, dtls,, . . .. dtkgk, bi]. If & is taken by C, this will mean that xi is Srue (notice the crosswise ~p~cation). This completes the const~~tion of G. It is easy to check that G is Eulerian. We next claim that G has a circuit C such that (V, E - C) is connected iff F is satisfiable. Suppose that such a C exists, C cannot use any of the double edges of G, because for any such C (V, E - C) is disconnected. Therefore C will be a subset of $he P,$, the Pxi ‘s and the [bi, %+I] edges. Can C consist of both PQ and Pxi, for some Xi? Recall that F contains a clause Cj =:(xiVXi)*The Cj=component is thus connected to the rest of the graph via O~Y Phi and z”jri edges. Thus (V, E - C) would disconnect the Cicomponent, contrary to our assumption. It follows that C consists of the [bi, ai+*]edges, and for each Xiof one of the Pxi or k4 paths. Take, as before, Phi c- C to mean that xi isfalse, and pirr 51 C to mean that xi is true. consider now any clause Cj. The Cjcomponent is connected to the rest of the graph. This means that, for some literal X E Ci, P&@ C. This however means that Ci is satisfied by the truth assignment implied by C. Hence F is satisfiable. ~onve~ely, if t is a truth silent satisf~g F,
Volume 12, number 4
INFORMATION PROCESSING LETTERS
it is easy to see that the simple circuit C=
{[bi, ai+l]: i = 1, .... em- l} U ([b,, aI]} U
u Ph, t(A)= forse
does not disconnect G. Sincein the proof of the Theorem we could appropriately direct all edges and still preserve the Eulerian property and the reduction, we have the following result : Corollary. Testing whether an Eulerian digraph is not
minimal is NPcomplete.
13 August 1981 a
[2] M. Grotschel and W. Padberg, On the symmetric travelling salesman problem, Math. Programming 16(3) (1979) 265-302. [ 31 E.L. Lawler, J.K. Lenstra and A.G.H. RinhooyKan, Eds., The Traveling Salesman Problem, forthcoming. [4] C.H. Papadimitriou, The adjacency relation on the traveling salesman polytope in NP-complete, Math. Programming 14 (1978) 312-324. [ 51 C.H. Papadimitriou, The traveling salesman problem with many visits to few cities, Proc. 1980 EURO Conference, Cambridge, England [6] C.H. Papadimitriou and K. Steiglitz, The complexity of local search for the traveling salesman problem, SIAM J. Comput. 60) (1977) 76-83. [7] C.H. Papadimitriou and K. Steiglitz, Some difficult examples of the traveling salesman problem, Oper. Res. 26(3) (1978) 434-443. [ 8) H. Psaraftis, A dynamic programming approach for . sequencing groups of small identical jobs, Oper. Res., forthcoming.
References [l ] N. Christofides, Worst-case analysis of a new heuristic
for the traveling salesman problem, Techn. Rep. GSIA, Carnegie-Mellorl University (1976).
205
Received 19 ~~v~rn~~ 1980; revisedversion received20 February1981
Suppose that we are @vena graph (; 2 [If, E) and we are asked whether there exists a cycle s ’ of G “-~on~dered as a subset C c E of the edger?t f4CT-such that (V, E - C) is connected. In the special case in which G is Eulerian (i.e., connected and witi. all or, in another version, all but two - vertices with c ttcstl degree) this problem cafl be restated as follows: given as Eule~~ graph, is it rn~~? That is, does it have an Eulerian proper subgraph? This problem has two important applications in rna~ernati~~ pr~~~~g. It has been obeyed sevI eral times that the special case of the traveling salesman problem (TSP) in which the distances satisfy the triarrngle irzequtlft,v(ATSP) has certain positive algorithmic propertie:a([I ,7] ; for an overviewof the TSP see the forcing hook 23)) Now, for the goners TSP there is an elemt polyto~l fo~nulation with many applicationsto the development of algorithms and the modeling cl’ltrcal search heuristics (see, e.g., [2,6])..We~on~de~beachtour of It pivots as a O-l vector tif the (~)-d~e~ion~ space, and take the canvex polytope which is the convex huil of these i(n - I)! points. It turns out that this polytope coma pletely raptures the TSP, and its pro~~ies are some* times sources of ~go~~~ ideas, or of further evil. dence about the difficulty of the TSP (see, eg., [2,4]). Is there such a polytopal formulation of the ATSP? It turns out that, under tha usual gumption that all
* Reslraarch supportedby NSF Grant ~~S79~896~
distances are positive, an indirect $1.ch formulation does exist, Consider the poly (Iape 111 in (~)~d~en~ sional mace ‘whoseve~ices aret not the tours~but the ~~i#i~l Eileen
alpha cmn nodes. Fcx WI
distance matrix D define its clogureD to be *.1c whose (i, jrh entry is the length of the shortest p&m from i to j under D, 6 by its very ~on~t~~*ion~a~~~~ fies the t~~~e inequity Fu~he~~r~ an ~pt~um tour of n points under 5 directly corresponds to a rn~~~ Eulerian graph under D (simply replace earth edge (i, j) by thr+~ortest path from i to j under D)* Thus thd p~~~~tope n does raptured in some ~en~clr, the ATSP. ?“he p_Dblsmof testing an Eulerian graph for minimality ii, therefore the problem of recognizingthe set sf vertices of lL Wenow iurn to the aced ~~l~~at~~ of tots Eule~~~~graphs for comity’ Consider the versionof the TSP: we wish to visit n cities, the f’irst kr times, the second kz, etc. This problem can be solved by dyn~ic prosing in time pr~~~ti~~ to n Cki e fI(ki + 1) [8]. By using a different techa nique, however, we can solve it in time e(n) *log Zki, thus rendering the ~omple~ty practically ~de~nd~n~ of the ki’s. Here e(e)is an e~ponenti~y hong fugue Zion [4,5]. This te~~que entails flog a tr~s~~aiioa problem for the optimal Eulerian graph correa Jpondirtgto each given degree sequence. The computae tion time would be reduce ~,~bst~ti~y if we could repeat this only for the og lal Eule*!an ~~ph%thst are rni~~~, Hence the nt 1 for an a ~~o~t~ that l
203
Volume 12, number4
INFOlRMATlON PRWESSZNG LETTERS
tests Eulerian graphs for minimality. Very ~~~rn~y, this problem is ~~ornplet~ (Theorem 1). The same result holds for the case of Eulerian d&mph, which is even more important with respect to the second application above. As a consequencc ll is to our knowledge the only natural polytope, *ACset of vertices of which is hard to recognize. Theorem 1. Testing whether a given Eulerian graph is not rninhal is NP4omplete. Proof.An Ederiaa graph G = (V, E) is not minimal iff (by def~tion) there is an Eulerian graph (V, E’), with E’ c E, Take E - E’; it is a non-empty graph with even degrees, and it is therefore the union of several simple cycles. Let C be one of these cycles. It follows that G is not minimal i$f there is a simple cycle C c E such that (V, E - C) is connected. This problem is certainly in MA To prove it NP=complete, we shall reduce the satisftibility problem to it. Suppose that we are given a Boolean fo.rmula F with variables xl, .... x, and clauses C1, l*., C,; flout loss of generality we may assume Sii;sithere are n clauses of the form (X*Vxi) for i -‘ 73 ..,, n in F. We shall construct an Eulerian graph G(& E) with the property that there is a simple cycle C of G such that (V, E - C) is connected iff F is storable. For each variable xi, G has two nodes ai and bi. They are, all 2n of them, connected to a node c via disjoint paths of length 2 (or double edges, as we shall call them, see Fig. 1). Double edges play an important role in our ~onst~ction. They cannot par~cipate in the cycle C, because their removal would isolate the
Figa I. 204
13 August 1981
Fig. 2.
~dpo~t. They do however enforce ~oMecti~ty and Eulerianness. We &O add the edges [bigai+r], i = 1, ..,, n - 1, as well as [bn,.al] - see Fig. 1. For each clause Ci with, say, ji literals we have ji nodes, namely dil, .... di$ They are CYC~C~Y connected via double edges (see Fig. 2). We call ‘thisset of nodes the Ci-component of G; the %‘s,hi’s and c make up ‘he central component. Notice that so far we have only taken into account the ‘syntax’ of F, that is, the integers n, m, and the ji’s. The precise structure of the clauses enters into play next. Suppose that the literal xi appears pl th in the qr th clause, ,.., prst in the 4rth clause. Ihen we add to E the path P5 = [q, dQaP1,.... dqrPr, bi]. This path will be taken by C jffxi is to be false_Also, suppose that the literal xi appears s1th in the t sth clause, ... . Sk* in the tk et clause. We add to E the path pjr = [Q, dtls,, . . .. dtkgk, bi]. If & is taken by C, this will mean that xi is Srue (notice the crosswise ~p~cation). This completes the const~~tion of G. It is easy to check that G is Eulerian. We next claim that G has a circuit C such that (V, E - C) is connected iff F is satisfiable. Suppose that such a C exists, C cannot use any of the double edges of G, because for any such C (V, E - C) is disconnected. Therefore C will be a subset of $he P,$, the Pxi ‘s and the [bi, %+I] edges. Can C consist of both PQ and Pxi, for some Xi? Recall that F contains a clause Cj =:(xiVXi)*The Cj=component is thus connected to the rest of the graph via O~Y Phi and z”jri edges. Thus (V, E - C) would disconnect the Cicomponent, contrary to our assumption. It follows that C consists of the [bi, ai+*]edges, and for each Xiof one of the Pxi or k4 paths. Take, as before, Phi c- C to mean that xi isfalse, and pirr 51 C to mean that xi is true. consider now any clause Cj. The Cjcomponent is connected to the rest of the graph. This means that, for some literal X E Ci, P&@ C. This however means that Ci is satisfied by the truth assignment implied by C. Hence F is satisfiable. ~onve~ely, if t is a truth silent satisf~g F,
Volume 12, number 4
INFORMATION PROCESSING LETTERS
it is easy to see that the simple circuit C=
{[bi, ai+l]: i = 1, .... em- l} U ([b,, aI]} U
u Ph, t(A)= forse
does not disconnect G. Sincein the proof of the Theorem we could appropriately direct all edges and still preserve the Eulerian property and the reduction, we have the following result : Corollary. Testing whether an Eulerian digraph is not
minimal is NPcomplete.
13 August 1981 a
[2] M. Grotschel and W. Padberg, On the symmetric travelling salesman problem, Math. Programming 16(3) (1979) 265-302. [ 31 E.L. Lawler, J.K. Lenstra and A.G.H. RinhooyKan, Eds., The Traveling Salesman Problem, forthcoming. [4] C.H. Papadimitriou, The adjacency relation on the traveling salesman polytope in NP-complete, Math. Programming 14 (1978) 312-324. [ 51 C.H. Papadimitriou, The traveling salesman problem with many visits to few cities, Proc. 1980 EURO Conference, Cambridge, England [6] C.H. Papadimitriou and K. Steiglitz, The complexity of local search for the traveling salesman problem, SIAM J. Comput. 60) (1977) 76-83. [7] C.H. Papadimitriou and K. Steiglitz, Some difficult examples of the traveling salesman problem, Oper. Res. 26(3) (1978) 434-443. [ 8) H. Psaraftis, A dynamic programming approach for . sequencing groups of small identical jobs, Oper. Res., forthcoming.
References [l ] N. Christofides, Worst-case analysis of a new heuristic
for the traveling salesman problem, Techn. Rep. GSIA, Carnegie-Mellorl University (1976).
205