Problems concerned with constructing a subgraph of a given graph satisfying certain connectivity requirements are among the most natural and occupy an important place in the field of approximation algorithms. In this paper, we consider the following fundamental connectivity problem in graphs: Given an undirected simple graph , find a -edge-connected spanning subgraph of with minimum number of edges. We briefly denote this problem by 2-ECSS. It has the following natural LP relaxation, where denotes the set of edges with one end in the cut and the other not in .
The dual of this LP is as follows.
This problem is a special case of many other network design problems, in particular the famous traveling salesman problem. It remains NP-hard and MAX SNP-hard even for subcubic graphs . Thus, approximation algorithms have been sought. The first result beating the factor came from Khuller and Vishkin , which is a -approximation algorithm. Cheriyan, Sebö and Szigeti  improved the factor to . Vempala and Vetta , and Jothi, Raghavachari and Varadarajan  claimed to have and approximations, respectively. Krysta and Kumar  went on to give a -approximation for some small assuming the result of Vempala and Vetta . A relatively recent paper by Sebö and Vygen  provides a -approximation algorithm given by using more elegant ear decompositions, and mentions that the aforementioned claimed approximation ratio has not appeared with a complete proof in a fully refereed publication. The result of Vempala and Vetta , which was initially incomplete, very recently re-appeared in . Given this, it is not clear if the ratio by Krysta and Kumar  still holds. To the best of our knowledge, the ratio stands as the current best factor. A paper by Boyd, Fu and Sun  shows that the integrality gap of satisfies for the class of subcubic bridgeless graphs. Thus, we have in general by adding the result of Sebö and Vygen  that .
We prove the following theorem.
There exists a polynomial-time -approximation algorithm for 2-ECSS. Furthermore, .
This improves the previous best factor , and gives the new bounds .
Our approach is based on a result of Frank , which states that certain ear decompositions can be computed efficiently. This was also previously used by Cheriyan, Sebö and Szigeti , and Sebö and Vygen  for 2-ECSS. These authors consider constructing finer ear decompositions beyond the one provided by the theorem of Frank. We do also construct a more sophisticated ear decomposition over the base solution, but this time including modifications involving -ears, too. More crucially in the analysis, we make use of a convex combination of two different lower bounds: one implied by an optimal ear decomposition with respect to the number of even ears; the other by the dual of the LP relaxation of the problem. This also allows us to bound the integrality gap.
In this section, we collect facts that will be used in our analysis. If a result already appears in the literature, we cite it without giving a proof and refer the reader to the relevant paper.
We first make the assumption that the input graph is -connected. The derived approximation ratio on each maximally -connected subgraph (i.e. blocks) of also holds for , since the optimal value for is the sum of the optimal values for the blocks. Thus, we can run any algorithm on the blocks separately.
An ear decomposition of a graph is a partition of into subgraphs, namely such that is the trivial path of length , and for , is either
a path with exactly both end vertices in , or
a cycle with exactly one vertex in .
An element of the set is called an ear. If it is of the first type above, it is called an open ear, otherwise a closed ear. Note that is always closed. The set of vertices of an ear, which are not end vertices are called the internal vertices of the ear. The set of internal vertices of an ear is denoted by . For a nonnegative integer , an ear with edges is called an -ear (i.e. . is called an even ear if is even, an odd ear otherwise. A -ear is also called a trivial ear. An -ear with such that none of its internal vertices is an end vertex of another -ear with is called a pendant ear. Note that if we disregard the trivial ears, all the internal vertices of a pendant ear have degree .
An open ear decomposition is one in which all the ears are open ears. It is well known by a result of Whitney  that a graph is -edge-connected if and only if it has an ear decomposition, and -connected if and only if it has an open ear decomposition. Given a -edge-connected graph , let denote the minimum number of even ears over all possible ear decompositions.
 Given a -edge-connected graph , one can compute an open ear decomposition with even ears in time .
Based on this result, the following is given by Cheriyan, Sebö and Szigeti .
 Given a -connected graph , an open ear decomposition of with even ears can be computed in time .
We call such an ear decomposition an optimal ear decomposition.
Let denote the optimum value of a 2-ECSS in . Let denote the optimum value of the LP on . Obviously, we have . Define .
We will interpret the lower bound implied by as a sum distributed over ears. Given an optimal ear decomposition , and an odd ear , we set . For an even ear , we set .
We argue inductively by considering the induction hypothesis on the solution obtained by removing a pendant ear. In the base case, there is a single pendant ear . By definition, if is an odd ear, then . Similarly, if is an even ear, then . As for the inductive step, let be the graph obtained by removing a pendant ear. If the removed pendant ear is an odd ear, then . If it is an even ear, then . This establishes the result. ∎
Given all these, we have
We will also use for lower bounding the optimum. In , a cut is synonymously called a dual. We will consider a feasible dual solution in which the dual values corresponding to certain vertices are positive, and all the others are . Abusing the notation, a dual consisting of a single vertex is denoted by instead of , and denote the dual variable corresponding to by .
3 Approximating 2-Ecss
The algorithm consists of several phases to construct a finer ear decomposition. At each phase, it either performs certain operations improving the solution by edge interchanges, or it rearranges the ears by terminating some and creating new ones. Instead of describing the algorithm in pseudocode, which would hinder the basic ideas behind it, we explain it by stating the conditions we require on the solution, which are successively stated to render the analysis more transparent. Each phase satisfies a condition, building on the solution filtered by the previous one so that configurations ruled out by previous cases are not considered in the foregoing one. In reality, it is possible to satisfy all the conditions together in an algorithmically more compact way, which is not the focus of this paper.
The algorithm starts by computing an optimal ear decomposition. Efficient implementation of this procedure is guaranteed by Proposition 3. Let be the solution returned by this phase. Before stating all the conditions, we first assume that there are no trivial ears, which can be satisfied without changing the parities of the ears (see Figure 0(a) for an illustration):
(0) Let be an ear on . Let be another open ear on . Then, there is no edge of such that and are the end vertices of .
Given an ear and another ear , if at least one end vertex of is in , we say that touches via its end vertices, which are called touched vertices of , and is touched by . Otherwise, we say that and are independent, is independent of , and vice versa.
(1) Let be a -ear on . Let be another open -ear on . Then, and are independent.
Consider a -ear and another open -ear touching . By assumption, the end vertices of cannot be adjacent vertices of . The algorithm checks if the deletion of an edge of other than the middle edge can be performed without violating feasibility. After the deletion, merging of the -ears to form a -ear is performed, which does not change the number of even ears (see Figures 0(b) and Figure 0(c) for examples).
(2) Let be a -ear on . Let be another open -ear on . Then, no vertex in is adjacent to a vertex in via the edges in .
To satisfy this, we iterate over the edges such that
and for distinct -ears and .
It modifies by performing what we call improvement steps via such . Let the end vertices of and be and , adjacent to and , respectively. The algorithm checks if selecting and deleting the edges and does not violate feasibility. If so, it performs the deletions and includes into the solution (see Figure 2 for illustrations of all the cases for open ears, where the included edge is in bold). Note that this does not change the number of even ears on .
In order to describe the remaining conditions more precisely, we consider a numbering of the vertices of a given -ear from to , from one end vertex to the other. Number and are called the side vertices of .
(3) Let be a -ear on . Let be another open -ear on . Then, and are independent.
This condition is satisfied as follows. Given a -ear , if a there is a single touched vertex in , which is a side vertex, then we terminate the -ears and declare a new ear of length . If the difference of the coordinates of the touched vertices is , we declare two new ears, one being an -ear, the other a -ear. If this difference is , we declare one -ear and one -ear. We then run the phases ensuring Condition (1) and Condition (2) to handle the new -ears. See Figure 3 for example illustrations, and note that the cases occurring upon translation of the coordinates of the touched vertices can be handled likewise as described. In this figure, and in the following ones describing rearrangements and improvements, the new ears are depicted with solid lines of varying thickness, the disregarded edges are shown using dotted lines.
(4) Let be a -ear touching a -ear via an end vertex of . Then, the other touched vertex of is not a side vertex of .
This is satisfied by terminating and , and declaring a new -ear as in Figure 0(c).
(5) Let be a -ear on . Let be another open -ear independent of or an open -ear on . Then, no side vertex of is adjacent to a vertex in via the edges in .
Similar to the case of Condition (2), we iterate over the edges such that
is a side vertex of and for distinct ears (a -ear) and (a -ear or a -ear).
We modify by performing improvement steps via such or rearrange the ears. All possible cases modulo the equivalence relation grouping the symmetric cases are depicted in Figure 4.
(6) Let be a -ear on . Then, there is no set of -ears touching via more than vertices.
We consider the case in which two -ears touch via more than vertices, and describe the rearrangement of the ears in Table 1 and Table 2, where we number the vertices of starting from the top end. In each case, we terminate , the two -ears touching , and declare at most two new ears. We denote the case in which the first -ear touches via and , and the second -ear touches via and by , where take values in . Note that by definition, the coordinates of a -ear cannot be . By Condition (0), we have that the difference between the coordinates of a -ear is at least . By Condition (4), the tuples and cannot appear. Furthermore, we only consider the cases for which . It is easy to see that the rest is essentially the same by a translation of all the coordinates of the -ears. We also do not show the case , which is the same as when the vertices of are numbered starting from the bottom end. There are in total cases. Notice that there are cases in which we declare new -ears. By construction however, there cannot be a set of -ears touching these -ears via more than vertices so that the condition is still satisfied. In all other cases, an ear of length at least is declared, possibly together with another one. If there is only one new ear, it is of length at least . If there are two new ears, their total length is at least .
(7) Let be a -ear on touched by a set of -ears. Let further the set of touched vertices contain one end vertex of . Then, there are two non-adjacent vertices , called free vertices, not adjacent to any inner vertex of a -ear or a -ear via the edges in .
By Condition (6), if there are at least two -ears touching , then the number of touched vertices of is at most . In other words, the -ears touching all touch via the same vertices. Thus, we consider improvement steps for the case of one -ear touching via two vertices. From this phase on, instead of describing what the improvement steps do, we point out the possibility of an improvement on an instance which we call a forbidden configuration. Assuming without loss of generality that the touched end vertex of is , we list the forbidden configurations and the corresponding improvements for the cases in which the touched vertices are and in Figure 5, Figure 6, and Figure 7. Note that the cases in which an inner vertex of is adjacent to an inner vertex of another -ear, there is at least vertices of that -ear on one side. In all of the cases, either a new -ear, or one -ear and one -ear, or one -ear and one -ear are created. Ruling out these forbidden cases, we see that there are two non-adjacent free vertices in .
(8) Let be a -ear touched by a set of -ears. Let further the touched vertices be in . Then, there is a vertex , called the free vertex, not adjacent to any inner vertex of a -ear or a -ear via the edges in .
There are essentially two main cases in which the touched vertices are and . We list the forbidden cases and the corresponding improvements in Figure 8 and Figure 9, where the possible lengths of the newly created ears are the same as in Condition (7).
(9) Let be a -ear touched by a set of -ears . Let further the touched vertices be in . Then, no vertex in is adjacent to an inner vertex of a -ear via the edges in , for .
The setting is the same as in Condition (8). The forbidden configurations and improvements are shown in Figure 10, where one -ear and one -ear are created in each case.
In any of the phases ensuring the conditions above, the rearrangement of the ears or the improvement steps do not create a configuration to be reconsidered in the phase so that the algorithm terminates. Furthermore, the maximum number of ears involved in any rearrangement or an improvement step is , implying that there are at most operations performed in total.
4 Proof of Theorem 1
Let be the total cost of the edges returned by the algorithm. All the operations performed to ensure Condition (3)-Condition (9) create either a single new -ear with , or two new ears of total length at least . In the latter case, let be the graph obtained by removing the inner vertices of these ears from . Clearly, . Then, in order to establish , it suffices to argue on the residual solution in :
Based on this assumption, we argue by giving a procedure, which at each step removes a set of ears from starting from pendant ears until there remains only the trivial path of length . It will be the case that the ratio of the cost of the removed edges with the lower bound associated to the ears (which we call the ear ratio) is not more than , establishing the approximation ratio. In doing this, we either use (1) or the fact that a feasible dual solution is a lower bound for the optimum, thus considering a convex combination of these two lower bounds.
If there is a pendant -ear with , we remove and note that for , by definition so that the ear ratio satisfies . If is a -ear or a -ear, then we have so that the ear ratio is , i.e. is optimally covered.
If there is a pendant -ear not touching any -ear, we remove , this time using the value of a feasible solution to as a lower bound. We assign for the distinct vertices , and . Notice that this is feasible in by Condition (1) and Condition (2): There are no edges in that has to be set to a value greater than to maintain feasibility. Then, the total dual value induced by is , which optimally covers it.
If a pendant -ear touches a -ear , then we remove together with all the -ears touching . By Condition (6), the set of -ears do not touch via more than vertices. Consider first the case in which one of the touched vertices of is an end vertex of . Then, by Condition (7), there are two non-adjacent free vertices . Assigning in , the total dual value induced by is , and the ear ratio is . The ear ratio of the removed -ears is by the same assignment in the previous paragraph. Note that by the definition of the free vertices, we can keep the values of the edge dual variables at except those of the -ears.
Consider now the case in which the touched vertices of is in and the number of -ears touching , which we denote by is at least . Then, we have two possible configurations that are not ruled out by Condition (8) and Condition (9), which are shown in Figure 10(a) and Figure 10(b). In these figures, the edges not in are shown using dashed lines, and the cuts corresponding to certain dual variables in are shown using dotted lines. In both of the configurations, we use the same assignment for -ears as in the previous paragraph, leading to a total dual value of for each. For the -ear , the free vertex whose existence is guaranteed by Condition (8) is assigned the value . The cut containing all the inner vertices of and the -ears is assigned the value . Observe that this does not violate feasibility by Condition (9). The ratio of the total cost of the edges considered with the total dual value we have assigned then satisfies
There remains the case in which the touched vertices of is in and there is a single -ear touching , which is depicted in Figure 10(c) and Figure 10(d). In this case, the free vertex of is assigned the value , and the two side vertices take the value . To maintain feasibility, the edge between the free vertex and one side vertex is assigned the value . The inner vertices of , and the cut containing all these vertices together with those of also take the value . Notice that we can keep all the edge dual variables except the aforementioned one at by the definition of the free vertex and the side vertices (i.e. Condition (4) and Condition (5)), together with Condition (9). The total cost of the edges is , whereas the total dual value is . The ratio satisfies . This establishes the approximation ratio. Noting the inequality (1), the integrality gap of is also bounded by , which completes the proof.
5 Tight example
A tight example for the algorithm is given in Figure 12. An optimal ear decomposition consists of a closed -ear together with all the -ears on which the algorithm does not perform any modifications. The cost of the solution returned by the algorithm is thus . An optimal solution is the Hamiltonian cycle of cost .
6 Final Remarks
For 2-ECSS, an optimal ear decomposition already gives a -approximation algorithm: In the argument we have used for the proof of Theorem 1, the ear ratio for a -ear is . Noting that even ears are optimally covered, in order to get a -approximation, one should reduce to the case in which we only have -ears. The basic idea we have introduced is that in the case of -ears touching -ears, which is the main bottleneck, we can make sure by rearranging the ears and performing certain improvements, that there are enough free vertices to yield a large value for the dual of the natural LP relaxation. It is an important question if it is possible to generalize this approach to the problems with metric and general costs. The approximation factors and in these cases, respectively, have resisted improvement over several decades. This would possibly involve a more general lower bound about ear decompositions as our approach gives a hint that using the LP relaxation alone might not be enough.
We would like to thank Christoph Hunkenschröder and Jens Vygen for giving information about the status of 2-ECSS.
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