A Proof of the Boyd-Carr Conjecture

A Pro of of the Bo yd-Carr Conjecture F rans Sc halek amp Da vid P . Williamson ∗ Ank e v an Zuylen † Octob er 22, 2018 Abstract Determining the precise in tegrality gap for the subtour LP relaxation of the tra veling sales- man problem is a significan t op en question, with little progress made in thirt y years in the general case of symmetric costs that ob ey triangle inequality . Bo yd and Carr [3] observe that w e do not even kno w the worst-case upper bound on the ratio of the optimal 2-matching to the subtour LP; they conjecture the ratio is at most 10/9. In this pap er, w e prov e the Boyd-Carr conjecture. In the case that a fractional 2-matching has no cut edge, w e can further prov e that an optimal 2-matching is at most 10/9 times the cost of the fractional 2-matching. ∗ Address: School of Op erations Research and Information Engineering, Cornell Universit y , Ithaca, NY 14853, USA. Email: dpw@cs.cornell.edu . This w ork was carried out while the author was on sabbatical at TU Berlin. Supp orted in part by the Berlin Mathematical Sc ho ol, the Alexander von Hum b oldt F oundation, and NSF grant CCF-1115256. † Address: Max-Planck-Institut f ¨ ur Informatik, Department 1: Algorithms and Complexity , Campus E1 4, Ro om 311c, 66123 Saarbr¨ uc ken, Germany . Email: anke@mpi-inf.mpg.de . 1 In tro duction The trav eling salesman problem (TSP) is the most famous problem in discrete optimization. Given a set of n cities and the costs c ( i, j ) of trav eling from city i to city j for all i, j , the goal of the problem is to find the least exp ensiv e tour that visits each cit y exactly once and returns to its starting point. An instance of the TSP is called symmetric if c ( i, j ) = c ( j, i ) for all i, j ; it is asymmetric otherwise. Costs ob ey the triangle ine quality if c ( i, j ) ≤ c ( i, k ) + c ( k , j ) for all i, j, k . The TSP is known to b e NP-hard, even in the case that instances are symmetric and ob ey the triangle inequality . F rom now on w e consider only these instances unless otherwise stated. Because of the NP-hardness of the trav eling salesman problem, researchers hav e considered appro ximation algorithms for the problem. The b est appro ximation algorithm currently known is a 3 2 -appro ximation algorithm given by Christofides in 1976 [7]. Better approximation algorithms are known for sp ecial cases. Exciting progress has been made recently in the case of the graphical TSP , in whic h costs c ( i, j ) are giv en b y shortest path distances in an unw eigh ted graph; M¨ omk e and Sv ensson [14] give a 1.461-approximation algorithm for this case. How ev er, to date, Christofides’ algorithm has the b est known p erformance guarantee for the general case. There is a w ell-known, natural direction for making progress whic h has also defied improv ement for nearly thirt y y ears. The following linear programming relaxation of the tra veling salesman problem was used b y Dantzig, F ulkerson, and Johnson [8] in 1954. F or simplicit y of notation, we let G = ( V , E ) b e a complete undirected graph on n vertices. In the LP relaxation, w e hav e a v ariable x ( e ) for all e = ( i, j ) that denotes whether we trav el directly b et ween cities i and j on our tour. Let c ( e ) = c ( i, j ), and let δ ( S ) denote the set of all e dges with exactly one endp oin t in S ⊆ V . Then the relaxation is Min X e ∈ E c ( e ) x ( e ) ( S U B T ) sub ject to: X e ∈ δ ( i ) x ( e ) = 2 , ∀ i ∈ V , (1) X e ∈ δ ( S ) x ( e ) ≥ 2 , ∀ S ⊂ V , 3 ≤ | S | ≤ | V | − 3 (2) 0 ≤ x ( e ) ≤ 1 , ∀ e ∈ E . (3) The first set of constraints (1) are called the de gr e e c onstr aints . The second set of constrain ts (2) are sometimes called subtour elimination c onstr aints or sometimes just subtour c onstr aints , since they preven t solutions in whic h there is a subtour of just the v ertices in S . As a result, the linear program is sometimes called the subtour LP . It is kno wn that the equality sign in the first set of constrain ts may be replaced b y ≥ in case the costs ob ey the triangle inequality (Goemans and Bertsimas [12]; see also Williamson [19]). The LP is known to give excellen t lo wer bounds on TSP instances in practice, coming within a percent or tw o of the length of the optimal tour (see, for instance, Johnson and McGeoch [13]). Ho wev er, its theoretical worst-case is not well understo o d. In 1980, W olsey [20] show ed that Christofides’ algorithm pro duces a solution whose v alue is at most 3 2 times the v alue of the subtour LP (also shown later b y Shmoys and Williamson [18]). This pro ves that the inte gr ality gap of the subtour LP is at most 3 2 ; the integralit y gap is the w orst-case ratio, tak en ov er all instances of the problem, of the v alue of the optimal tour to the v alue of the subtour LP , or the ratio of the optimal in teger solution to the optimal fractional solution. The integralit y gap of the LP is known to b e at least 4 3 via a sp ecific class of instances. Ho wev er, no instance is kno wn that has integralit y gap w orse than this, and it has b een conjectured for some time that the integralit y gap is at most 4 3 1 Figure 1: Illustration of the w orst example known for the ratio of 2-matc hings to the subtour LP . The figure on the left shows the instance; all edges in the graph hav e cost 1, all other edges hav e cost 2. The figure in the center giv es the subtour LP solution, in which the dotted edges hav e v alue 1 2 , and the solid edges hav e v alue 1; this is also an optimal fractional 2-matching. The figure on the right gives an optimal 2-matching, whic h is also the optimal tour. (see, for instance, Go emans [11]). The results of M¨ omk e and Sv ensson [14] sho w that in the case of the graphical TSP , the in tegralit y gap is at most 1.461; if the graph is cubic, Bo yd, Sitters, v an der Ster, and Stougie [6] show that the gap is 4 3 , and M¨ omke and Sv ensson extend this b ound to sub cubic graphs as well. There is some evidence that the conjecture might b e true. Benoit and Boyd [2] hav e shown via computational metho ds that the conjecture holds for n ≤ 10, and Boyd and Elliot-Magwoo d [5] ha ve extended this to n ≤ 12. In a 1995 pap er, Go emans [11] show ed that adding an y class of v alid inequalities known at the time to the subtour LP could increase the v alue of the LP by at most 4 3 ; this is necessary for the conjecture to b e true. Somewhat weak er evidence is as follows. A 2-matching is an in teger solution to the subtour LP obeying only the degree constrain ts (1) and the b ounds constraints (3). 1 A fr actional 2-matching is a 2-matching without the integralit y constrain ts. Bo yd and Carr [4] hav e sho wn that the integralit y gap for the 2-matching problem is at most 4 3 . F urthermore, Boyd and Carr [3] hav e sho wn that if the subtour LP solution is half-integral (that is, x ( i, j ) ∈ { 0 , 1 2 , 1 } for all i, j ∈ V ) and has a particular structure then there is a tour of cost at most 4 3 times the v alue of the subtour LP . Not only do w e not know the integralit y gap of the subtour LP , Bo yd and Carr ha ve observ ed that we don’t even kno w the worst-case ratio of the optimal 2-matching to the v alue of the subtour LP , whic h is surprising b ecause 2-matchings are w ell understoo d and w ell c haracterized. They mak e the following conjecture. Conjecture 1 (Bo yd and Carr [3]) The worst-c ase r atio of an optimal 2-matching to an opti- mal solution to the subtour LP is at most 10 9 . It is known that there are cases for which the cost of an optimal 2-matching is at least 10 9 times the optimal solution to the subtour LP; see Figure 1. Boyd and Carr hav e shown that the conjecture is true if the solution to the subtour LP has a very sp ecial structure: namely , all v ariables x ( e ) ∈ { 0 , 1 2 , 1 } , the cycles formed b y the edges e with x ( e ) = 1 2 all hav e the same o dd size k , and the supp ort is ( k − 1)-edge-connected. 2 In the general case, the only b ound on this ratio that w e kno w of is the Bo yd and Carr b ound on the in tegrality gap of 2-matchings; since the constraints of the subtour LP are a sup erset of the fractional 2-matc hing constrain ts, this implies the ratio is at most 4 3 . The work of Go emans [11] has some b earing on this conjecture. He studies the following linear 1 W e note that what we refer to here as 2-matc hings, are also sometimes called 2-factors. 2 In fact, they show in this case the optimal 2-matching has cost at most 3 k +1 3 k times the subtour LP . 2 program whic h is essentially same as the subtour LP in the case edge costs ob ey triangle inequalit y: Min X e ∈ E c ( e ) x ( e ) ( S U B T 0 ) sub ject to: X e ∈ δ ( S ) x ( e ) ≥ 2 , ∀ S ⊂ V , S 6 = ∅ , (4) x ( e ) ≥ 0 , ∀ e ∈ E . (5) Go emans sho ws (among other things) that adding comb inequalities to this LP can increase the LP v alue by at most 10 9 ; more precisely , he shows that if x is a feasible solution to ( S U B T 0 ), then 10 9 x is feasible for the LP obtained by adding com b inequalities to ( S U B T 0 ). It is kno wn that adding a subset of the comb inequalities to the degree constraints (1) and bounds (3) giv es the 2-matc hing p olytope. This would imply the Boyd-Carr conjecture if it were known that there is an optimal solution that ob eys the degree constrain ts when the com b inequalities are added to ( S U B T 0 ); as men tioned ab o v e, it can b e shown that there is an optimal solution for ( S U B T 0 ) that ob eys the degree constraints when the edge costs ob ey the triangle inequalit y . But w e do not kno w whether there is an optimal solution that obeys the degree constraints if the comb inequalities are added. 3 The contribution of this paper is to improv e our state of knowledge for the subtour LP by pro ving Conjecture 1. W e start b y sho wing that in some cases the cost of an optimal 2-matching is at most 10 9 the cost of a fractional 2-matc hing, which is a stronger statement than Conjecture 1; in particular, w e sho w this is true whenev er the supp ort of the fractional 2-matching has no cut edge. The example in Figure 1 shows that the ratio can b e at least 10 9 in such cases, so this result is tight. As the first step in this pro of, we giv e a simplification of the Boyd and Carr result b ounding the in tegrality gap for 2-matc hings b y 4 3 . In the case that the supp ort of an optimal fractional 2-matc hing has no cut edge, the pro of b ecomes quite simple. The p erfect matching p olytope plays a crucial role in the pro of: w e use the matc hing edges to show us which edges to remov e from the solution in addition to sho wing us which edges to add. W e note that this idea w as indep enden tly developed in the recen t work of M¨ omk e and Svensson, but also previously app eared in the reduction of the 2-matc hing p olytop e to the matching polytop e; see, for instance, Schrijv er [17, Section 30.7]. W e also use a notion from Boyd and Carr [4] of a gr aphic al 2-matc hing: in a graphical 2-matching, eac h v ertex has degree either 2 or 4, each edge has 0, 1, or 2 copies, and each component has size at least three. Giv en the triangle inequality , we can shortcut any graphical 2-matc hing to a 2-matching of no greater cost. T o obtain our pro of of the Boyd-Carr conjecture, we giv e a p olyhedral formulation of the graphical 2-matching problem, and use it to pro ve Conjecture 1. If x is a feasible solution for the subtour LP , then, roughly sp eaking, we sho w that 10 9 x is feasible for the graphical 2-matching p olytope. Our previous results give us intuition for the precise mapping of v ariables that we need. Using the graphical 2-matc hing p olytop e allows us to ov ercome the issues with the degree constrain ts faced in trying to use Go emans’ results. All the results ab o v e can b e made algorithmic and hav e p olynomial-time algorithms, though w e do not explicitly determine running times. W e conclude by p osing a new conjecture, namely that the worst-case in tegrality gap is ac hieved for solutions to the subtour LP that are fractional 2-matchings (that is, for instances suc h that 3 T o quote Goemans [11, p. 348]: “One migh t wonder whether the worst-case improv emen ts remain unc hanged when one adds the degree constrain ts x ( δ { i } ) = 2 for all i ∈ V and restricts one’s attention to cost functions satisfying the triangle inequalit y . W e b eliev e so but hav e b een unable to pro ve it. The result w ould follow immediately if one could prov e that the degree constraints nev er affect the v alue of the relaxation when the cost function satisfies the triangle inequality .” 3 adding the subtour constraints to the degree constraints and the b ounds on the v ariables do es not c hange the ob jective function v alue). In a companion pap er, Qian, Sc halek amp, Williamson, and v an Zuylen [16] show that the pro of of the Boyd-Carr conjecture can b e used to help b ound the in tegrality gap of the subtour LP for the 1,2-TSP . They show that the gap is at most 106 81 ≈ 1 . 3086 < 4 3 . They also give a pro of that the cost of the optimal 2-matching is at most 10 9 times the cost of a fractional 2-matc hing in the case that c ( i, j ) ∈ { 1 , 2 } , whic h giv es an alternate pro of of the Boyd-Carr conjecture in this case. Our pap er is structured as follo ws. W e introduce basic terms and notation in Section 2. In Section 3, we rederiv e the Bo yd-Carr in tegralit y gap for 2-matchings, and sho w that the gap is at most 10 9 in the case the fractional 2-matching has no cut edge. In Section 4, w e giv e the polytop e for graphical 2-matchings and show ho w to use it to prov e the Boyd-Carr conjecture. Finally , we close with our new conjecture in Section 5. 2 Preliminaries W e will w ork extensively with fractional 2-matc hings; that is, optimal solutions x to the LP ( S U B T ) with only constrain ts (1) and (3). F or con venience we will abbreviate “fractional 2-matc hing” b y F2M and “2-matching” b y 2M. F2Ms hav e the following w ell-known structure (attributed to Balinski [1]). Eac h connected comp onen t of the supp ort graph (that is, the edges e for whic h x ( e ) > 0) is either a cycle on at least three v ertices with x ( e ) = 1 for all edges e in the cycle, or consists of o dd-sized cycles with x ( e ) = 1 2 for all edges e in the cycle connected by paths of edges e with x ( e ) = 1 for eac h edge e in the path (the center figure in Figure 1 is an example). W e call the former comp onen ts inte ger c omp onents and the latter fr actional c omp onents . Many of our results fo cus on transforming an F2M into a 2M, in which all comp onen ts are integer. F or that reason, we will often fo cus solely on how to transform the fractional comp onen ts into integer comp onen ts. W e then call the edges of fractional comp onents for which x ( e ) = 1 2 cycle e dges and the edges for which x ( e ) = 1 p ath e dges . Note that removing a cycle edge can never disconnect a fractional comp onen t. If remo ving a path edge disconnects a fractional comp onen t, w e call it a cut e dge . The asso ciated path of the path edge w e will call a cut p ath , since ev ery edge in it will b e a cut edge. W e will say that a fractional 2-matc hing is c onne cte d if it has a single comp onen t. W e will use a concept introduced by Bo yd and Carr [4] of a gr aphic al 2-matching (G2M). As stated ab ov e, in a graphical 2-matching, each vertex has degree either 2 or 4, eac h edge has 0, 1, or 2 copies, and each comp onen t has size at least three. Given the triangle inequality , we can shortcut an y G2M to a 2M of no greater cost. Our techniques for transforming an F2M to a 2M actually find G2Ms. W e will often need to find minimum-cost p erfect matchings. By a result of Edmonds [9], the p erfect matching p olytope is defined by the follo wing linear program ( M ): Min X e ∈ E c ( e ) x ( e ) ( M ) sub ject to: X e ∈ δ ( i ) x ( e ) = 1 , ∀ i ∈ V , (6) X e ∈ δ ( S ) x ( e ) ≥ 1 , ∀ S ⊂ V , | S | o dd , (7) x ( e ) ≥ 0 , ∀ e ∈ E . (8) 4 3 2-matc hing In tegralit y Gaps In this section, we b ound the cost of a G2M in terms of an F2M via combinatorial metho ds. W e start by giving a pro of of a result of Bo yd and Carr [4] that there is a G2M of cost at most 4 3 the cost of an F2M. Our pro of is somewhat simpler than theirs, but more imp ortan tly , it introduces the main ideas that we will need to obtain other results. W e then show that if the F2M has no cut edges, w e can improv e the b ound from 4 3 to 10 9 . The main idea of this section is that given an F2M, w e define a matc hing problem and compute a p erfect matc hing. The p erfect matc hing tells us how to mo dify the fractional comp onen ts b y either duplicating or removing edges so that we obtain a G2M. W e then relate the cost of the p erfect matc hing found to the F2M b y providing a feasible solution to the p erfect matching LP ( M ). W e will need the follo wing result of Naddef and Pulleyblank [15] ; w e give the pro of since we will use some of its ideas later on. Lemma 3.1 (Naddef and Pulleyblank [15]) L et G b e a cubic, 2-e dge-c onne cte d gr aph with e dge c osts c ( e ) for al l e ∈ E . Then ther e exists a p erfe ct matching in G of c ost at most 1 3 P e ∈ E c ( e ) . Pr o of : The main idea is to show that x ( e ) = 1 3 is a feasible solution to the matching polytop e ( M ). The lemma then follows from the fact that ( M ) has integer extreme p oin ts. Since G is cubic, | V | m ust b e even, and P e ∈ δ ( i ) x ( e ) = 1. Now consider any S ⊂ V with | S | odd. Because G is cubic, it m ust be that | δ ( S ) | is o dd, and since G is 2-edge-connected, | δ ( S ) | ≥ 2. Therefore | δ ( S ) | ≥ 3, and P e ∈ δ ( S ) x ( e ) ≥ 1. Theorem 3.2 Ther e exists a G2M of c ost at most 4 3 times the c ost of an F2M if the F2M has no cut e dge. Pr o of : As describ ed ab o ve, it is sufficient to fo cus on a single fractional comp onen t of the F2M. Let G b e the support graph of this component. T o find the G2M, w e find a minimum-cost p erfect matching on the graph G 0 w e obtain b y replacing each path in G by a single edge, which w e will call (at the risk of some confusion) a path edge. W e set the cost of this edge to b e the cost of the path in G , and we set the cost of a cycle edge in G 0 to the ne gative of the cost of the cycle edge in G . Note that G 0 is cubic and 2-edge-connected b ecause the supp ort graph G of the F2M has no cut edge. Giv en a minim um-cost perfect matc hing in G 0 , w e construct a G2M in G b y first including all paths from G . If a path edge is in the matching in G 0 , we double the path in G . If a cycle edge is not in the matc hing in G 0 , then w e include the cycle edge in the G2M in G , otherwise we omit the cycle edge. W e first show that this indeed defines a G2M: for each vertex, the degree is four if the p erfect matc hing contains the path edge inciden t on the v ertex (since in that case, the tw o cycle edges on the vertex cannot b e in the p erfect matching, and hence b oth are added to the G2M together with t wo copies of the path), and it is tw o otherwise (since one cycle edge is in the p erfect matching and hence only the other cycle edge and one cop y of the path are added to the graphical 2-matching). Note that an y connected comp onen t indeed has at least three no des, since for an y doubled path, w e also tak e the four cycle edges inciden t on the endp oin ts. W e let C denote the sum of the costs of the cycle edges, and P the cost of the paths. Note that the cost of the F2M solution is 1 2 C + P . The cost of the G2M is equal to the cost of all edges in the supp ort graph ( P + C ) plus the cost of the p erfect matching. Because G 0 is cubic and 2-edge-connected, we can inv oke Lemma 3.1 to sho w that the p erfect matc hing has cost at most a 5 pattern 1 pattern 2 pattern 3 Figure 2: Illustrations of patterns for ` = 9. third the cost of the edges in G 0 , or at most 1 3 P − 1 3 C . Hence the cost of the G2M is at most P + C + 1 3 P − 1 3 C = 4 3 P + 2 3 C = 4 3  P + 1 2 C  , or at most 4 3 the cost of the F2M solution, as claimed. The idea of using edges from a p erfect matching to decide which edges to include in a matching and which edges to remo ve has also b een used recently by M¨ omk e and Sv ensson [14]. W e now mo dify the pro of of the theorem abov e so that the result extends to the case in which the F2M has cut edges. Theorem 3.3 (Bo yd and Carr [4]) Ther e exists a G2M of c ost at most 4 3 times the c ost of an F2M. Pr o of : As describ ed ab o ve, it is sufficient to fo cus on a single fractional comp onen t of the F2M, and we let G b e the supp ort graph of this comp onen t. W e once again create a new graph G 0 from G , so that we can later define a matching problem in G 0 . The matching will again sho w us how to create a G2M in G . W e extend the previous construction to deal with the case when the supp ort graph has cut paths. W e introduce a gadget in G 0 for eac h cut path in G , which replaces the cut path and its tw o endpoints. The other paths in G are again replaced by single edges in G 0 of cost equal to the cost of the path. Each cycle edge in G is also in G 0 with cost equal to the negative of its cost in G . T o introduce the cut-path gadget, we b egin b y using an idea of Boyd and Carr [4]; namely , that w e only need to consider three p atterns to get an almost feasible graphical 2-matching on the cut path, when w e allow ourselves to increase the cost b y a third compared to the F2M. Suppose the cut path has ` edges and ` + 1 no des, and let k = b `/ 3 c . W e can remo ve ev ery third edge, double the remaining edges to obtain groups of no des that are 2-edge-connected, where we get k groups of three no des that are G2M comp onen ts, plus one group of ` − 3 k ∈ { 0 , 1 , 2 } no des. Alternatively , w e could remov e every third edge, starting from the first edge and double the remaining edges, in whic h case the first group has one no de, the next k or k − 1 groups ha ve three nodes and the last group again has one or tw o no des. The final pattern remo ves ev ery third edge, starting from the second edge, so that the first group has t wo no des, the next k or k − 1 groups ha ve three no des, and, again, the last group has one or tw o no des. Figure 2 illustrates the three patterns for ` = 9. T o get a G2M that contains a certain pattern, w e will ensure that if a group has size less than three, the G2M will include the tw o cycle edges inciden t on the first no de (if the group is at the start of the pattern) or last node (if the group is at the end of the pattern). W e remark that there is exactly one pattern that starts with a group of size one, tw o and three, and hence t wo patterns need the G2M to include t wo cycle edges inciden t on the first no de of the cut path. On the other hand, there is also exactly one pattern that ends with a group of size one, 6 pattern 3 pattern 1 pattern 2 Figure 3: Pattern gadget for ` = 9. t wo and three (the length of the cut path determines which of the three patterns ends with a group of size three: it is the second pattern if ` (mod 3) = 0, the third pattern if ` (mo d 3) = 1 and the first pattern if ` (mo d 3) = 2), and hence there are also tw o patterns that need the G2M to include the tw o cycle edges incident on the last no de of the cut path. W e are now ready to define the cut-path gadget. W e replace eac h endp oint of the cut path in G b y a path of length t wo in G 0 ; eac h of these new edges will ha ve cost 0. Eac h no de on the path will b e connected to a p attern e dge corresp onding to one of the three patterns. The middle no de is connected to the pattern edge corresp onding to the pattern which do es not need tw o cycle edges incident on the endp oin t of the cut path (i.e. the pattern for whic h the group containing the endp oin t has size three). W e set the cost of a pattern edge to the cost of the edges in the corresp onding pattern. See Figure 3 for an illustration of the gadget when ` = 9. If w e replace eac h cut path in G by a cut-path gadget in G 0 , once again G 0 will b e a cubic graph. It is not hard to c hec k that G 0 is also 2-edge-connected because we ha v e replaced the cut path in G with three pattern edges crossing the cut in G 0 . W e argue that there is a minim um-cost p erfect matching that uses exactly one edge from each cut-path gadget. Note that the fact that we replace only the cut paths in G b y a cut gadget in G 0 means that a perfect matc hing in G 0 con tains an o dd n umber of pattern edges in a gadget. If it contains three pattern edges, then w e could find a matching of no greater cost b y choosing only one pattern edge, namely the pattern edge that is not inciden t on the middle no de for the either one of its endp oints. Note that we can add tw o edges of cost 0 that connect the four no des incident on the other t wo pattern edges, to again hav e a perfect matc hing without increasing the cost. No w we sho w how to obtain a G2M in G from the minimum-cost perfect matching in G 0 . In the G2M w e include all edges from G that are in paths which are not cut paths, the cycle edges in G whic h are not chosen by the p erfect matching, duplicates of edges in paths in G that are chosen by the p erfect matching, and the edges in a pattern if the corresp onding pattern edge is in the p erfect matc hing. W e argue that this set of edges is a G2M in G . Note that if the p erfect matching contains only the pattern edge incident on the middle no de, then the tw o cycle edges that are adjacen t to the gadget are also in the matching. Hence the corresp onding endp oin t in G of the cut path has no cycle edges inciden t on it in the G2M, but since the pattern edge is incident on the middle node, the corresponding pattern ensures that the node has degree tw o and is in a connected comp onen t of size three. If the perfect matching con tains the pattern edge inciden t on a no de other than the middle no de, then neither of the t wo cycle edges that are adjacent to the gadget in G 0 are in the p erfect matching. Hence the corresp onding endp oin t of the cut path in G has b oth of these cycle edges incident on it in the G2M, and zero or tw o edges from the pattern corresponding to the c hosen pattern edge. Hence the no de has degree tw o or four and it is in a connected comp onen t of size at least three. As b efore, b ecause G 0 is cubic and 2-edge-connected, we can apply Lemma 3.1 to b ound the cost of the p erfect matc hing in G 0 . Let P 1 b e the cost of the paths in G that are not cut paths, and P 2 the cost of the cut paths in G , so that the cost of the F2M is P 1 + P 2 + 1 2 C . Note that the 7 cost of the three pattern edges in the gadget corresp onding to a cut path sums up to four times the cost of the cut path. Thus the total cost of the edges in G 0 is P 1 + 4 P 2 − C . By Lemma 3.1, the cost of the p erfect matching in G 0 is at most 1 3 P 1 + 4 3 P 2 − 1 3 C . The cost of the G2M corresp onding to the minimum-cost p erfect matc hing is therefore at most P 1 + 1 3 P 1 + 4 3 P 2 + C − 1 3 C = 4 3 P + 2 3 C = 4 3  P + 1 2 C  as claimed. W e no w show ho w to use the ideas b ehind the cut-path gadget to obtain a b etter G2M if no cut paths exist. Theorem 3.4 If an F2M has no cut e dge, then ther e exists a G2M of c ost at most 10 9 times the c ost of the F2M. Pr o of : Once again w e define a new graph G 0 from the supp ort graph G of a fractional component of the optimal F2M. Each cycle edge in G is in G 0 with cost that is the negative of its cost in G . Eac h path in G and its tw o endp oin ts are replaced by the cut-path gadget used in the pro of of Theorem 3.3. The costs of the pattern edges in G 0 are slightly different than in the previous pro of: w e subtract the cost of the original path from the cost of each pattern edge in its gadget. In other w ords, the cost of a pattern edge in G 0 is obtained by adding once the cost of the edges that app ear t wice in the pattern and subtracting the cost of the edges that do not app ear in the pattern. Note that the sum of the costs of the three pattern edges in G 0 is equal to the cost of the original path in G . Also, note that the sum of the costs of an y t w o pattern edges in G 0 is nonnegative: an edge on the path contributes its cost either p ositiv ely to one pattern and negatively to the other, or p ositiv ely to b oth patterns. W e first argue that there is a minimum-cost p erfect matching that chooses either zero or one pattern edge in eac h cut-path gadget. Supp ose the p erfect matching contains tw o pattern edges in a gadget. Note that on b oth sides of the gadget these pattern edges must be incident on the middle no de, otherwise some middle no de is not matched. Hence the four endp oin ts of the tw o pattern edges are connected in G 0 b y tw o edges of cost zero. By the observ ation ab o v e, the cost of the tw o pattern edges is nonnegativ e, and so we can remov e the t w o pattern edges from the matching and add the tw o edges of cost zero without increasing the cost of the matching. By the same argument, w e can handle the case that the p erfect matching contains three pattern edges from a gadget by c ho osing the pattern edge that is not incident on the middle no de on b oth sides of the gadget, and replacing the other tw o pattern edges in the matc hing b y the cost zero edges that connect their endp oin ts. Therefore, w e can assume the p erfect matching chooses either zero or one pattern edge in a gadget. If it chooses zero pattern edges, then w e add the path from G to the G2M. Otherwise, the pattern corresp onding to the c hosen pattern edge is added to the G2M. W e also add the cycle edges to the G2M corresp onding to the cycle edges that are not in the perfect matc hing. By almost the same arguments as b efore, the solution constructed is indeed a G2M. The only case not cov ered by previous arguments is the case in which zero pattern edges are chosen in G 0 . Then it must b e the case that one of the tw o cycle edges is c hosen in G 0 and the other is not, so that one of the tw o cycle edges is included in the G2M and the other not. Since we include the path from G in the G2M if no pattern edges are chosen, the endp oin t of the path will ha v e degree t wo. T o argue ab out the cost of the minimum-cost perfect matching in G 0 , we create a feasible solution for the matching linear program ( M ). T o do this, for eac h pattern edge e , we set x ( e ) = 1 9 , 8 and for every other edge e 0 , w e set x ( e 0 ) = 4 9 . W e will show this is a feasible solution in a momen t. Let P b e the cost of the path edges in the F2M, and C the cost of the cycle edges, so that the F2M has cost P + 1 2 C . Since the sum of the cost of the pattern edges in a gadget is equal to the cost of the path, the cost of this solution for ( M ) is 1 9 P − 4 9 C , and there exists a p erfect matching of cost at most this m uch. Thus the cost of the G2M is at most P + 1 9 P + C − 4 9 C = 10 9 P + 5 9 C = 10 9  P + 1 2 C  , as claimed. T o see that x is a feasible solution for ( M ), consider any cut such that the num b er of no des on each side of the cut is o dd. If there exists a cycle from the F2M such that not all nodes in the gadgets for the no des in the cycle are on the same side of the cut, then there are tw o edges crossing the cut with v alue 4 9 . Since G 0 is cubic, if the cut has o dd size, then the total n umber of edges crossing the cut is o dd, and there m ust be at least one more edge in the cut with v alue at least 1 9 . Hence the total v alue on the edges crossing the cut is at least one. F or an y other cut, since there is no cut path in G , there are at least three gadgets crossing the cut in G 0 . Since eac h gadget con tains three pattern edges, the v alue of the edges crossing the cut is again at least one. 4 A P olyhedral Pro of of the Bo yd-Carr Conjecture W e will generalize the result in Theorem 3.4 and sho w that the ratio betw een the cost of the optimal 2-matc hing and the subtour LP is at most 10 9 . In the combinatorial proofs of the previous section, we hea vily used the fact that F2Ms ha ve a nice simple structure, and, unfortunately , this do es not hold for the subtour LP solution. W e therefore turn to a p olyhedral rather than a combinatorial pro of. W e derive a polyhedral description for graphical 2-matc hings, and we then use this description to construct a feasible (fractional) G2M solution from an y solution to the subtour LP of cost not more than 10 9 times the v alue of the subtour LP . The manner in which the feasible G2M solution is defined based on a solution to ( S U B T ) is a generalization of the proof of Theorem 3.4. W e start b y giving a p olyhedral description of a generalization of 2-matc hing, where the no de set consists of “mandatory no des” ( V man ) and “optional no des” ( V opt ). The former need to hav e degree 2 in the solution, whereas the latter can ha ve degree 0 or 2. W e will refer to this problem as the 2-Matching with Optional Nodes Problem (2MO). Theorem 4.1 L et G = ( V man ∪ V opt , E ) b e a 2MO instanc e. The c onvex hul l of inte ger 2MO solutions is given by the fol lowing p olytop e: X e ∈ δ ( i ) y ( e ) = 2 , ∀ i ∈ V man , (9) X e ∈ δ ( i ) y ( e ) ≤ 2 , ∀ i ∈ V opt , (10) X e ∈ δ ( S ) \ F y ( e ) + X e ∈ F (1 − y ( e )) ≥ 1 , ∀ S ⊆ V , F ⊆ δ ( S ) , F matching , | F | o dd, (11) 0 ≤ y ( e ) ≤ 1 , ∀ e ∈ E . (12) The pro of of Theorem 4.1 is similar to the pro of of the p olyhedral description of the 2-matching p olytope (Theorem 30.8) in Schrijv er [17], and is deferred to App endix A. 9 Recall the definition of a graphical 2-matc hing (G2M): (i) each vertex has degree either 2 or 4, (ii) each edge has 0, 1, or 2 copies, and (iii) each comp onen t has size at least three. W e will (for the moment) relax the second condition so that each edge has at most 3 copies. Lemma 4.2 We c an r e duc e a G2M instanc e G = ( V , E ) to a 2MO instanc e G 0 = ( V 0 , E 0 ) as fol lows: L et V 0 man = { i m : i ∈ V } , V 0 opt = { i o : i ∈ V } , V 0 = V 0 man ∪ V 0 opt , E 0 = { ( i m , j m ) : ( i, j ) ∈ E } ∪ { ( i m , j o ) : ( i, j ) ∈ E } . We add an e dge { i, j } to the (r elaxe d) G2M solution for e ach e dge ( i m , j m ) , ( i o , j m ) and ( i m , j o ) that is in the asso ciate d 2MO solution. Pr o of : Note that condition (i) for no de i directly follows from the degree constraints for no des i m and i o in the reduction. Relaxed condition (ii) follo ws from the fact that for every edge in the G2M instance there are three asso ciated edges in the 2MO instance. Finally , since each no de i m has degree 2 in the 2MO solution, there cannot b e a comp onen t of size 1. Supp ose there there is a comp onen t of size 2. Then this must b e an isolated doubled or quadrupled edge, say ( i, j ), b ecause of the degree constraints. Clearly w e can’t ha ve a quadrupled edge since there are at most three copies of edge ( i, j ) in the 2MO solution. W e also can’t hav e an isolated doubled edge: in order for the edge to b e isolated, w e w ould need ( i m , j m ) and ( i m , j o ) to b e in the 2MO solution. But then j o m ust ha ve degree 2, and its second edge must b e ( j o , k m ) for some k 6 = i, j , since there are no edges ( i o , j o ) or ( j o , j m ) in the 2MO instance. If the edges ha ve nonnegative costs, we may assume with loss of generalit y that eac h edge app ears at most twice in an optimal G2M solution: if any edge app ears three times, we can remov e t wo copies of it without affecting the parity of its endp oin ts, and the cost cannot increase. W e will now use a solution to the subtour LP on G = ( V , E ) to define a feasible solution to the 2MO instance G 0 = ( V 0 , E 0 ) asso ciated with the graphical 2-matc hing problem on G . It will b e instructive to first consider the case when the subtour LP solution x is an F2M with no cut edge. In that case, the pro of of Theorem 3.4 giv es us a wa y to construct a G2M solution. In fact, it allows us to find a probabilit y distribution on G2Ms, suc h that the exp ected cost of the G2M is exactly 10 9 times the cost of the F2M solution. This probability distribution has a n umber of sp ecial prop erties: (i) if a G2M has p ositiv e probabilit y , then eac h doubled edge is a path edge with x -v alue 1, and has exactly one endp oint that has degree 4; (ii) for eac h path edge ( i, j ) with x -v alue 1, the probability that it o ccurs t wice and i has degree 4 is 1 9 , and the exp ected n umber of times ( i, j ) o ccurs is 10 9 . These observ ations give a hin t as to ho w w e should define a 2MO solution based on a subtour LP solution x . W e think of the edge ( i m , j m ) as the first copy of the edge ( i, j ), and ( i m , j o ) as the second copy if j has degree 4, and ( i o , j m ) as the second copy if i has degree 4. Then the probability of ( i m , j o ) and ( i o , j m ) is 1 9 x ( i, j ) if x ( i, j ) = 1, and the probabilit y of ( i m , j m ) is 8 9 x ( i, j ). This interpretation do es not quite work for the cycle edges ( i, j ) with x -v alue 1 2 , since at most one cop y o ccurs in the G2M. A b etter interpretation is that we consider a Eulerian w alk on each comp onen t of the G2M solution, and asso ciate i m with the first time w e enter and leav e no de i , and i o with the second time we en ter and leav e no de i (if i has degree 4). If w e direct the walk in eac h of the tw o p ossible directions with probability 1 2 , then the probabilit y w e use edge ( i m , j m ) is 8 9 x ( i, j ) and the probabilit y w e use edge ( i m , j o ) is 1 9 x ( i, j ). W e argue this as follo ws. F or a path edge ( i, j ) with x ( i, j ) = 1, the probability that w e use edge ( i m , j o ) is 1 9 , since if j has degree 4, we kno w b y the construction that ( i, j ) is a doubled edge, and i has degree 2. Hence, if j has degree 4, then ( i m , j o ) is in the w alk, and the probability that j has degree 4 is 1 9 . A similar argumen t shows that w e use ( i o , j m ) with probability 1 9 . Also, the exp ected num b er of times we use edge ( i, j ) in the G2M is 10 9 , so the probabilit y of using ( i m , j m ) in the w alk must b e 8 9 . 10 F or a cycle edge ( i, j ) with x ( i, j ) = 1 2 , the probability that w e use ( i m , j o ) is 1 9 · 1 2 , since if j has degree 4, then the G2M con tains a doubled path edge ( j, k ) where x ( j, k ) = 1 and k has degree 2. Hence the probability that we use ( i m , j o ) is the probability that the walk is directed in suc h a wa y that we visit i b efore the lo op from j to k and back, and this happ ens with probability 1 2 . Similarly , the probability that we use ( i o , j m ) is 1 18 , and the fact that the exp ected num b er of times w e use an edge with x -v alue 1 2 in the G2M is 5 9 , sho ws that the probabilit y of using ( i m , j m ) in the w alk m ust b e 4 9 . The follo wing lemma states that using the probabilities 8 9 x ( i, j ) and 1 9 x ( i, j ) to define a fractional solution to the 2MO instance corresp onding to the G2M instance G also yields a feasible solution if, rather than an F2M with no cut edge, x is a feasible solution to the subtour LP on G . Lemma 4.3 Given a gr aph G = ( V , E ) , let x b e a fe asible solution to the subtour LP for G . Then the fol lowing solution is a fe asible solution to the 2MO instanc e G 0 = ( V 0 , E 0 ) asso ciate d with the gr aphic al 2-matching instanc e given by G for α = 1 9 : y ( i m , j m ) = (1 − α ) x ( i, j ) y ( i m , j o ) = αx ( i, j ) y ( i o , j m ) = αx ( i, j ) for al l ( i, j ) ∈ E . Note that the cost of the constructed G2M solution is exactly 10 9 times the cost of the solution of the subtour LP . Thus our result follo ws immediately from the lemma. Corollary 4.4 Ther e exists a G2M of c ost at most 10 9 times the value of the subtour LP. Pro of of Lemma 4.3 : W e need to sho w that y satisfies the constraints (9)-(12) on G 0 , where G 0 is defined as in Lemma 4.2. Constrain ts (9), (10) and (12) are ob viously met, and w e only need to sho w that constraints (11) are met. T o this end, fix S ⊆ V 0 , F ⊆ δ ( S ) where F is a matc hing and | F | is o dd. W e define z ( e 0 ) = y ( e 0 ) if e 0 ∈ δ ( S ) \ F and z ( e 0 ) = 1 − y ( e 0 ) if e 0 ∈ F . F or simplicit y , for an y set of edges X ⊆ E 0 , we define z ( X ) = P e 0 ∈ X z ( e 0 ). Then we need to sho w that z ( δ ( S )) ≥ 1. First, supp ose S do es not contain any no de i m for an y i ∈ V . F or an y j o ∈ S , we hav e that z ( δ ( S ) ∩ δ ( j o )) = z ( { ( i m , j o ) : i ∈ V } ). Since | F | ≥ 1, there exists some j o ∈ S such that F con tains some edge inciden t on j o , say ( i 0 m , j o ). Then, z ( { ( i m , j o ) : i ∈ V } ) = 1 − α x ( i 0 , j ) + P i ∈ V : i 6 = i 0 αx ( i, j ) = αx ( δ ( j )) + 1 − 2 α x ( i 0 , j ). No w, note that x ( δ ( j )) = 2 and x ( i 0 , j ) ≤ 1, hence z ( δ ( S ) ∩ δ ( j o )) ≥ 1. By symmetry , it remains to consider the case when b oth S and V 0 \ S con tain a no de i m for some i ∈ V . W e consider an edge e = ( i, j ) ∈ G suc h that at least one of the three edges ( i o , j m ) , ( j m , i m ) , ( i m , j o ) crosses the cut S in G 0 . Note that there are 2 3 − 1 = 7 p ossible choices for the edges that cross the cut. W e discern five different types of edges in G for which at least one of the three corresp onding edges crosses the cut (type I I and t yp e V each cov er 2 of the p ossible choices): (I) The edge ( i m , j m ) crosses the cut. (I I) The edges ( i o , j m ) and ( j m , i m ) or the edges ( j m , i m ) and ( i m , j o ) cross the cut. (I II) The edges ( i o , j m ) , ( j m , i m ) and ( i m , j o ) cross the cut. (IV) The edges ( i o , j m ) , ( i m , j o ) cross the cut. (V) The edge ( i o , j m ) or the edge ( i m , j o ) crosses the cut. 11 i o i m j m j o (a) Type I. i o i m j m j o (b) Type II. i o i m j m j o (c) Type II I. i o i m j m j o (d) Type IV. i o i m j m j o (e) Type V. Figure 4: Illustrations of the five types of cuts of the edges in the reduction. The y -v alue on the top and b ottom edge is αx ( i, j ) and the y -v alue on the middle edge is (1 − α ) x ( i, j ). Figure 4 illustrates the five types. W e use the notation i ∗ to denote either i m or i o , and we will say an edge e 0 = ( i ∗ , j ∗ ) ∈ G 0 is in a gadget of t yp e I, I I, . . . , V, if the edge ( i, j ) ∈ G is an edge of that t yp e. W e now consider three different cases, dep ending on the set F . Claim 4.5 If F c ontains an e dge in a gadget of typ e IV or V, then z ( δ ( S )) ≥ 1 . Pr o of : Let e 0 ∈ F be con tained in a gadget of type IV or V. Note that e 0 has one endp oin t in V 0 man and one endp oin t in V 0 opt . Let e 0 = ( i o , j m ). Since ( j m , i m ) do es not cross the cut, i o and i m are on differen t sides of the cut. Hence, the paths { ( i o , j 0 m ) , ( j 0 m , i m ) } cross the cut S for every j 0 ∈ V . Each of these paths th us con tribute at least αx ( i, j 0 ) to z ( δ ( S )) for j 0 6 = j . Also, since e 0 = ( i o , j m ) ∈ F , z ( e 0 ) = 1 − αx ( i, j ). W e th us get that z ( δ ( S )) ≥ P j 0 6 = j αx ( i, j 0 ) + 1 − αx ( i, j ) = P j 0 αx ( i, j 0 ) + 1 − 2 αx ( i, j ) ≥ 1, where the last inequality follo ws since P j 0 x ( i, j 0 ) = 2 by the degree constrain ts, and x ( i, j ) ≤ 1. F or the remaining cases, we asso ciate a cut R in the graph G with the cut S in G 0 : let R = { i ∈ V : i m ∈ S } . Note that R, V \ R are not empt y . Note that if e is of type I, II, or II I, then the edge ( i m , j m ) crosses the cut, and hence, the edge e crosses the cut R in G . In the remainder of this proof, w e will write z ( δ ( S )) = y ( δ ( S )) + | F | − 2 y ( F ), and w e will give a low er b ound on y ( δ ( S )) to show that z ( δ ( S )) ≥ 1. In order to giv e a lo wer b ound on y ( δ ( S )), we need to use the fact that x satisfies degree constraints for each no de, and that x ( δ ( R )) ≥ 2. It will therefore b e con venien t to relate the contribution to y ( δ ( S )) of the three edges ( i o , j m ), ( j m , i m ), and ( i m , j o ) to the edge ( i, j ) ∈ G , if ( i, j ) ∈ δ ( R ), but also to the no des i and j for certain types of no des i, j ∈ V . In particular, we say a no de i ∈ V is a lonely no de if |{ i m , i o } ∩ S | = 1. W e let L b e the set of lonely no des. W e assign each lonely no de i an amount of αx ( i, j ), for each edge ( i, j ) of t yp e I, I I, . . . , V. Note that for eac h lonely no de i , the paths { ( i o , j m ) , ( j m , i m ) } cross the cut for all j ∈ V , 12 and hence, each lonely no de gets assigned α P j x ( i, j ), which by the degree constrain ts is equal to 2 α . (I) F or an edge ( i, j ) of type I, the total contribution of the three edges ( i o , j m ) , ( j m , i m ) , ( i m , j o ) to y ( δ ( S )) is (1 − α ) x ( i, j ). Note that b oth i and j are lonely no des. W e assign (1 − 3 α ) x ( i, j ) to the edge ( i, j ), and α x ( i, j ) eac h to nodes i and j . (I I) F or an edge ( i, j ) of type I I, the total contribution of the three edges ( i o , j m ) , ( j m , i m ) , ( i m , j o ) to y ( δ ( S )) is x ( i, j ). Note that only one of i, j is a lonely node, and w e therefore assign (1 − α ) x ( i, j ) to the edge ( i, j ), and αx ( i, j ) to the lonely no de among i, j . (I II) F or an edge ( i, j ) of type II I, the total con tribution of the three edges ( i o , j m ) , ( j m , i m ) , ( i m , j o ) to y ( δ ( S )) is (1+ α ) x ( i, j ), and neither i nor j is a lonely no de. W e therefore assign (1+ α ) x ( i, j ) to the edge ( i, j ). (IV) F or an edge ( i, j ) of type IV, the total con tribution of the three edges ( i o , j m ) , ( j m , i m ) , ( i m , j o ) to y ( δ ( S )) is 2 αx ( i, j ). Since ( i, j ) 6∈ δ ( R ) and b oth i and j are lonely nodes, we assign 0 to ( i, j ) and αx ( i, j ) eac h to i and j . (V) F or an edge ( i, j ) of type V, the total con tribution of the three edges ( i o , j m ) , ( j m , i m ) , ( i m , j o ) to y ( δ ( S )) is αx ( i, j ). Since ( i, j ) 6∈ δ ( R ) and only one of i and j is a lonely no de, w e can assign 0 to ( i, j ) and α x ( i, j ) to the lonely no de. By the argumen t ab o ve, we hav e assigned 2 α to each lonely no de. W e now show how this fact, com bined with the fact that x ( δ ( R )) ≥ 2 and the assignmen t of v alues to the edges in δ ( R ), allows us to conclude that z ( δ ( S )) ≥ 1. Claim 4.6 If | F | = 1 , then z ( δ ( S )) ≥ 1 . Pr o of : Let F = { e 0 } . Let ( i, j ) b e suc h that e 0 = ( i ∗ , j ∗ ). W e will show that z ( δ ( S )) = y ( δ ( S )) + 1 − 2 y ( e 0 ) ≥ 1. Note that 2 y ( e 0 ) ≤ 2(1 − α ) x ( i, j ) ≤ 2 − 2 α , so it is enough to sho w that y ( δ ( S )) ≥ 2 − 2 α . First, supp ose that | L | ≤ 1. Then, there is no edge of t yp e I, so to each edge e ∈ δ ( R ), we assigned at least (1 − α ) x ( e ). Hence, y ( δ ( S )) ≥ (1 − α ) x ( δ ( R )) ≥ 2 − 2 α , since x ( δ ( R )) ≥ 2 by the subtour elimination constraints. If | L | ≥ 2, then we assigned 2 α to each no de in L , giving at least 4 α . W e assigned at least (1 − 3 α ) x ( e ) to each edge e ∈ δ ( R ). Therefore, y ( δ ( S )) ≥ 4 α + (1 − 3 α ) x ( δ ( R )) ≥ 2 − 2 α , where w e again use that x ( δ ( R )) ≥ 2. Claim 4.7 If | F | ≥ 3 , then z ( δ ( S )) ≥ 1 . Pr o of : By Claim 4.5, we may assume that all edges in F are contained in a gadget of type I, I I or I II, and hence, that the corresp onding edges in e ∈ G are in δ ( R ). Let E 1 , E 2 , E 3 b e the edges in δ ( R ) of t yp e I, I I and I I I, resp ectiv ely , for whic h the gadget con tains one or more edges in F . Note that a lonely no de i can b e inciden t on at most one edge in E 1 ∪ E 2 ∪ E 3 : Only the edges ( i, j ) ∈ E 1 ∪ E 2 can b e incident on a lonely no de i , and in the first case, ( i m , j m ) must b e in F , and in the second case, either ( i m , j o ) or ( i m , j m ) is in F , since these are the only edges that cross the cut for these t yp es. Now, since F is a matc hing, it can hav e at most one edge inciden t on i m and hence i can b e incident on at most one edge in E 1 ∪ E 2 ∪ E 3 . W e therefore ha ve that y ( δ ( S )) ≥ (1 − 3 α ) x ( E 1 ) + 4 α | E 1 | + (1 − α ) x ( E 2 ) + 2 α | E 2 | + (1 + α ) x ( E 3 ) . 13 On the other hand, since F is a matching, only the gadgets for edges of t yp e I I I can contain t wo edges in F . Hence, | F | = | E 1 | + | E 2 | + (1 + β ) | E 3 | , where β is the fraction of edges in E 3 for whic h t w o edges in the corresp onding gadget are con tained in F . Also, y ( F ) ≤ (1 − α ) ( x ( E 1 ) + x ( E 2 ) + x ( E 3 )), since y (( i ∗ , j ∗ )) ≤ (1 − α ) x ( i, j ), and, if t w o edges in the gadget for e ∈ E 3 are con tained in F , then these edges b oth hav e y -v alue αx ( e ), and since α ≤ 1 3 , 2 αx ( e ) ≤ (1 − α ) x ( e ). Hence, we get that z ( δ ( S )) = y ( δ ( S )) + | F | − 2 y ( F ) ≥ (1 + 4 α ) | E 1 | + ( − 1 − α ) x ( E 1 ) + (1 + 2 α ) | E 2 | + ( − 1 + α ) x ( E 2 ) + | E 3 | + ( − 1 + 3 α ) x ( E 3 ) + β | E 3 | ≥ 3 α ( | E 1 | + | E 2 | + | E 3 | ) + β | E 3 | ≥ 3 α | F | , where the p en ultimate inequalit y follo ws from the fact that x ( E k ) ≤ | E k | and α ≤ 1 3 , and the last inequalit y from the fact that α ≤ 1 3 . Hence, if w e choose α = 1 9 , then z ( δ ( S )) ≥ 1. 5 Conjectures and Conclusions I c onje ctur e that ther e is no [p olynomial-time] algorithm for the tr aveling salesman pr ob- lem. My r e asons ar e the same as for any mathematic al c onje ctur e: (1) It is a le gitimate mathematic al p ossibility, and (2) I do not know. — Edmonds [10] W e conclude our pap er with a conjecture. W e do so in the spirit of Jack Edmonds, quoted ab o v e; w e do not kno w whether the conjecture is true or not, but w e think that ev en a pro of that this conjecture is false w ould be interesting. Our conjecture says that the in tegrality gap (or w orst-case ratio) of the subtour LP is obtained for specific kinds of v ertices of the subtour p olytop e; namely , ones in which the subtour LP solution has no subtour constraint as part of the dual basis, or, restated a differen t w a y , for costs c such that an optimal subtour LP solution for c is the same as an optimal fractional 2-matching for c . Let us call such costs c fr actional 2-matching c osts for the subtour LP . Note that for suc h solutions of the subtour LP , the fractional 2-matc hing will hav e no cut edge. Conjecture 2 The inte gr ality gap for the subtour LP is attaine d for a fr actional 2-matching c ost for the subtour LP. W e could make a similar conjecture for the ratio of the cost of the optimal 2-matc hing to the subtour LP , but by Theorem 3.4 and Corollary 4.4, we already kno w that the conjecture is true. Ho wev er, its truth does not shed any ligh t on the conjecture ab ov e. In a companion pap er, Qian et al. [16] sho w that if an analogous conjecture for edge costs c ( i, j ) ∈ { 1 , 2 } is true, then the integralit y gap for 1,2-TSP is at most 7 6 . They conjecture that the in tegrality gap for the 1,2-TSP is at most 10 9 ; it is known that it can b e no smaller than 10 9 . It w ould b e nice to sho w that if the analogous conjecture is true then the in tegrality gap for 1,2-TSP is at most 10 9 . In terestingly , w e app ear to kno w almost nothing ab out the consequences of Conjecture 2. Ev en for this v ery restricted set of cost functions, we do not kno w a better upp er bound on the in tegrality gap of the subtour LP other than the b ound of 3 2 . Note that the low er b ound of 4 3 is attained for a 14 fractional 2-matching cost. It would b e v ery interesting to prov e that for such costs the integralit y gap is indeed 4 3 . Bo yd and Carr [3] hav e sho wn this for some fractional 2-matching costs in whic h all the cycles of the fractional 2-matching hav e size 3; this result also follo ws from the technique of Theorem 3.2, since the resulting graphical 2-matc hing is Eulerian if all cycles hav e size 3 and the fractional 2-matching has a single comp onent (the graphical 2-matc hing ma y not b e connected if there are cycles of size 5). Ac kno wledgements W e thank Sylvia Bo yd for useful and encouraging discussions; we also thank her for giving us p oin ters on her v arious results. Gyula Pap made some useful suggestions regarding the p olyhedral form ulation of graphical 2-matchings. References [1] M. L. Balinski. Integer programming: Metho ds, uses, computation. Management Scienc e , 12:253–313, 1965. [2] G. Benoit and S. Bo yd. Finding the exact in tegrality gap for small trav eling salesman problems. Mathematics of Op er ations R ese ar ch , 33:921–931, 2008. [3] S. Bo yd and R. Carr. Finding lo w cost TSP and 2-matching solutions us- ing certain half-in teger subtour vertices. T o app ear in Discr ete Optimization . See http://dx.doi.org/10.1016/j.disopt.2011.05.002 . Prior version a v ailable at http://www.site.uottawa.ca/ ∼ sylvia/recentpapers/halftri.pdf . Accessed June 27, 2011. [4] S. Boyd and R. Carr. A new b ound for the ratio b et ween the 2-matching problem and its linear programming relaxation. Mathematic al Pr o gr amming , 86:499–514, 1999. [5] S. Boyd and P . Elliott-Magwoo d. Structure of the extreme p oin ts of the subtour elimina- tion polytop e of the STSP. In S. Iw ata, editor, Combinatorial Optimization and Discr ete A lgorithms , volume B23 of RIMS Kˆ okyˆ ur oku Bessatsu , pages 33–47. Research Institute for Mathematical Sciences, Kyoto Univ ersity , Kyoto, Japan, 2010. [6] S. Boyd, R. Sitters, S. v an der Ster, and L. Stougie. TSP on cubic and sub cubic graphs. In O. G ¨ unl¨ uk and G. J. W o eginger, editors, Inte ger Pr o gr amming and Combinatorial Optimiza- tion, 15th International Confer enc e, IPCO 2011 , num b er 6655 in Lecture Notes in Computer Science, pages 65–77. Springer, Berlin, Germany , 2011. [7] N. Christofides. W orst case analysis of a new heuristic for the tra veling salesman problem. Rep ort 388, Graduate School of Industrial Administration, Carnegie-Mellon Universit y , Pitts- burgh, P A, 1976. [8] G. Dantzig, R. F ulk erson, and S. Johnson. Solution of a large-scale trav eling-salesman problem. Op er ations R ese ar ch , 2:393–410, 1954. [9] J. Edmonds. Maxim um matching and a p olyhedron with (0,1) vertices. J. R es. Nat. Bur. Standar ds Se ct. B , 69B:125–130, 1965. 15 [10] J. Edmonds. Optimum branc hings. Journal of R ese ar ch of the National Bur e au of Standar ds B , 71B:233–240, 1967. [11] M. X. Go emans. W orst-case comparison of v alid inequalities for the TSP. Mathematic al Pr o gr amming , 69:335–349, 1995. [12] M. X. Go emans and D. J. Bertsimas. Surviv able netw orks, linear programming relaxations, and the parsimonious prop ert y . Mathematic al Pr o gr amming , 60:145–166, 1990. [13] D. S. Johnson and L. A. McGeo c h. Exp erimental analysis of heuristics for the STSP. In G. Gutin and A. P . Punnen, editors, The T r aveling Salesman Pr oblem and Its V ariants , pages 369–444. Kluw er Academic Publishers, Dordrect, The Netherlands, 2002. [14] T. M¨ omk e and O. Svensson. Approximating graphic TSP b y matchings. CoRR , abs/1104.3090, 2011. Av ailable at . Accessed Ma y 10, 2011. [15] D. Naddef and W. R. Pulleyblank. Matchings in regular graphs. Discr ete Mathematics , 34:283–291, 1981. [16] J. Qian, F. Sc halek amp, D. P . Williamson, and A. v an Zuylen. On the integralit y gap of the subtour LP for the 1,2-TSP. Manuscript, 2011. [17] A. Schrijv er. Combinatorial Optimization: Polyhe dr a and Efficiency . Springer, Berlin, Ger- man y , 2003. [18] D. B. Shmo ys and D. P . Williamson. Analyzing the Held-Karp TSP bound: A monotonicity prop ert y with application. Information Pr o c essing L etters , 35:281–285, 1990. [19] D. P . Williamson. Analysis of the He ld-Karp heuristic for the tra veling salesman problem. Master’s thesis, MIT, Cambridge, MA, June 1990. Also appears as T ech Report MIT/LCS/TR- 479. [20] L. A. W olsey . Heuristic analysis, linear programming and branc h and b ound. Mathematic al Pr o gr amming Study , 13:121–134, 1980. A P olyhedral description of 2MO W e rep eat Theorem 4.1 for sake of completeness. Theorem A.1 L et G = ( V man ∪ V opt , E ) b e a 2MO instanc e. The c onvex hul l of inte ger 2MO solutions is given by the fol lowing p olytop e: X e ∈ δ ( i ) x ( e ) = 2 , ∀ i ∈ V man (13) X e ∈ δ ( i ) x ( e ) ≤ 2 , ∀ i ∈ V opt (14) X e ∈ δ ( S ) \ F x ( e ) + X e ∈ F (1 − x ( e )) ≥ 1 , ∀ S ⊆ V , F ⊆ δ ( S ) , F matching , | F | o dd, (15) 0 ≤ x ( e ) ≤ 1 , ∀ e ∈ E . (16) 16 Pr o of : The proof that we present here is similar to the pro of of the p olyhedral description of the 2-matc hing p olytope (Theorem 30.8) in Sc hrijver [17]. W e will first show that any 2MO solution is con tained in the p olytop e, and next show that the extreme points of the polytop e coincide with the 2MO solutions. Constrain ts (13), (14) and (16) ob viously hold for a 2MO solution. T o sho w that constrain t (15) is satisfied, we consider tw o cases: (case 1) There is a ¯ e ∈ F with x ( ¯ e ) = 0. This makes the left hand side of constraint (15) at least 1, since x ( e ) ≥ 0 for all e . (case 2) x ( e ) = 1 for all e ∈ F . Since | F | is o dd, and each no de is incident to an ev en n umber of edges in an 2MO solution, it follows that there has to be an edge in the solution in δ ( S ) that is not in F . So the constraint also holds in this case. The p olytope thus con tains all 2MO solutions. W e will now sho w that its extreme p oin ts coincide with 2MO solutions, b y reducing 2MO instances to matching instances, for whic h p erfect matc hings corresp ond to 2MO solutions. W e will show that an y feasible p oint in the 2MO polytop e corresp onds to a feasible p oint in the p erfect matching p olytop e. Because an y p oint in the p erfect matc hing p olytope can b e written as a conv ex combination of p erfect matchings this implies that an y p oint in the 2MO p olytop e can b e written as a con vex combination of 2MO solutions, and therefore all extreme p oin ts of the 2MO polytop e corresp ond to 2MO solutions. Before w e consider the reduction to p erfect matchings, we will first sho w that adding con- strain t (15) for all F ⊆ E of o dd cardinality do es not c hange the 2MO p olytop e. These additional constrain ts will b e conv enien t when sho wing that a feasible p oin t in the 2MO p olytop e is in the p erfect matching p olytope. W e pro v e this by induction on | F | . Consider ¯ S and ¯ F ⊆ δ ( ¯ S ) so that F is not a matching, i.e. | ¯ F ∩ δ ( i ) | ≥ 2 for some i ∈ V . W e consider three cases. • (Case 1) | ¯ F ∩ δ ( i ) | ≥ 3. Then X e ∈ δ ( ¯ S ) \ ¯ F x ( e ) + X e ∈ ¯ F (1 − x ( e )) ≥ X e ∈ ¯ F (1 − x ( e )) ≥ X e ∈ ¯ F ∩ δ ( i ) (1 − x ( e )) ≥ 3 − X e ∈ ¯ F ∩ δ ( i ) x ( e ) ≥ 3 − X e ∈ δ ( i ) x ( e ) ≥ 3 − 2 ≥ 1 . • (Case 2) | ¯ F ∩ δ ( i ) | = 2 and i ∈ ¯ S . Let F 0 = ¯ F \ δ ( i ) and let S 0 = ¯ S \ { i } . Then X e ∈ δ ( ¯ S ) \ ¯ F x ( e ) + X e ∈ ¯ F (1 − x ( e )) ≥ X e ∈ δ ( S 0 ) \ F 0 x ( e ) − X e ∈ δ ( i ) x ( e ) + X e ∈ δ ( i ) ∩ ¯ F x ( e ) + X e ∈ F 0 (1 − x ( e )) + X e ∈ δ ( i ) ∩ ¯ F (1 − x ( e )) = X e ∈ δ ( S 0 ) \ F 0 x ( e ) + X e ∈ F 0 (1 − x ( e )) − X e ∈ δ ( i ) x ( e ) + 2 . By induction and the degree b ound for i , this quan tity is at least 1. • (Case 3) | ¯ F ∩ δ ( i ) | = 2 and i 6∈ ¯ S . Let F 0 = ¯ F \ δ ( i ) as in the previous case, but now let S 0 = ¯ S ∪ { i } . Then the exact same string of inequalities as in the previous case holds. W e no w use the usual reduction from 2-matchings to matchings (see Theorem 30.7 in Schrijv er, the notation of whic h we will also follow): for each no de i in the 2MO, there will b e tw o no des in the matching instance: i 0 and i 00 . F or eac h edge e = ( i, j ) in the 2MO instance, there will b e 17 i ' i " p ei p ej j ' j " Figure 5: Illustration of the reduction from 2MO to matchings. The part of the matc hing instance is drawn which corresp onds to an edge b et ween a mandatory no de i , and an optional node j . t wo no des and five edges in the matching instance: no des p e,i and p e,j , and edges ( i 0 , p e,i ), ( i 00 , p e,i ), ( p e,i , p e,j ), ( j 0 , p e,j ), and ( j 00 , p e,j ). The only difference b et ween the reduction from 2-matc hings to matc hings, and the reduction from 2MO to matchings is that for optional no des we also add an edge b et ween no des i 0 and i 00 . An illustration of the reduction is giv en in Figure 5, where the part of the matching instance is giv en whic h corresponds to an edge betw een a mandatory no de i , and an optional no de j . Giv en a (fractional) solution x to a 2MO instance, w e define a solution y to the corresp onding matc hing instance as follows: y ( i 0 , p e,i ) = y ( i 00 , p e,i ) = 1 2 x ( e ) and y ( p e,i , p e,j ) = 1 − x ( e ) for all e = ( i, j ) ∈ E , and y ( i 0 , i 00 ) = 1 − 1 2 X e ∈ δ ( i ) x ( e ) for all i ∈ V opt . W e will no w sho w that this solution is indeed in the p erfect matching p olytop e giv en b y the constrain ts (6), (7) and (8) of the linear program ( M ) in Section 2 (where the v ariables are here called y instead of x ). F or no des p e,i , the degree b ound constraints (6) follow directly from the definition of y (there are three edges incident on p e,i with y -v alues 1 2 x ( e ), 1 2 x ( e ) and 1 − x ( e ), whic h sum to 1). F or the other no des, constrain t (6) follows directly from the degree b ound constrain ts (13) or (14) in the 2MO instance and the definition of y . Constraints (8) follow directly from constraints (16). W e will no w prov e that constraints (7) also hold for all subsets of no des of o dd cardinality in our reduction. Let S 0 b e such a subset. W e consider four cases. • (Case 1) |{ i 0 , i 00 } ∩ S 0 | = 1 for some i ∈ V . Note that we hav e edges ( i 0 , p e,i ) and ( i 00 , p e,i ) in the reduction b oth of which ha ve y -v alue 1 2 x ( e ), and of which exactly one will b e in δ ( S 0 ). F urthermore, ( i 0 , i 00 ) is in δ ( S 0 ) if i is in V opt . Therefore P e 0 ∈ δ ( S 0 ) y ( e 0 ) ≥ P e ∈ δ ( i ) 1 2 x ( e ) = 1 if i ∈ V man b y the degree b ound (13). Similarly P e 0 ∈ δ ( S 0 ) y ( e 0 ) ≥ P e ∈ δ ( i ) 1 2 x ( e ) + (1 − 1 2 P e ∈ δ ( i ) x ( e )) = 1 if i ∈ V opt b y the degree b ound (14). • (Case 2) F or some e = ( i, j ) ∈ E , p e,i ∈ S 0 , p e,j 6∈ S 0 and { i 0 , i 00 } ∩ S 0 = ∅ . Let p = p e,i . Then P e 0 ∈ δ ( S 0 ) y ( e 0 ) ≥ y ( p, i 0 ) + y ( p, i 00 ) + y ( p, p e,j ) = 1 2 x ( e ) + 1 2 x ( e ) + 1 − x ( e ) = 1. 18 • (Case 3) F or some e = ( i, j ) ∈ E , p e,i ∈ S 0 , p e,j 6∈ S 0 and { j 0 , j 00 } ⊆ S 0 . Let p = p e,j . Then P e 0 ∈ δ ( S 0 ) y ( e 0 ) ≥ y ( p, j 0 ) + y ( p, j 00 ) + y ( p, p e,i ) = 1 2 x ( e ) + 1 2 x ( e ) + 1 − x ( e ) = 1. • (Case 4) W e ma y now assume that S 0 is such that |{ i 0 , i 00 } ∩ S 0 | is even for all i ∈ V , and that { i 0 , i 00 } ∈ S 0 and { j 0 , j 00 } ∩ S 0 = ∅ if p e,i ∈ S 0 and p e,j 6∈ S 0 , b ecause otherwise we are in one of the previous cases. Define ¯ S = { i ∈ V : i 0 ∈ S 0 and i 00 ∈ S 0 } and ¯ F = { e = { i, j } ∈ E : p e,i ∈ S 0 and p e,j 6∈ S 0 } . Note that the previous argument implies that ¯ F ⊆ δ ( ¯ S ). Consider e = ( i, j ) ∈ δ ( ¯ S ) in the 2MO instance, and assume without loss of generality that i ∈ ¯ S . By definition of ¯ S , this means { j 0 , j 00 } ∩ S 0 = ∅ . W e consider e ∈ ¯ F and e 6∈ ¯ F separately . First of all, assume e ∈ ¯ F . Since we are not in the previous cases this means that p e,i ∈ S 0 and p e,j 6∈ S 0 . So for eac h such e in the 2MO instance, w e hav e ( p e,i , p e,j ) ∈ δ ( S 0 ) in the matc hing instance, with an y -v alue of 1 − x ( e ). Second, assume e 6∈ ¯ F . By definition of ¯ F , w e kno w that either p e,i and p e,j are both in S 0 , or both not in S 0 . So for eac h suc h e in the 2MO instance, w e ha ve either { ( i 0 , p e,i ) , ( i 00 , p e,i ) } ⊆ δ ( S 0 ) or { ( j 0 , p e,j ) , ( j 00 , p e,j ) } ⊆ δ ( S 0 ) in the matching instance, each of whic h carry a total y -v alue of x ( e ). W e thus get P e 0 ∈ δ ( S 0 ) y ( e 0 ) ≥ P e ∈ δ ( ¯ S ) \ ¯ F x ( e ) + P e ∈ ¯ F (1 − x ( e )). W e then note that | ¯ F | is equal to the num b er of no des of the type p e,i in S 0 , which implies that the parit y of | ¯ F | and | S 0 | are alwa ys the same, as the other no des in S 0 app ear in pairs. Th us since | S 0 | is o dd, | ¯ F | is o dd, and w e hav e P e 0 ∈ δ ( S 0 ) y ( e 0 ) ≥ P e ∈ δ ( ¯ S ) \ ¯ F x ( e ) + P e ∈ ¯ F (1 − x ( e )) ≥ 1 by the feasibility of x for constrain ts (15). W e conclude the pro of by noting that a p erfect matc hing in the constructed instance corresponds to the 2MO solution consisting of all edges e = ( i, j ) for whic h ( p e,i , p e,j ) is not in the p erfect matc hing solution. 19