A 4/3 -Approximation Algorithm for the Minimum 2 -Edge Connected Multisubgraph Problem in the Half-Integral Case
S.Boyd, J.Cheriyan, R.Cummings, L.Grout, S.Ibrahimpur, Z.Szigeti, L.Wang
aa r X i v : . [ c s . D S ] A ug A 4 / ∗ Sylvia Boyd † Joseph Cheriyan ‡ Robert Cummings ‡ Logan Grout ‡ Sharat Ibrahimpur ‡ Zolt´an Szigeti § Lu Wang ‡ August 11, 2020
Abstract
Given a connected undirected graph G on n vertices, and non-negative edge costs c ,the problem is that of finding a 2-edge connected spanning multisubgraph of G of minimum cost. The natural linear program (LP) for , which coincides with thesubtour LP for the Traveling Salesman Problem on the metric closure of G , gives a lowerbound on the optimal cost. For instances where this LP is optimized by a half-integralsolution x , Carr and Ravi (1998) showed that the integrality gap is at most : theyshow that the vector x dominates a convex combination of incidence vectors of 2-edgeconnected spanning multisubgraphs of G .We present a simpler proof of the result due to Carr and Ravi by applying an exten-sion of Lov´asz’s splitting-off theorem. Our proof naturally leads to a -approximationalgorithm for half-integral instances. Given a half-integral solution x to the LP for ,we give an O ( n )-time algorithm to obtain a 2-edge connected spanning multisubgraphof G whose cost is at most c T x . The 2-edge connected multisubgraph ( ) problem is a fundamental problem in survivablenetwork design where one wants to be resilient against a single edge failure. In this problem,we are given an undirected graph G = ( V , E ) with non-negative edge costs c and we wantto find a 2-edge connected spanning multisubgraph of G of minimum cost. Below we give aninteger linear program for . The variable x e denotes the number of copies of edge e thatare used in a feasible solution. For any S ⊂ V , δ ( S ) := { e = uv ∈ E : u ∈ S, v / ∈ S } denotesthe cut induced by S . For any F ⊆ E and vector x ∈ R E , we use x ( F ) as a shorthand for P e ∈ F x e . Also, for any graph H with edge costs c , we sometimes use c ( H ) as a shorthandfor c ( E ( H )). ∗ A preliminary version of this paper will appear in the Proceedings of APPROX 2020. † [email protected] . School of Electrical Engineering and Computer Science, University of Ottawa, Ottawa,Canada ‡ { jcheriyan,sharat.ibrahimpur } @uwaterloo.ca . Dept. of Combinatorics and Optimization, Universityof Waterloo, Waterloo, Canada § [email protected] . University Grenoble Alpes, CNRS, G-SCOP, Grenoble, France - IP ) min X e ∈ E c e x e (1)subject to x ( δ ( S )) ≥ ∀ ∅ ( S ( V , (2) x e ≥ ∀ e ∈ E, (3) x e integral ∀ e ∈ E. (4)It is easy to see that an optimal solution for never has to use more than two copiesof an edge. As is discussed in [CR98], since we are allowed to use more than one copy of anedge, without loss of generality, we may assume that G is complete by performing the metriccompletion: for each u, v ∈ V we set the new cost of the edge uv to be the shortest pathdistance between u and v in G . In the sequel, we assume that G is a complete graph and thatthe cost function c is metric i.e., c ≥ u, v, w ∈ V , we have c uw ≤ c uv + c vw .The linear relaxation ( - LP ) for is obtained by dropping the integrality con-straints given by (4). By a result due to Goemans and Bertsimas [GB93] called the par-simonious property , adding the constraint x ( δ ( v )) = 2 for each v ∈ V to ( - LP ) doesnot increase the optimal solution value; here, we require the assumption that the costs forma metric. So, the optimal value of ( - LP ) is the same as the optimal value for thewell-known subtour elimination LP ( Subtour - LP ) for the Traveling Salesman Problem ( TSP )defined below. Due to this connection, we often refer to an optimal solution for ( - LP )as an optimal solution to ( Subtour - LP ), and vice versa. Another consequence of the parsimo-nious property is that for graphs with at least 3 vertices, the constraint x e ≤ Subtour - LP ): for any e = uv , we have 2 x e = x ( δ ( u )) + x ( δ ( v )) − x ( δ ( { u, v } )) ≤ Subtour - LP ) min X e ∈ E c e x e (5)subject to x ( δ ( S )) ≥ ∀ ∅ ( S ( V , (6) x ( δ ( v )) = 2 ∀ v ∈ V , (7) x e ≥ ∀ e ∈ E. (8)A long-standing open problem called the “four-thirds conjecture” states that the inte-grality gap of ( Subtour - LP ) is . Besides the importance of in the field of survivablenetwork design, the connection between ( - LP ) and ( Subtour - LP ) has spurred interestin determining the integrality gap for ( - LP ) as a means to gaining useful lower boundson the integrality gap for ( Subtour - LP ). The general version of metric TSP has resisted allattempts at proving an upper bound better than on the integrality gap, so a great deal ofresearch has focused on obtaining improvements for important special cases. In [SWvZ14],the authors conjecture that the integrality gap for ( Subtour - LP ) is achieved on instanceswhere an optimal (fractional) solution to ( Subtour - LP ) is half integral i.e., 2 x e ∈ Z ≥ for all e .We refer to such instances as half integral instances . More than two decades ago, Carr andRavi [CR98] proved that the integrality gap of ( - LP ) is at most in the half-integralcase. They show that x dominates a convex combination of 2-edge connected spanningmultisubgraphs of G . This supports the four-thirds conjecture for TSP since the (integer)optimal value for lower bounds the (integer) optimal value for
TSP . However, theproof of Carr and Ravi does not give a polynomial-time algorithm for . Very recently,2n [KKG20], Karlin, Klein, and Oveis Gharan gave a randomized approximation algorithmfor half-integral instances of
TSP whose (expected) approximation factor is − . -approximation algorithm, albeit randomized, for as well.We note that the result of Carr and Ravi mentioned above does not apply to the strictvariant of (henceforth denoted by ) where we are allowed to pick at most one copyof an edge in G , i.e. where we are considering subgraphs of G rather than multisubgraphs;similarly, our main result does not apply to . Our main contribution is a deterministic approximation algorithm for on half-integralinstances that matches the existence result in [CR98].
Theorem 1.
Let x denote an optimal half-integral solution to an instance ( G, c ) of ( Subtour - LP ) (and ( - LP ) ). There is an O ( | V ( G ) | ) -time algorithm for computing a -edge connectedspanning multisubgraph of G with cost at most c T x . We can strengthen the above result by using the meta-rounding algorithm of Carr andVempala [CV02]. Under some mild assumptions, the meta-algorithm uses an LP-based α -approximation algorithm as a black-box and gives an efficient procedure to obtain a convexcombination of integer solutions that is dominated by αx , where x is a (feasible) fractionalLP solution. We defer our proof of the following result to a subsequent full version of ourpaper.Let G = ( V , E ) be a complete graph on n vertices. Let x ∈ R E ≥ be a fractionalhalf-integral solution to ( Subtour - LP ) (or equivalently, ( - LP )) i.e., x satisfiesthe constraints (6)-(8). In poly ( n )-time, we can obtain 2-edge connected spanningmultisubgraphs H , . . . , H k and nonnegative real numbers µ , . . . , µ k , P ki =1 µ i = 1satisfying P ki =1 µ i χ E ( H i ) ≤ x .Given a half-integral solution x to ( Subtour - LP ) for G , let G = ( V, E ) denote the multi-graph induced by 2 x . Formally, the vertex-set V := V , and for each edge e ∈ E , the edge-set E has 2 x e copies of the edge e . Note that if | V | ≥
3, then 2 x e ∈ { , , } for all e ∈ E , andif | V | = 2, then 2 x e = 4 for the unique edge e ∈ E . With a slight abuse of notation, we usethe same cost function c to denote the edge costs in G i.e., c f := c e where e ∈ E gave rise tothe edge f ∈ E . By (7) and (6), G is a 4-regular 4-edge connected multigraph. Theorem 1follows from the following result applied to the graph G induced by 2 x . Theorem 2.
Let G = ( V, E ) be a -regular -edge connected multigraph on n vertices. Let c : E → R be an arbitrary cost function on the edges of G (negative costs on the edges areallowed), and let e be an arbitrary edge in G . Then, in O ( n ) time, we can find a -edgeconnected spanning subgraph H of G − e satisfying:(i) c ( H ) ≤ c ( G − e ) ; and(ii) each multiedge of G appears at most once in H (multiedges may arise in H due tomultiedges in G ). F ⊆ E , let χ F ∈ { , } E denote the characteristic vector of F : χ Fe = 1 if and onlyif e ∈ F . Note that distinct multiedges in E correspond to distinct coordinates in χ F . Asmentioned before, Carr and Ravi prove the existence of such a subgraph H by showing thatfor any 4-regular 4-edge connected multigraph G , there exists a finite collection H , . . . , H k of 2-edge connected spanning subgraphs of G such that χ E ( G ) \{ e } lies in the convex hull of { χ H i } i . At a high level, their proof is inductive and splits into two cases based on whether G has a certain kind of a tight set (a cut of size 4). In the first case they construct twosmaller instances of the problem by contracting each of the shores of the tight set, and inthe second case they perform two distinct splitting-off operations at a designated vertex toobtain two smaller instances of the problem. In either case, the convex combinations from thetwo subinstances are merged to obtain a convex combination for G . The first case requiresgluing since the subgraphs obtained from the two subinstances need to agree on a (tight) cut.Merging the convex combinations arising from the second case is rather straightforward asthe two subinstances are more or less independent.Our first insight in this work is that the case from Carr and Ravi’s proof that requiresthe gluing step can be completely avoided, thereby unifying the analysis. This is discussedin Section 2. Our proof relies on an extension of Lov´asz’s splitting-off theorem that is due toBang-Jensen et al., [BJGJS99]. For further discussion on splitting-off theorems, see [Fra11,Chapter 8]. The challenge in efficiently finding a cheap subgraph H from the above convexcombination construction is that each inductive step requires solving two subinstances of theproblem, each with one fewer vertex, leading to an exponential-time algorithm. Having saidthat, an (expected) polynomial-time Las Vegas randomized algorithm can be easily designedthat randomly recurses on one of the two subinstances and produces a 2-edge connectedspanning subgraph whose expected cost is at most c ( G − e ). Our second insight, which isused in derandomizing the above procedure, is that it is easy to recognize which of the twosubinstances leads to a “cheaper” solution, so we recurse only on the cheaper subinstance.Complementing this step, we lift the solution back to the original instance. This operation canlead to two different outcomes so the cost analysis must account for the worst outcome. Thereis a choice of defining the costs in the subinstance such that the cost of the lifted subgraphis the same irrespective of the outcome. Such a choice can lead to negative costs, but this isnot a hindrance for our inductive step because Theorem 2 allows arbitrary real-valued edgecosts. This generality of cost functions is crucial to our algorithm.In Section 4 we consider a well-studied special case of the problem. We present asimple O ( n )-time algorithm that given a 3-regular 3-edge connected graph G , finds a 2-edgeconnected spanning multisubgraph of cost at most c ( G ) (see Theorem 10). The proof isinspired by that of Haddadan, Newman, and Ravi in [HNR19] where they give a polynomial-time algorithm for this problem with a factor ( > ). In [HN18, Theorem 1.1], Haddadanand Newman improve this result to a factor , and very recently, in [Had20, Theorem 1.20],Haddadan claims a stronger result with a factor of = − . We remark that theseproofs are longer and/or more complicated than that of Theorem 10. Another motivation forSection 4 is to illustrate the potential of Theorem 2 in giving simpler proofs for results thatmay not have any explicit half-integrality restrictions.4 .2 Related Work The problem has been intensively studied in network design and several works havetried to bound the integrality gap α of ( - LP ). For the general case with metriccosts, we have ≤ α ≤ , where the lower bound is from [ABEM06] and the upperbound follows from the polyhedral analysis of Wolsey [Wol80] and Shmoys and Williamson[SW90] (this analysis also gives a -approximation algorithm). It is generally conjecturedthat α = , however in [ABEM06], Alexander et al., study α and conjecture that α = based on their findings. As mentioned before, Carr and Ravi [CR98] show thatthe integrality gap of ( - LP ) is at most in the half-integral case. In [BL17] Boyd andLegault consider a more restrictive collection of instances called half-triangle instances wherethe optimal LP solution is half-integral and the graph induced by the half-edges is a collectionof disjoint triangles. They prove that α = in this setting. Half-triangle solutions areof interest as there is evidence that the integrality gap of ( - LP ) is attained at suchsolutions (see [ABEM06]). When the costs come from a graphic metric (i.e., we want to finda minimum-size 2-edge connected spanning multisubgraph of a given unweighted graph), wehave ≤ α ≤ (see [BFS16, SV14]). In this section, we give a simplified proof of the following result from [CR98]. As mentionedbefore, avoiding the case involving the gluing operation is useful for our algorithm in Section 3.For notational convenience, for any subgraph K of some graph, we use χ K to denote χ E ( K ) whenever the underlying graph is clear from the context. Theorem 3 (Statement 1 from [CR98]) . Let G = ( V, E ) be a -regular -edge connectedmultigraph and e = uv be an arbitrary edge in this graph. There exists a finite collection { H , . . . , H k } of -edge connected spanning subgraphs of G − e such that for some nonnegative µ , . . . , µ k with P i µ i = 1 , we have χ E \{ e } = P ki =1 µ i χ H i . Moreover, we may assume thatnone of the H i ’s use more than one copy of an edge in E ; H i may have multiedges as longas they come from distinct edges in G . The following tools on the splitting-off operation will be useful. In keeping with standardterminology, we designate a vertex v (one of the endpoints of e in the theorem statement) atwhich the splitting-off operation is applied. For a multigraph H = ( V, E ) and x, y ∈ V , let λ H ( x, y ) denote the size of a minimum ( x, y )-cut in H , and let deg H ( x ) denote the degree of x in H . Note that each multiedge is counted separately towards the degree of a vertex andthe size of a cut. Definition 4.
Given a multigraph G and two edges sv and vt that share an endpoint v , thegraph G s,t obtained by splitting off the pair ( sv, vt ) at v is given by G + st − sv − vt . Definition 5.
Given a multigraph G and a vertex v of G of even degree, a complete splittingat v is a sequence of deg G ( v ) splitting off operations that result in vertex v having degreezero in the resulting graph. efinition 6. Let k ≥ be an integer and let G be a multigraph such that for all x, y ∈ V \{ v } , λ G ( x, y ) ≥ k . Let e = sv and vt be two edges incident to v . We say that the pair ( sv, vt ) is admissible if for all x, y ∈ V \ { v } , λ G s,t ( x, y ) ≥ k , and for a particular edge e ∈ δ ( v ) , we let A e denote the set of edges f ∈ δ ( v ) \ { e } such that ( e, f ) is an admissible pair. The following result due to Bang-Jensen et al., [BJGJS99] shows that in our setting witha 4-regular 4-edge connected multigraph at least two distinct edges incident to v form anadmissible pair with e = uv . Using this we can perform a complete splitting at v in twodistinct ways. Lemma 7 (Theorem 2.12 from [BJGJS99]) . Let k ≥ be an even integer. Let G be amultigraph such that for all x, y ∈ V \ { v } , λ G ( x, y ) ≥ k . Let deg G ( v ) be even (each multiedgeis counted separately towards the degree). Then, | A uv | ≥ deg G ( v ) . Lemma 8.
Let G be a -regular -edge connected multigraph and e = vx be an edge incidentto v . Then, (i) | A e | ≥ ; and (ii) if ( e, f ) is an admissible pair for some f = vy ∈ δ ( v ) \ { e } ,then the remaining two edges in δ ( v ) \ { e, f } form an admissible pair in G x,y .Proof. Conclusion (i) follows from Lemma 7 since G is 4-regular and 4-edge connected. Forconclusion (ii), let f ∈ δ ( v ) \ { e } be such that ( e, f ) forms an admissible pair in G . Let G x,y denote the graph obtained by splitting off the pair ( e = vx, f = vy ) i.e., G x,y = G − vx − vy + xy .Observe that the hypothesis of Lemma 7 still holds for G x,y with k = 4 because (a) weperformed a splitting off operation using an admissible pair of edges; and (b) deg G x,y ( v ) = 2is even. Let g denote one of the two remaining edges in δ ( v ) \ { e, f } . By Lemma 7, the otherunique edge h ∈ δ ( v ) \ { e, f, g } forms an admissible pair with g in G x,y .Equipped with the above tools, we give a proof of Theorem 3. Proof of Theorem 3.
Let G = ( V, E ) be a 4-regular 4-edge connected multigraph and let e = uv be an arbitrary edge in G . We prove this theorem via induction on n := | V ( G ) | .The base case n = 2 corresponds to a pair of vertices having four parallel edges, call them e, f, g, h . Observe that χ E \{ e } = (cid:0) χ { f,g } + χ { f,h } + χ { g,h } (cid:1) , so the induction hypothesis istrue for the base case.For the induction step, suppose that n ≥ n − e ∈ E . Consider a4-regular 4-edge connected multigraph G on n vertices and an arbitrary edge e = uv ∈ E ( G ).Besides e , let vx, vy, vz be the other three edges incident to v . With a relabeling of vertices,by Lemma 8, we may assume that ( uv, vx ) and ( uv, vy ) form an admissible pair in G (seeFigure 1).By the second conclusion of Lemma 8, ( vy, vz ) is an admissible pair in G u,x , and ( vx, vz )is an admissible pair in G u,y . Consider the graph G obtained by splitting off the pair ( vy, vz )in G u,x i.e., G = G − v + { ux, yz } ; it is customary to drop the vertex v after all its edgeshave been split off. Similarly, let G be the graph obtained by splitting off the pair ( vx, vz )in G u,y i.e., G = G − v + { uy, xz } .Since we only split off admissible pairs, both G and G are 4-regular 4-edge connectedmultigraphs on n − K of some graph, χ K is a6 vx y ze (a) v has fourdistinct neighbors | A e | ∈ { , } . u = zvx ye (b) v has twoparallel edges with u A e = { vx, vy } . uvx = y ze (c) v has twoparallel edges with x, x = u A e = { vx, vy } . u = zvx = ye (d) v has twoparallel edges toeach of { u, x } A e = { vx, vy } . Figure 1: Four configurations of edges in δ ( v ) = { uv, vx, vy, vz } that can arise in our proof.shorthand for χ E ( K ) whenever the underlying graph is clear from the context. Applying theinduction hypothesis to G with the designated edge e = ux gives:23 · χ E ( G ) \{ e } = 23 · χ ( E \ δ ( v )) ∪{ yz } = k X i =1 µ i χ H i , (ConvexComb- G )where { µ i } i denote the coefficients in a convex combination, and { H i } i are 2-edge connectedspanning subgraphs of G such that none of them use more than one copy of an edge in G .Repeating the same argument for G with the designated edge e = uy gives:23 · χ E ( G ) \{ e } = 23 · χ ( E \ δ ( v )) ∪{ xz } = k X i =1 µ i χ H i , (ConvexComb- G )where { µ i } i denote the coefficients in the other convex combination arising from { H i } i . It re-mains to combine (ConvexComb- G ) and (ConvexComb- G ) to obtain such a representationfor G with the designated edge e . We mimic the strategy from [CR98].For each i ∈ { , . . . , k } , we lift H i to a spanning subgraph ˆ H i of G − e . Define ˆ H i asfollows: ˆ H i := ( H i − yz + vy + vz if yz ∈ E ( H i ) ,H i + vy + vx if yz / ∈ E ( H i ) . (Lift- G )Similarly, for each i ∈ { , . . . , k } , we define ˆ H i as the following spanning subgraph of G − e : ˆ H i := ( H i − xz + vx + vz if xz ∈ E ( H i ) ,H i + vx + vy if xz / ∈ E ( H i ) . (Lift- G )We finish the proof of Theorem 3 by arguing that the following convex combination meetsall the requirements: q := 12 k X i =1 µ i χ ˆ H i + 12 k X i =1 µ i χ ˆ H i . (ConvexComb- G )7any of our arguments are the same for G and G so we just mention them in thecontext of G . First of all, by the induction hypothesis and (Lift- G ) it is clear that e (= uv ) , yz, ux / ∈ E ( ˆ H i ), where yz and ux refer to the edges that originated from the splittingoff operations applied at v . Next, we argue that ˆ H i is a spanning subgraph of G that uses nomore than one copy of any edge in G . By the induction hypothesis, none of the subgraphs H i use more than one copy of an edge in G , and H i spans V \ { v } . By the way we lift H i to ˆ H i , it is clear that ˆ H i uses no more than one copy of any multiedge in G , and that it isspanning. To see that ˆ H i is 2-edge connected, observe that the two cases of lifting may beviewed as either (i) subdividing the edge yz by a node v when yz ∈ E ( H i ), or (ii) adding anedge yx and subdividing it by a node v when yz / ∈ E ( H i ). Clearly, these operations preserve2-edge connectivity, hence, ˆ H i is 2-edge connected.It remains to argue that the vector q in the expression (ConvexComb- G ) matches thevector χ E ( G ) \{ e } . Since { µ i } i and { µ i } i denote coefficients in a convex combination, takingan unweighted average of these two combinations gives us another convex combination. Sincenone of the edges in E ( G ) \ δ ( v ) are modified in the lifting step, q f = 2 / f . Next, consider the edge vy . Observe that ˆ H i always contains the edge vy , whereasˆ H i contains vy only when xz / ∈ E ( H i ) (this happens with weight 1 / q vy = · · = . The analysis for vx is symmetric. Lastly, consider the edge vz . It appears inˆ H i ( ˆ H i ) if and only if yz ∈ E ( H i ) (respectively, xz ∈ E ( H i ). Therefore, q vz = · + · = .This completes the proof of Theorem 3. In this section we give a proof of Theorem 2 which we use to obtain a -approximationalgorithm for on half-integral instances (Theorem 1). We apply the same splitting-offtheorem of [BJGJS99] together with an induction scheme that is captured in Theorem 2. Akey feature of this theorem is that we allow edges of negative cost, although the edge costsin any instance of are non-negative.Consider a 4-regular 4-edge connected multigraph G = ( V, E ) on n vertices, and let e = uv be an edge in G . Let c : E → R be an arbitrary real-valued cost function. Our goal is toobtain a 2-edge connected spanning subgraph H of G whose cost is at most c ( G − e ) whileensuring that H uses no more than one copy of any multiedge in G . Observe that if we hadaccess to the collection { H , . . . , H k } of 2-edge connected spanning subgraphs from Theorem 3for some k that is polynomial in | V ( G ) | , then we would be done: for any cost function c , thecheapest subgraph in this collection (w.r.t. cost c ) is one such desired subgraph. It is notclear how to efficiently obtain such a collection; a naive algorithm that follows the proof ofTheorem 3 does not run in polynomial time.As alluded to before, for the purposes of obtaining a cheap 2-edge connected subgraph, itsuffices to only recurse on one of the two subinstances that arise in the proof of Theorem 3.This insight comes from working backwards from (ConvexComb- G ). Since this convex com-bination for G is a simple average of the convex combinations from the two subinstances (see(ConvexComb- G ) and (ConvexComb- G )), it is judicious to only recurse on the “cheaper”subinstance. Combining (ConvexComb- G ) and (Lift- G ), we get that the first subinstancegives rise to a convex combination for χ E ( G ) \{ e } + ( χ { vy } − χ { vx } ). On the other hand,the second subinstance gives rise to a convex combination for χ E ( G ) \{ e } + ( χ { vx } − χ { vy } ).8hus, we should recurse on G if c vx ≥ c vy , and G otherwise. For the sake of argument,suppose that we are recursing on G . So far, we have ignored an important detail in therecursion: the splitting-off operation creates a new edge yz that was not originally presentin G , so we need to assign it some cost to apply the algorithm recursively. Depending onhow we choose the cost of yz , it might either be included or excluded from the subgraphobtained for the smaller instance, so to bound the cost of the lifted solution we must havea handle on both outcomes of the lift operation. Setting c yz := c vz − c vx balances the costof both outcomes. Note that c yz could possibly be negative, but this is permissible since thestatement of Theorem 2 allows for arbitrary edge costs. We formalize the above ideas.In the recursive step, we pick one end vertex v of e and apply a complete splitting offoperation at v to obtain a 4-regular 4-edge connected graph on n − O ( n ) time. The running time of the algorithm is O ( n ), since we apply theinduction step O ( n ) times. We remark that the running time of the algorithm in Theorem 2can be improved to O ( n o (1) ) by using the results for maintaining 3-edge connectivity fromthe work of Jin and Sun [JS20]; we defer the details to a subsequent full version of our paper.Let T = { v, x, y, z } be the four neighbors of v and let e = uv . Recall that A e denotes theset of edges f ∈ δ ( v ) \ { e } such that ( e, f ) is an admissible pair (see Definition 6). Lemma 9.
For vx ∈ δ ( v ) \ { e } , we can check whether vx ∈ A e in O ( n ) time.Proof. We may suppose that the elements of the set T of neighbors of v are all distinct.Otherwise, by Lemma 8, we know exactly which pairs are admissible, see Figure 1. Considerthe graph ˆ G = ( G u,x ) y,z obtained by splitting off the pairs ( uv, vx ) and ( yv, vz ) at v . Let G ∗ be the graph obtained from ˆ G by contracting ux to a single vertex s and contracting yz to asingle vertex t . Then we apply a max s − t flow computation to check whether G ∗ has ≥ s − t paths; otherwise, G ∗ has an ( s, t )-cut δ ( S ) of size ≤
3. In the latter case,it is clear that our trial splitting is not admissible.In the former case, we claim that our trial splitting is admissible. Suppose that ˆ G isnot 4-edge connected. Then there exists a non-empty, proper vertex set S in ˆ G such that | T ∩ S | ≤ | T \ S | and | δ ˆ G ( S ) | <
4. Clearly, | S ∩ T | ≤
2, and if | S ∩ T | = 2, then we have | S ∩ { u, x }| = 1 and | S ∩ { y, z }| = 1 (otherwise, S would give an ( s, t )-cut of G ∗ of size ≤ S is the same in G and in ˆ G , we have, by 4-edge connectivity of G , 4 > | δ ˆ G ( S ) | = | δ G ( S ) | ≥
4, a contradiction.To see that the running time is linear, observe that G ∗ has ≤ n edges, an s − t flow ofvalue ≥ Proof of Theorem 2.
First, consider the base case in the recursion when n = 2. The onlysuch 4-regular 4-edge connected multigraph is given by four parallel edges between u and v ,of which e is one. Picking the two cheapest edges from the remaining three edges gives thedesired subgraph.For the induction step, suppose that n ≥ n − e . Consider a 4-regular 4-edge connected multigraph G on n vertices and an edge e = uv in G . Our algorithm proceeds as follows. By Lemmas 8 and 9, we can find in O ( n )-time twoneighbors of v , say x and y , such that vx, vy ∈ A e and c vx ≥ c vy . Next, we construct the9raph ˆ G := ( G u,x ) y,z = G − v + { ux, yz } and extend the cost function c to the new edge yz as c yz := c vz − c vx (note that the cost of ux is inconsequential and that c yz may be negativeor non-negative). We recursively find a 2-edge connected spanning subgraph ˆ H of ˆ G withcost at most c ( ˆ G − ux ). Then, we lift ˆ H to obtain a spanning subgraph H of G : H := ( ˆ H − yz + vy + vz if yz ∈ E ( ˆ H ) , ˆ H + vy + vx if yz / ∈ E ( ˆ H ) . We analyze the cost of this subgraph. Regardless of the cases above, our choice of c yz implies that c ( H ) = c ( ˆ H ) + c vy + c vx . Therefore, c ( H ) ≤ c ( ˆ G − ux ) + c vy + c vx = 23 { c ( G − e ) − c vx − c vy − c vz + ( c vz − c vx ) } + c vy + c vx = 23 c ( G − e ) + 13 ( c vy − c vx ) ≤ c ( G − e ) , where the last inequality follows from our choice of vx, vy to satisfy c vx ≥ c vy .It remains to argue that H is a 2-edge connected spanning subgraph of G − e that uses nomore than one copy of any multiedge in G . It is clear that the following hold: (a) e / ∈ E ( H );(b) H is a spanning subgraph of G ; and (c) each multiedge of G appears at most once in H . Since ˆ H is 2-edge connected and adding and/or subdividing an edge preserves 2-edgeconnectivity, H is 2-edge connected. Overall, in O ( n )-time we have constructed a 2-edgeconnected spanning subgraph H of G − e whose cost is at most c ( G − e ), thereby provingTheorem 2.Using Theorem 2, we give a deterministic -approximation algorithm for on half-integral instances. Proof of Theorem 1.
Let x be an optimal half-integral solution to ( Subtour - LP ) (and ( - LP ))for an instance given by an n -vertex graph G = ( V , E ) and a metric cost function c . Let G = ( V, E ) denote the graph induced by 2 x where for each e ∈ E we include 2 x e copies ofthe edge e in G . Since x has (fractional) degree 2 at each vertex and it is fractionally 2-edgeconnected, G is a 4-regular 4-edge connected multigraph. With a slight abuse of notation, weuse the same cost function for the edges of E : for any e ∈ E , c e := c f , where f denotes theedge in E that gave rise to e . We invoke Theorem 2 on G and some edge e ∈ E . This givesus a 2-edge connected spanning subgraph H of G − e satisfying c ( H ) ≤ c ( G − e ). Liftingthe subgraph H to G gives a 2-edge connected spanning multisubgraph H (of G ); note that H uses at most two copies of any edge in G . By the first conclusion of Theorem 2 and thenon-negativity of c , c ( H ) = c ( H ) ≤ c ( G − e ) ≤ c ( G ) = c T x , where the last equalityfollows by recalling that G is induced by 2 x . Besides invoking Theorem 2 we only performtrivial graph operations so the running time is O ( n ). for 3-Regular 3-Edge Connected Graphs Let G = ( V, E ) be a 3-regular 3-edge connected graph with non-negative edge costs c ∈ R E ≥ .In this section we consider an analogous problem to that of Theorem 2, namely the problem offinding a polynomial-time algorithm which gives a 2-edge connected spanning multisubgraph10f G of cost at most βc ( G ) for some β ≥
0. Note that the everywhere vector for G isfeasible for ( Subtour - LP ). For any costs c for which the everywhere vector is also optimalfor ( Subtour - LP ) (such as for the graphic metric), such an algorithm would provide a β -approximation for . The conjecture that α = would then imply β = · = should be possible, and the conjecture for α would imply β = · = should bepossible. In [BL17] a constructive algorithm for β = is given, however it does not run inpolynomial time.In [HNR19, Theorem 2], Haddadan, Newman, and Ravi show that it is possible to dobetter than for this problem, and provide an efficient algorithm for β = . In fact,they show that the everywhere vector can be expressed as a convex combination of 2-edge connected spanning multisubgraphs of G and this convex combination can be found inpolynomial time. They remark that combining their ideas with an efficient algorithm forTheorem 3 would imply the result for β = ( < ). Although a polynomial-time algorithmfor Theorem 3 is not currently known, it is possible to use our result in Theorem 2 to obtain β = , as follows. Theorem 10.
Let G = ( V, E ) be a -regular -edge connected graph on n vertices with non-negative edge costs c ∈ R E ≥ . Then in O ( n ) -time we can find a -edge connected spanningmultisubgraph H of G such that c ( H ) ≤ c ( G ) .Proof. Let F be a 2-factor of G that intersects all of the 3-edge cuts and 4-edge cuts of G . Such a 2-factor can be found in O ( n )-time (see [BIT13, Theorem 5.4]). Let G ′ be thegraph obtained by contracting the cycles of F and removing any resulting loops, and let M := E ( G ′ ). Clearly G ′ is 5-edge connected (by choice of F ), and thus the vector y ∈ R M ≥ defined by y e := for all e ∈ M is feasible for ( - LP ) for G ′ . It then follows from thepolyhedral analysis of Wolsey [Wol80] and Shmoys and Williamson [SW90] of the ( Subtour - LP )that we can find a 2-edge connected spanning multisubgraph of G ′ with edge set R satisfying c ( R ) ≤ c T y = c ( M ). Then the graph H induced by F ∪ R is a 2-edge connected spanningmultisubgraph of G such that c ( H ) ≤ c ( F ) + 35 c ( M ) ≤ c ( F ) + 35 c ( E \ F ) . (9)Now consider the vector z ∈ R E ≥ where z e := 1 / e ∈ F , and z e := 1 otherwise.Vector z is a feasible half-integer solution for ( Subtour - LP ), and thus by Theorem 2 and theideas used in the proof of Theorem 1, in O ( n )-time we can find a 2-edge connected spanningmultisubgraph H of G such that c ( H ) ≤ c ( F ) + 43 c ( E \ F ) . (10)We complete the proof by showing that either H or H has cost at most c ( G ). Using(9) and (10) we have:min( c ( H ) , c ( H )) ≤ c ( H )+ 38 c ( H ) ≤ (cid:0) c ( F )+ 35 c ( E \ F ) (cid:1) + 38 (cid:0) c ( F )+ 43 c ( E \ F ) (cid:1) = 78 c ( G ) . cknowledgments We thank Chaitanya Swamy for pointing us to [CV02].
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