Theoretical Computer Science

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1 Theoretical Computer Science 410 (2009) Contents lists available at ScienceDirect Theoretical Computer Science journal homepage: A push relabel approximation algorithm or approximating the minimum-degree MST problem and its generalization to matroids Kamalika Chaudhuri a,1, Satish Rao b, Samantha Rieseneld b, Kunal Talwar c,,2 a U.C. San Diego, United States b U.C. Berkeley, United States c Microsot Research, Mountain View, CA, United States a r t i c l e i n o a b s t r a c t Keywords: Approximation algorithms Push relabel Degree-bounded network design Spanning trees In the minimum-degree minimum spanning tree (MDMST) problem, we are given a graph G, and the goal is to ind a minimum spanning tree (MST) T, such that the maximum degree o T is as small as possible. This problem is NP-hard and generalizes the Hamiltonian path problem. We give an algorithm that outputs an MST o degree at most 2 opt (G) + o( opt (G)), where opt (G) denotes the degree o the optimal tree. This result improves on a previous result o Fischer [T. Fischer, Optimizing the degree o minimum weight spanning trees. Technical Report 14853, Dept. o Computer Science, Cornell University, Ithaca, NY, 1993] that inds an MST o degree at most b opt (G) + log b n, or any b > 1. The MDMST problem is a special case o the ollowing problem: given a k-ary hypergraph G = (V, E) and weighted matroid M with E as its ground set, ind a minimumcost basis (MCB) T o M such that the degree o T in G is as small as possible. Our algorithm immediately generalizes to this problem, inding an MCB o degree at most k 2 opt (G, M) + O(k k opt (G, M)). We use the push relabel ramework developed by Goldberg [A. V. Goldberg, A new max-low algorithm, Technical Report MIT/LCS/TM-291, Massachusetts Institute o Technology, 1985 (Technical Report)] or the maximum-low problem. To our knowledge, this is the irst use o the push relabel technique in an approximation algorithm or an NP-hard problem. The MDMST problem is closely connected to the bounded-degree minimum spanning tree (BDMST) problem. Given a graph G and degree bound B on its nodes, the BDMST problem is to ind a minimum cost spanning tree among the spanning trees with maximum degree B. Previous algorithms or this problem by Könemann and Ravi [J. Könemann, R. Ravi, A matter o degree: Improved approximation algorithms or degree-bounded minimum spanning trees, SIAM Journal on Computing 31(6) (2002) ; J. Könemann, R. Ravi, Primal-dual meets local search: Approximating MST s with nonuniorm degree bounds, in: Proceedings o the Thirty-Fith ACM Symposium on Theory o Computing, 2003, pp ] and by Chaudhuri et al. [K. Chaudhuri, S. Rao, S. Rieseneld, K. Talwar, What would Edmonds do? Augmenting paths and witnesses or bounded degree MSTs, in: Proceedings o APPROX/RANDOM, 2005, pp ] incur a near-logarithmic additive Corresponding author. addresses: kamalika@soe.ucsd.edu (K. Chaudhuri), satishr@cs.berkeley.edu (S. Rao), samr@cs.berkeley.edu (S. Rieseneld), kunal@microsot.com (K. Talwar). 1 Research done at UC Berkeley. 2 Research partially done at UC Berkeley /$ see ront matter 2009 Elsevier B.V. All rights reserved. doi: /j.tcs

2 4490 K. Chaudhuri et al. / Theoretical Computer Science 410 (2009) error in the degree. We give the irst BDMST algorithm that approximates both the degree and the cost to within a constant actor o the optimum. These results generalize to the case o nonuniorm degree bounds Elsevier B.V. All rights reserved. 1. Introduction Given a weighted graph G = (V, E, c), the minimum-degree minimum spanning tree (MDMST) problem is to ind a minimum spanning tree (MST) o G that minimizes the maximum degree. This problem, even in the unweighted case, generalizes the Hamiltonian Path problem and is thereore NP-hard. In this paper we give a polynomial-time algorithm that, given a weighted graph G, outputs an MST o degree at most 2 opt (G)+O( opt (G)), where opt (G) denotes the degree o an optimal solution. This is the irst constant-actor approximation to this problem. The MDMST problem requires us to optimize the degree in a graph G o a minimum-cost base in the graphical matroid o G. We consider a more general setting where the (hyper)graph and the matroid are not necessarily related. Given a k-ary hypergraph G = (V, E) and a weighted matroid M with E as the ground set, the minimum-degree minimum-cost base (MDMCB) problem is to ind a minimum-cost base T o M that minimizes the degree o T in G. (See Section 6.1 or a more complete deinition.) Unlike in the MDMST problem, where the vertices have a very speciic relation to the elements o the matroid, this setting allows the nodes o the hypergraph to correspond to arbitrary subsets o the elements. The only restriction is that each element o the matroid occurs in at most k subsets. As a concrete example o the MDMCB problem, consider a network in which each link is controlled by a subset o a set o autonomous entities, with the restriction that no link is controlled by more than k entities. The goal is to build an MST o the network such that the maximum number o links controlled by a single entity is minimized. Other natural combinatorial optimization problems can also be ormalized as instances o the MDMCB problem. Our MDMST algorithm generalizes in a straightorward way to the MDMCB problem. Given a k-ary hypergraph G = (V, E) and weighted matroid M = (E, I, c), it outputs an MCB o M that has degree in G at most k 2 opt (G, M) + O(k 3 2 opt (G, M)), where opt (G, M) is the degree o an optimal solution. The MDMST and MDMCB algorithms use the push relabel ramework invented by Goldberg [9] (and ully developed by Goldberg and Tarjan [10]) or the max-low problem. To our knowledge, this work is the irst use o the push relabel technique in an approximation algorithm or an NP-hard problem. Subsequent to the publication o a previous version o this work [4], Goemans [8] gave an algorithm or the MDMST problem that outputs an MST o degree at most opt (G) + 2. Finally Singh and Lau [22] gave a MDMST algorithm that outputs a tree o degree at most opt (G) + 1, which is optimal unless P = NP. Both these works are based on polyhedral techniques showing that extremal solutions to the natural linear-programming relaxation have particular structure. Using a lemma o Goemans [8], we show that running our push relabel algorithm on the set o tight edges in an extremal solution gives a ( opt (G) + 2)-algorithm as well. While these recent results dominate our results or the MDMST problem, the techniques developed in this paper may be o independent interest. One o the motivations or studying the MDMST problem is its connection to the bounded-degree minimum spanning tree (BDMST) problem. Given a graph and upper bounds on the degrees o its nodes, the BDMST problem is to ind a spanning tree o minimum cost, among the ones that obey the degree bounds. This bi-criteria optimization problem generalizes several combinatorial problems, including the Traveling Salesman Path Problem (TSPP), which corresponds to the case when degrees are restricted to 2 uniormly. Since we do not assume the triangle inequality, approximations or the BDMST problem must relax the degree constraint, unless P equals NP. Let c opt (B) be the cost o an optimal solution to the BDMST problem, given input graph G and uniorm degree bound B. We call a BDMST algorithm an (α, (B))-approximation algorithm i, given graph G and bound B, it produces a spanning tree that has cost at most α c opt (B) and maximum degree (B). Könemann and Ravi give, to our knowledge, the irst BDMST approximation scheme [15]: a polynomial-time (1 + 1 β, bb(1 + β) + log b n)-approximation algorithm or any b > 1, β > 0. They illustrate the close relationship between the BDMST and MDMST problems. Using a novel cost-bounding technique based on Lagrangean duality, Könemann and Ravi show that the MDMST problem can essentially be used as a black box in an algorithm or the BDMST problem. In a subsequent paper, [16], they use primal dual techniques and give similar results or nonuniorm degree bounds. Könemann and Ravi rely on an MDMST algorithm due to Fischer [6]. Given a graph G or which the MDMST solution is opt (G), Fischer s algorithm inds an MST o G o degree at most b opt (G) + log b n or any b > 1. In a recent paper, [3], the authors give an improved MDMST algorithm based on inding augmenting paths o swaps. The algorithm simultaneously enorces upper and lower bounds on degrees, which, by using linear programming duality and techniques o [15,5], is shown b to result in an optimal-cost (1, B + O(log 2 b b n))-approximation BDMST algorithm or any b (1, 2). At the expense o quasipolynomial time, the authors [3] also give an algorithm that produces an MST with degree at most opt (G)+O( log n ), log log n leading to a quasipolynomial-time (1, B + O( log n ))-approximation algorithm or the BDMST problem. log log n

3 K. Chaudhuri et al. / Theoretical Computer Science 410 (2009) The push relabel MDMST algorithm in this paper also implies a polynomial-time (1 + 1 β, 2B(1 + β) + O( B(1 + β)))- approximation scheme or the BDMST problem, or any β > 0. Thus we give the irst algorithm that approximates both degree and cost to within a constant actor o the optimum. For example, or B = 2 (i.e. or the TSPP), all previous algorithms would produce a tree with near-logarithmic degree and cost within a constant actor o the optimum; our algorithm, in contrast, gives a tree o cost within a (1 + ɛ)-actor o the optimal solution and o maximum degree O( 1 ) or any ɛ > 0. Our work does not assume the triangle inequality; when ɛ the triangle inequality holds, Hoogeveen [11] gives a 3 -approximation to the TSPP based on Christoides algorithm. The 2 Euclidean version o the BDMST problem has also been widely studied. See, or example, [19,13,2,12]. For the sake o a simpler exposition, we describe our BDMST results in the setting o uniorm degree bounds. Our techniques imply analogous results even in the case o more general nonuniorm degree bounds. Though our BDMST algorithm does not simultaneously enorce upper and lower degree bounds, our techniques here do apply to a version o the BDMST problem in which lower bounds on node degrees must be respected, which may be o independent interest. The BDMST and MDMST problem are dierent generalizations o the same unweighted problem: given an unweighted graph G = (V, E), ind a spanning tree o G o minimum maximum degree. Fürer and Raghavachari [7] give a lovely algorithm or this problem that outputs an MST with degree opt (G)+1. Their algorithm inds a sequence o swaps in a laminar amily o subtrees o G such that the sequence results in an improvement to the degree o some high-degree node, without creating any new high-degree nodes. The laminar structure relies on the property that an edge e E that is not in a spanning tree T can replace any tree edge on the induced cycle o T {e}. This property is not maintained in weighted graphs because a non-tree edge can only replace other tree edges o equal cost. The structure o an improving sequence o swaps in a weighted graph can thereore be signiicantly more complicated. While Fischer s MDMST solution is locally optimal with respect to single edge-swaps, our algorithm explores a more general set o moves that may consist o long sequences o branching, interdependent changes to the tree. Surprisingly, the push relabel ramework can be delicately adapted to explore these sequences. The basic idea that we borrow rom Goldberg [9] is to give each node a label and permit excess to low rom a higher-labeled node to lower-labeled nodes. Nodes are allowed to increase their label when they are unable to get rid o their excess. For max-low, the excess was a prelow, while in our case, the excess reers to excess degree. We are intrigued by the possibility that the push relabel ramework may be extended to search what may appear to be complicated neighborhood structures or other optimization problems. Independent o our work, Ravi and Singh [20] give an algorithm or the MDMST problem with an additive error o p, where p is the number o distinct weight classes. We note that this bound is incomparable to the one presented here and does not improve previous results or the BDMST problem. The aorementioned works o Goemans [8] and Singh and Lau [22] also give algorithms or the BDMST problem that achieve additive errors o 2 and 1, respectively, while giving optimal cost. These iterative rounding techniques have been urther generalized to other degree constrained network design problems [1,14,17,18]. Notably, our results on MDMCB have been improved by Király, Lau and Singh [14], who give an algorithm that outputs an minimum-cost base which violates the degree constraint by an additive (k 1), or k-ary hypergraphs Techniques All known algorithms or the MDMST problem begin with an arbitrary MST T, and repeatedly update T by swapping a non-tree edge e E with a tree edge e T o the same weight, where e is on the induced cycle in T {e}. Fischer proceeds by executing any swap that improves a degree-d node without introducing new degree-d nodes, or selected high values o d. He shows that when the tree is locally optimal, the maximum degree o the tree is at most b opt (G) + log b n, or any b > 1, where opt (G) is the degree o the optimal MDMST solution or G. Moreover, this analysis is tight [3]. To illustrate the diiculty o the MDMST problem, we describe here a pathological MST T in a graph G (see Fig. 1): the tree T has a long path consisting o O(n) nodes ending in a node u o degree d. The children o u each have degree (d 1); the children o the degree-(d 1) nodes have degree (d 2), and so on until we get to the leaves. Each edge on the path has cost ɛ, and an edge rom a degree-(d i + 1) node to its degree-(d i) child has cost i. In addition, each o the degree-(d i) nodes has a cost-i edge to one o the nodes on the path. For some d with d = O(log n/ log log n), the number o nodes in the graph is O(n). Note that an MST o G with optimal degree consists o the path along with the non-tree edges and has maximum degree 3. On the other hand, every cost-neutral swap that improves the degree o a degree-(d i) node in the current tree increases the degree o a degree-(d i 1) node. Hence the tree T is locally optimal or the algorithms o [6,15]. Moreover, all the improving edges are incident on a single component o low-degree nodes; one can veriy that the algorithm o Chaudhuri et al. [3] starting with this tree will not be able to improve the maximum degree. In act, a slightly modiied instance, G, where several o the non-tree edges are incident on the same node on the path, is not improvable beyond O(d). Previous techniques do not discriminate between dierent nodes with degree less than d 1 and hence cannot distinguish between G and G. On the other hand, our MDMST algorithm, described in Section 3, may perorm a swap that improves the degree o a degree-d node by creating one or more new degree-d nodes. In turn, it attempts to improve the degree o these new degree-

4 4492 K. Chaudhuri et al. / Theoretical Computer Science 410 (2009) Fig. 1. Graph G and a locally optimal tree. The shaded triangles at each level represent subtrees identical to the explicit subtree (on the let) rooted at the same level. The bold nodes represent a path and the bold dotted edges correspond to a set o edges going to similar nodes in the subtrees denoted by shaded triangles. d nodes, which cannot necessarily be improved independently since their improvements may rely on the same edge or use edges that are incident to the same node. Moreover, this eect snowballs as more and more degree-d nodes are created. As previously mentioned, Goldberg s push relabel ramework helps us tame this beast o a process. A high-degree node may only relieve a unit o excess degree using a non-tree edge that is incident to nodes o lower labels. Thus, while two high-degree nodes may be created by a swap, at least they are guaranteed to have lower labels than the label o the node initiating the swap. While the algorithm may end up undoing a previous swap, the labels ensure that this process cannot continue indeinitely. We deine a notion o a easible labeling and prove that our MDMST algorithm maintains one. During the course o the algorithm, there is eventually a label p such that the number o nodes with label at least p is not much larger than the number o nodes with label at least p + 1. We use easibility to show that all nodes with labels p and higher must have high average degree in any MST, thus obtaining a lower bound on opt (G). This degree lower bound also holds or any ractional MST o the graph G. Combining our MDMST algorithm with the cost-bounding techniques o Könemann and Ravi [15] gives our result or the BDMST problem (Section 7) Organization o the paper The rest o the paper is organized as ollows. Section 2 deines the notation used throughout the paper, and Section 3 describes our push relabel algorithm or the MDMST problem. We give two dierent (and incomparable) bounds on the perormance o the algorithm in Sections 4 and 5. In Section 5.2, we use a result o Goemans to derive an algorithm with an additive error o 2. Section 6 generalizes our results to the MDMCB problem. Finally, in Section 7, we give our results or the BDMST problem. 2. Deinitions and notation For a graph G = (V, E) and an MST T o G, the degree (T) o T is deined to be the maximum over nodes u in V, o the degree o u in T. When T is obvious rom context, we simply write its degree as. For a subset F E o edges and a subset U V o nodes, let i F (U) denote the set o edges in F that have both endpoints in U, and let ϑ F (U) denote the set o edges in F incident on U, i.e. i F (U) = {(u, v) F : u, v U} and ϑ F (U) = {(u, v) F : u U or v U}. Finally, δ F (u) denotes the set o edges in F incident on a vertex u, i.e. δ F (u) = {e F : u e}. Let N be the set o nonnegative integers. A labeling l o the nodes is a unction l : V N. For a labeling l and an integer p, let level p be deined as the set {v : l(v) = p} o nodes that have label p, and let W p = {v : l(v) p} be the set o nodes with labels at least p. For a real number µ 1, level p is called µ-sparse i Wp µ Wp+1. Given an MST T, we deine a swap in T to be a pair o edges (e, e ) such that e T, e T, c(e) = c(e ), and e lies on the unique cycle o T { e }. Note that i (e, e ) is a swap in T, then T \ {e} { e } is also an MST o G. For a node u and a tree T, let S T u denote the set o swaps (e, e ) in T such that e is incident on u and e is not incident on u. We call a swap (e, e ) in S T u useul or u because it can be used to decrease the degree o u; i.e. the degree o u in T \ {e} { e } is one less than that in T. Given a labeling l on V, we extend it to a labeling on E by deining l(e) = max{l(u), l(v)} or e = (u, v). We say that a labeling l is easible or a tree T i or all nodes u V, or every swap (e, e ) S T u, l(e) l(e ) + 1. Given a labeling l and an MST T, a swap (e, e ) S T u is called permissible or u i l(u) l(e ) + 1. We show in Section 3.2 that our algorithm maintains a easible labeling. Consequently, each permissible swap (e, e ) S T u satisies l(u) = l(e ) + 1. We deer the deinitions or the MDMCB problem to Section 6.1.

5 K. Chaudhuri et al. / Theoretical Computer Science 410 (2009) Algorithm pr-mdmst(g, µ) T arbitrary MST o G. Repeat maximum degree over nodes in T. Initialize labels to 0. Put excess o 1 on nodes with degree. Put excess o 0 on all other nodes. Repeat p lowest level that contains an overloaded node. Select the set U p overloaded nodes with label p. I there is a node u U p that has a permissible, useul swap (e, e ) where e = (u, v) T T \ {e} {e }; Set excess on the endpoints o e to 0; For each endpoint o e that has degree. 3 set its excess to 1. else Relabel all nodes in U p to p + 1. until there are no more overloaded nodes or there is a µ-sparse level. until there is a µ-sparse level p. Let F T be the edges in T not incident on W p +1. Output tree T and the pair W p = (F, W p ). Fig. 2. A push relabel algorithm or the MDMST problem. 3. Minimum-degree MSTs MDMST problem: Given a weighted graph G = (V, E, c), ind an MST T o G such that max v V {deg T (v)} is minimized The push relabel MDMST algorithm Starting with an arbitrary MST o the graph, our algorithm runs in phases. The idea is to reduce the maximum degree o the tree in each phase using a push relabel technique. I we ail to make an improvement in some phase, we ind a certiicate o near-optimality. More ormally, let i be the maximum degree o any node in the tree T i at the beginning o phase i, also called the i - phase. During the i -phase, either we modiy T i to get T i+1 such that the maximum degree in T i+1 is less than i, or we output a certiicate that i is close to optimal. For a general phase o the algorithm, let T be the tree at the beginning o the phase, and let be the degree o T. The algorithm maintains a labeling l. The algorithm maintains the invariant that the labeling l is easible with respect to the current tree T. This notion o easibility is crucial in establishing a lower bound on the optimal degree when the algorithm terminates. In addition, each node is given an initial excess. The excess o a node is one i its degree is, and zero otherwise. As we show in Lemma 5, the algorithm maintains the invariant that the degree o each node is at most. For notational convenience, we call a node overloaded i it has positive excess. The pr-mdmst algorithm takes as input a graph G and a real-valued parameter µ 1. The parameter µ determines the termination condition o the main loop o the algorithm. In Sections 4 and 5, we derive two dierent bounds on the approximation ratio o the algorithm; the parameter µ is chosen appropriately in the two cases. We now describe a general phase o the algorithm. See Fig. 2 or a ormal description. The phase proceeds as ollows: The label l(u) o each node u is initialized to zero. The excess o each node o degree is initialized to one; the excess o every other node is initialized to zero. Let p be the label o the lowest level containing overloaded nodes. I there is an overloaded node u in level p that has a permissible, useul swap (e, e ) S T u, modiy T by deleting e = (u, v) and adding e = (u, v ). Then decrease the excess on u by one; i v has positive excess, decrement its excess as well. I u now has degree or more, add one to its excess; i v has degree or more, add one to its excess. I no overloaded node in level p has a permissible, useul swap, then relabel to p + 1 all overloaded nodes in level p. Repeat this loop until either there are no overloaded nodes or there is a µ-sparse level. Note that i the phase ends or the ormer reason, then the tree at the end o the phase has maximum degree at most 1. In the latter case, we show that is close to the optimal degree. 3 We prove in Lemma 5 that the degree o any node never goes above.

6 4494 K. Chaudhuri et al. / Theoretical Computer Science 410 (2009) a c b Fig. 3. Proo o Lemma 1. d I some node gets label n, there is guaranteed to be a 1-sparse level. Thus each node gets relabeled at most n times per phase, or any choice o the input parameter µ 1. The total number o iterations o the inner loop in any phase o the algorithm is thereore bounded by n 2. Since each phase (except the last) decreases the maximum degree o T by one, there are at most n phases. The algorithm thereore runs in polynomial time. The algorithm outputs a tree T and a pair W p = (F, X), where F is a orest on G and X is a subset o nodes. In the rest o this section, we show how to interpret (F, X) as a certiicate that the degree o T is close to opt (G). We do this in two dierent ways, in Sections 4 and 5, leading to two incomparable bounds on the approximation ratio o the algorithm. Remark: In Section 4, we set µ to a constant larger than 2. In this case, the number o relabels per node is bounded by log 2 n, resulting in a aster algorithm Feasibility We irst prove a crucial lemma. Lemma 1. The algorithm always maintains a easible labeling. Proo. We prove this by induction on the number o iterations in a phase. At the beginning o any phase, all labels are zero, which is a easible labeling. In one step o the algorithm, we either update a label or perorm a permissible swap (e, e ). Since we increment the label o a node only when it has no permissible swaps, easibility is maintained in the irst case. In the second case, since we change the structure o the tree, the set o available swaps may change. Consider a easible swap (e, e ), where e = (u, v) and e = (u, v ), and the swap is permissible or u or v (or both). Let T be the tree beore the (e, e ) swap, and let T be the tree ater the swap. Consider a swap (, ) in T. To show that easibility is maintained, we need to show that l( ) l( ) + 1. I the swap (, ) already exists in T, easibility holds inductively. However, the swap may have been missing in the tree T, but may appear in tree T or one o the ollowing three reasons. T and hence not available or the swap: See Fig. 3(a). In this case, = e. The cycle ormed by adding to T includes e and, otherwise T already has a cycle. Moreover c( ) = c( ) = c(e ). Thereore (, e ) is a swap in T, and easibility in T implies that l( ) l(e ) + 1. On the other hand, since (e, e ) is a permissible swap in T, l(e) l(e ) + 1. Thus l( ) l(e ) + 1 l(e) = l( ). T and hence not available or the swap: See Fig. 3(b). In this case, = e. As (e, e ) is a swap in T and (e, ) is a swap in T, the cycle ormed by adding to T includes e, and (e, ) is a valid swap in T. l(e) l( ) + 1 by easibility o T, and l(e) = l(e ) + 1 by permissibility. Thus l( ) = l(e ) l( ). T and T : See Fig. 3(c) and 3(d). Since (e, e ) is a swap in T, there is a unique cycle in T e that contains e. I does not lie on this cycle, as illustrated in Fig. 3(c), the swap (, ) already exists in T and the claim holds by induction. Otherwise the cycle in T e contains, and the only structure in which this happens is illustrated in Fig. 3(d). Since e is the missing edge in T o the cycle in T e, it must be the case that c(e ) c( ) = c( ), or else T \ { } {e } would have strictly smaller cost than the MST T. Similarly, since is the missing edge o the cycle in T, c( ) c(e) = c(e). Thereore c(e) = c(e ) = c( ) = c( ). Moreover, (, e ) and (e, ) are both available swaps in T. Thus l( ) l(e ) + 1 = l(e) l( ) + 1. We have shown that, in all cases, the swap (, ) is easible. Hence the induction holds.

7 K. Chaudhuri et al. / Theoretical Computer Science 410 (2009) The witness In this section, we introduce the notion o a witness. A witness is a combinatorial structure produced by our algorithms at termination to guarantee the near-optimality o the output. Our witness consists o a orest F E that is contained in some MST o G, along with a subset X V. A pair (F, X) is a witness i it has the ollowing property: For every MST T o G containing F, every edge in T \ F is incident on X. Lemma 2 ([6]). Let W = (F, X) be a witness or a graph G = (V, E) as deined above. Then any (ractional) minimum spanning tree o G has maximum degree at least V F 1. X Proo. Consider an MST T o G, and let T be an MST containing F that has maximal intersection with T. By the exchange property, T \ F is contained in T. The witness property implies that every edge in T \ F is incident on X. Since there are V F 1 edges in T \ F, the average degree o X in T \ F, and thereore in T, is at least V F 1. Since a ractional MST X is a convex combination o integral ones, the claim ollows. We now show that the pair (F, X) output by the algorithm pr-mdmst is a witness. Lemma 3. Let T be the MST o G and l : V N the labeling when the algorithm pr-mdmst terminates. For any integer p, let F be the subset o edges in T that are not incident on W p+1, and let X = W p. Then W p = (F, X) is a witness. Proo. Assume the contrary, and let T be an MST o G that contains F and also contains an edge e F not incident on W p. By the exchange property, there is an edge e T \ F such that (e, e ) is a swap in T. Since e T \ F, it is incident on W p+1 and thus l(e) p + 1. On the other hand, e is not incident on W p and thus l(e ) p 1. This, however, contradicts the easibility o the labeling Involuntary losses Let p be the µ-sparse level used by the algorithm to compute a witness. From Lemma 2 and Lemma 3, it ollows that any (ractional) MST o G has degree at least V F 1. This ratio can be rewritten as ( V F 1) W p +1. Note that the numerator W p W p +1 W p o the irst term is precisely the number o edges incident on W p +1 in T. The second term is bounded by 1, where µ is the µ sparseness o level p. The next lemma ollows immediately. Lemma 4. Let T be the MST o G, l : V N the labeling, and p the µ-sparse level when the algorithm pr-mdmst terminates. Let ϑ T (W p +1) be the set o edges in T incident on W p +1. Then any (ractional) MST o G has degree at least 1 µ ϑ T (W p +1 ). W p +1 Thus, to prove a lower bound on opt (G), we need to lower bound ϑ T (W p +1). Towards this end, we distinguish between two dierent ways a node can lose degree during the course o the algorithm. We say that a swap (e, e ) executed by the algorithm causes a loss in degree to a node u i e is incident on u. A loss in degree to a node u that is caused by a swap (e, e ) is called a voluntary loss i u is overloaded beore the swap is executed; otherwise it is called an involuntary loss. By deinition, voluntary losses do not decrease the degree o a node below 1. Note that every swap (e, e ) executed by the algorithm causes a voluntary loss to at least one endpoint o e (and an involuntary loss to at most one endpoint o e). Suppose the algorithm terminates with a µ-sparse level in the -phase. The last time it is relabeled, each node in W p +1 has degree at least and is thereore overloaded. I each node in W p +1 suered only voluntary losses in degree since its last relabeling, then its degree in T would be at least 1. However, a node in W p +1 may also suer rom involuntarily losses, which may decrease its degree arbitrarily. Hence a node-by-node analysis is insuicient. To get a lower bound on the average degree o W p +1 in T, we instead bound the total number o involuntary losses to nodes in W p +1. We do this in two dierent ways in the next two sections. 4. A constant-actor approximation In this section, we show that the algorithm outputs a tree T o degree 2 opt (G) + O( ). To bound the number o involuntary losses, we deine a partitioning o the swaps executed by the algorithm into cascades. Each cascade can be charged to a relabel, which enables us to bound the number o involuntary losses to W p +1 in terms o the size o this set Cascades Recall that, or an integer p, U p is deined to be the set o overloaded nodes in level p. For the purpose o analysis, we introduce the notion o lagging a node. The lag indicates that the node has been relabeled but its excess has not been

8 4496 K. Chaudhuri et al. / Theoretical Computer Science 410 (2009) removed. In addition, we give each node an overloading-swap ield. The overloading-swap ield o a node u points to the swap that put excess on it ater its last relabel. We start with all the lags cleared and all overloading-swap ields set to null. In each iteration o the -phase o the algorithm, we ind the lowest p such that U p is non-empty, i.e. there is an overloaded node with label p. I we can ind any swap (e, e ) that is permissible and useul or a node in U p, we execute the swap and clear lags (i set) on the endpoints o e. Moreover, or each endpoint u o e that is now overloaded, we set the overloadingswap ield o u to (e, e ). I there is no swap that is permissible and useul or any node in U p, we increment the label, set the lag, and clear the overloading-swap ield or each node in U p. The ollowing lemma shows that no node has excess larger than one during the course o the algorithm, which implies, in particular, that the overloading-swap ield is never overwritten beore it is cleared to null. Lemma 5. During the -phase, no node ever has degree more than. Proo. We use induction on the number o swaps executed during the phase. In the beginning o the phase, the maximum degree is. Any swap (e, e ) decreases the degree o a node in U p and adds at most one to the degree o a node with strictly lower label. By choice o p, all nodes with lower labels have degree at most 1 beore the swap. Since a swap adds at most one to the degree o any vertex, the induction holds. The lemma ollows. We deine the label o a swap (e, e ) to be the label o e when the swap is executed. We call a swap a root swap i it is useul or a lagged node. Note that a lagged node has its overloading-swap ield set to null. Let (e, e ) be a non-root swap that occurs in the sequence o swaps executed by the algorithm. The swap (e, e ) is executed in order to relieve the excess o an endpoint u o e. Let (, ) be the swap pointed to by the overloading-swap ield or node u when swap (e, e ) is executed. Thus (, ) is the last swap in the sequence that increases the degree o u and precedes (e, e ). We call (, ) the parent swap o (e, e ). Note that the lagging procedure ensures that every non-root swap executed by the algorithm has a parent. A label-p swap, by deinition, reduces the degree o a node with label p. Since excess lows rom a higher-labeled node to a lower-labeled node, the label o every non-root swap is strictly smaller than the label o its parent. The parent relation naturally deines a directed graph on the set o swaps, each component o which is an in-tree rooted at one o the root swaps. We deine a cascade to be the set o swaps in a component o this Directed Acyclic Graph. In other words, a cascade corresponds to the set o swaps sharing the same root swap as an ancestor. Note that the cascades may be interleaved in the sequence o swaps executed by the algorithm. The label o a cascade is deined to be the label o the root swap in it. Each swap is the parent o at most two swaps which each have a strictly smaller label. Thus it ollows that: Lemma 6. A label-p cascade contains at most 2 p q label-q swaps. We say that an involuntary loss is contained in a cascade i some swap in the cascade causes it. Since each swap causes at most one involuntary loss, the lemma above implies: Corollary 7. A label-p cascade contains at most (2 p q+1 1) involuntary losses to nodes with labels at least q. Proo. An involuntary loss to a label-r node must be caused by a swap with label at least r. The bound ollows by summing the number o swaps with label r, or r between q and p Computing the approximation ratio Armed with the bound o Corollary 7, we now proceed to lower bound opt (G). Recall rom Section 3.4 that it suices to lower bound the average degree o W p +1 in T, where W p is a µ-sparse level and T is the tree output by the algorithm. Lemma 8. Let p be an integer greater than p. Then Wp > µ Wp+1. Proo. Each iteration o a phase o the algorithm decreases the size o at most one level p and increases the size o the level p + 1. Thus the only level that can go rom not being µ-sparse to being µ-sparse is level p. Since the algorithm terminates as soon as it inds a µ-sparse level, it terminates with exactly one µ-sparse level. Lemma 9. The number o involuntary losses to nodes in W p +1 is at most 2 W p +1 ( µ µ 2 ). Proo. Each involuntary loss to a node in W p +1 occurs in a cascade, and by Corollary 7, the number o involuntary losses to nodes in W p +1 in a label-p cascade is at most 2 p p. The total number o involuntary losses to nodes in W p +1 during the course o the phase is at most 2 p p W p < Wp 2 p p +1 p p +1 p p +1 µ p p 1 ) p p 1 = ( 2 Wp 2 +1 p p µ +1 ( ) µ = 2 W p +1. µ 2

9 K. Chaudhuri et al. / Theoretical Computer Science 410 (2009) We are now ready to establish the approximation ratio o the algorithm. Theorem 10. Given a graph G and a constant µ > 2, the pr-mdmst algorithm obtains in polynomial time an MST o degree, where µ opt (G) µ µ 2. Proo. The pr-mdmst algorithm, when executed on graph G, terminates with a tree T and a pair W p = (F, X). We now compute the number ϑ T (W p +1) o edges incident on Wp +1 in T. Each node in W p +1 has degree at least ( 1) ater its last voluntary loss, and it may then suer some involuntary losses. Using the bound rom Lemma 9, the sum o degrees o nodes in W p +1 in T is at least ( 1 2µ ) W µ 2 p +1. Since there are at most Wp +1 1 edges in T that have both endpoints in W p +1, the number o edges in T that are incident on W p +1 is at least ( 2 2µ ) W µ 2 p +1. Thus rom Lemma 4, ( opt (G) 2 2µ ) 1 µ 2 µ. Rearranging, we get µ opt (G) µ µ 2. Setting µ to be 2 + 2, we get opt (G) 2 opt (G) + 4 opt (G) + 4 Corollary 11. Given a graph G, there is a polynomial-time algorithm that outputs an MST o degree, where 2 opt (G) + O( opt (G)). 5. An additive approximation In this section, we show a dierent upper bound on the perormance o the algorithm that is better than the bound in the previous section i the graph G is everywhere-sparse, i.e. or every subset U o nodes, the induced subgraph on U is sparse. More precisely, or a graph G, let the local density s(g) be deined as the density o the densest subgraph o G: s(g) = }. We show that on input G and µ = 1, the pr-mdmst algorithm outputs an MST o degree at most opt (G) { ie (U) max U V U + s(g). In Section 5.2, we combine this result with a lemma rom Goemans [8] to get a ( opt (G) + 2)-algorithm A density-based bound Let T be the tree and W p the witness output by the pr-mdmst algorithm on input G and µ = 1. Let be the degree o T. Since µ = 1, the sets W p and W p +1 must be equal, and hence, level p is empty when the algorithm terminates. The ollowing lemma bounds ϑ T (W p +1). Lemma 12. Let T be the MST, l the labeling, and p the empty level when the algorithm pr-mdmst terminates. Let i E (W p +1) be the set o edges in G that have both endpoints in W p +1, and let ϑ T (W p +1) be the set o edges in T that are incident on W p +1. Then ϑ T (W p +1) ( 1) Wp ie (W p +1). Proo. For a node u W p +1, let δ T (u) T be the set o edges in T incident on u, and let δ L (u) E be the set o edges that node u loses involuntarily ater its last voluntary loss. Since each node has degree at least 1 ater its last voluntary loss, it ollows that δ T (u) 1 δ L (u) \ δ T (u). Moreover, i v is the last node to be relabeled, δ T (v). Thus the sum o degrees o nodes in W p +1 in T is at least ( 1) Wp u W p +1 δ L (u) \ δ T (u). An edge (u, v) may be in δ L (u) or δ L (v) but not both. It ollows that u W p +1 δ L (u) \ δ T (u) u Wp +1 (δ L (u) \ δ T (u)). Recall that every involuntary loss to a node in W p +1 comes rom an edge in i E (W p +1). Thus each edge in u Wp +1 (δ L (u) \ δ T (u)) is in i E (W p +1) \ i T (W p +1). An edge (u, v) in i T (W p +1) contributes to both δ T (u) and δ T (v). Thus the number o edges in T incident on W p +1 is δ T (u) it (W p +1) ( 1) Wp u Wp +1 (δ L (u) \ δ T (u)) it (W p +1) u W p +1 ( 1) Wp ie (W p +1) \ i T (W p +1) it (W p +1) ( 1) Wp ie (W p +1).

10 4498 K. Chaudhuri et al. / Theoretical Computer Science 410 (2009) We now use the bound on ϑ T (W p +1) in Lemma 12 to prove a lower bound on opt (G). Theorem 13. Let T and W p be the output o the pr-mdmst algorithm given a graph G and µ = 1. Then the degree o T is bounded by opt (G) + s(g). Proo. By Lemma 12, ϑ T (W p +1) ( 1) Wp ie (W p +1) ie (W p +1 ). By the deinition o local density, s(g). W p +1 Lemma 4 then implies that opt (G) s(g) Since opt (G) is an integer, we conclude that opt (G) W p +1 s(g) An additive actor o 2 Goemans [8] shows that the support o the natural linear program rom the MDMST problem is sparse. He considers the ollowing linear program: min c(x) = e c e x e subject to: x(i E (S)) S 1 x(e) = V 1 x(δ E (v)) k x e 0 S V v V e E The above linear program is easible or an integer k i and only i opt (G) k. Let x be an optimal extreme-point solution to the above linear program or a graph G and or k = opt (G). Let E denote the support o x, i.e. E = {e E : x e > 0}. Theorem 5 in Goemans [8] can be paraphrased as: Theorem 14 (Goemans [8, Theorem 5]). The local density o the graph G = (V, E ) is less than 2. We irst argue that opt (G) = opt (G ). Since x e = 0 or all e E, it ollows that x is a easible solution to the linear program or G = (V, E ), and so opt (G ) opt (G). Since G is a subgraph o G, opt (G ) opt (G). We conclude that opt (G) = opt (G ). Since the linear program can be solved eiciently (see, e.g., Schrijver [21], Section 40.3), we can compute the graph G in polynomial time. Our ( opt (G) + 2)-algorithm computes the graph G and then runs the algorithm pr-mdmst with input G and µ = 1. By Theorem 13, the tree T output by the algorithm has degree most opt (G) + 2. Theorem 15 summarizes. Theorem 15. Given a graph G, there is a polynomial-time algorithm that computes an MST o G with degree at most opt (G) Minimum-degree minimum-cost base in a matroid In this section, we consider a generalization o the MDMST problem Deinitions Recall that a matroid is deined to be a pair M = (E, I), where E is a ground set o elements and I is a amily o independent sets such that (i) I, (ii) A I, B A imply that B I, and (iii) A, B I, A > B imply that there exists e A \ B with B {e} I. A maximum-cardinality independent set o M is called a base o M. Let c : E R + be a non-negative cost unction on the ground set o M. A base T o M that minimizes the cost c(t) = e T c(e) is reerred to as a minimum-cost base (MCB). Let xt {0, 1} E be the incidence vector o an MCB T. We say a vector x [0, 1] E is a ractional MCB i it is the convex combination o incidence vectors o MCB s. Recall that a hypergraph G = (V, E) consists o a set o nodes V and hyperedges E 2 2V, i.e. each hyperedge is a subset o vertices. We say G is a k-ary hypergraph i the cardinality o each edge in E is at most k. I u e, we say that node u is an endpoint o e and that e is incident on u. For a subset F E o edges and a node u V, let δ F (u) be the set {e F : u E} o edges incident on u. We deine the degree o u in F to be δ F (u). The degree o F is then deined as the maximum over u V o the degree o u in F, i.e. max u {e F : u e}. Further, or a subset o vertices U V, let i F (U) and ϑ F (U) denote the sets {e F : e U} and {e F : e U }, respectively. We can now ormally deine the minimum-degree minimum-cost base (MDMCB) problem. The input to the problem is a k-ary hypergraph G = (V, E) and a matroid M with E as its ground set.

11 K. Chaudhuri et al. / Theoretical Computer Science 410 (2009) MDMCB problem: Given a k-ary hypergraph G = (V, E) and a weighted matroid M = (E, I, c), ind an MCB o M that has minimum degree in G. Let opt (G, M) denote the optimal degree or an instance o MDMCB The MDMCB algorithm Our algorithm or MDMCB is a generalization o the pr-mdmst algorithm. To deine it ormally, we irst deine some analogous concepts. Given an MCB T o a matroid M, a pair (e, e ) o elements in E is a swap in T i e T, e T, and T \ {e} { e } is also an MCB o M. The label and excess o a node in V can be deined exactly as in Section 3. A labeling l : V N is extended to a labeling on the ground set E as ollows: or e E, l(e) = max u e l(u). For a node u V, we let S T u denote the set o swaps (e, e ) in T such that e is incident on u but e in not incident on u. We call a swap (e, e ) in S T u useul or u because it can be used to decrease the degree o u. We say a swap (e, e ) in T is easible or a labeling l i l(e) l(e ) + 1. As in Section 3, a labeling l : V N is deined to be easible or an MCB T i or all nodes u V, or all swaps (e, e ) S T u, l(e) l(e ) + 1. Given a labeling l and an MCB T, a swap (e, e ) S T u is called permissible or u i l(u) l(e ) + 1. Our pr-mdmcb algorithm is deined exactly as the push relabel algorithm described in Fig. 2, except that it takes as input a k-ary hypergraph G = (V, E) and a weighted matroid M on the ground set E, in addition to the parameter µ. To prove that the pr-mdmcb algorithm works correctly, we irst show that easibility is maintained in any iteration o the algorithm Feasibility Lemma 16. The pr-mdmcb algorithm always maintains a easible labeling. Proo. As in Lemma 1, we show this by induction on the number o iterations in a phase. In the beginning o any phase o the algorithm, all nodes and hence all edges e have label 0, which is a easible labeling. In one step o the algorithm, we either update a label or execute a permissible swap (e, e ). Since we increment the label o a node only when it has no permissible swaps, easibility is maintained in the irst case. To prove that easibility is maintained when a permissible swap is executed, we make use o the rank unction r : 2 E N o the matroid M. For a subset E E, the rank r(e ) is deined to be max F E :F I F. For any subsets A, B E, the rank unction r satisies (i) r(a) A, (ii) r(a) r(b) i A B, and (iii) r(a B) + r(a B) r(a) + r(b). Note that A is a base o M i and only i r(a) = r(m). Let T be the current MCB ater executing a sequence o swaps, so that the labeling l is easible or T. Let (e, e ) be the permissible swap executed in the current iteration, and let (, ) be a swap in T = T \ {e} { e }. I (, ) is a swap in T, then by the inductive condition, l( ) l( ) + 1. Thus it remains to consider a pair (, ) that is a swap in T but not in T. There are three cases: T and hence not available or the swap in T : In this case, e =. We observe that T \ { } { e } = (T \ {e} { e } ) \ { } { } = T \ { } { }, which is an MCB o M since (, ) is a swap in T. Thus the pair (, e ) is a swap in T. By easibility o (, e ) and permissibility o (e, e ), we conclude that l( ) l(e ) + 1 = l(e) = l( ). T and hence not available or the swap in T : In this case e =. We observe that T \ {e} { } = (T \ {e} { e } ) \ { } { } = T \ { } { }, which is an MCB o M since (, ) is a swap in T. Thus the pair (e, ) is a swap in T. By permissibility o (e, e ) and easibility o (e, ), we conclude that l( ) = l(e ) = l(e) 1 l( ). T and T : Note that since (, ) is not a swap in T but is a swap in T, T \ { } { } is not a base o M. Thus r(t \ { } { } ) = r(m) 1. We irst claim that T \{e} { } is a base o M. Suppose not. Then r(t \{e} { } ) = r(m) 1. By submodularity o the rank unction, r(t \ {e, } { } ) + r(t { } ) r(t \ {e} { } ) + r(t \ { } { } ). Thus r(t \ {e, } { } ) = r(m) 2 so that r(t \ {e, } { e, } ) r(m) 1, which contradicts that act that (, ) is a swap in T. We now argue that T \ { } { e } is also a base o M. Suppose not. Then r(t \ { } { e } ) = r(m) 1. Once again submodularity tells us that r(t \ { } { e, } ) + r(t \ { }) r(t \ { } { e } ) + r(t \ { } { } ) 2r(M) 2. Thus r(t \ { } { e, } ) r(m) 1. This then implies that r(m) = r(t \ { } { } ) = r(t \ {e, } { e, } ) r(m) 1, which again contradicts the act that (, ) is a swap in T. Since T, T \{e} { }, and T \{ } { e } are all bases o M, and T is an MCB, it ollows that c( ) c(e) and c(e ) c( ). Since c(e) = c(e ) and c( ) = c( ), we conclude that c(e) = c( ). Thus (e, ) and (, e ) are both swaps in T. By easibility o these swaps and permissibility o (e, e ), we conclude that l( ) l(e ) + 1 = l(e) l( ) + 1. We have shown that, in all cases, the swap (, ) is easible. Hence the induction holds.

12 4500 K. Chaudhuri et al. / Theoretical Computer Science 410 (2009) The witness The concept o the witness generalizes to the MDMCB problem. We deine a witness to be a pair (F, X) where F E is a subset o some MCB and X V has the property that in any MCB T o M containing F, each edge in T \ F is incident on X in G. The ollowing lemma is analogous to Lemma 2 and ollows rom essentially the same arguments. Lemma 17. Let W = (F, X) be a witness as deined above or a hypergraph G and a matroid M. Then any (ractional) MCB o M has maximum degree at least r(m) F in G. X 6.5. A constant-actor approximation We now give generalizations o other results rom Sections 3 and 4. Generalizations o Lemmas 3 5 and 8 are immediate. The observation that each swap is a parent o at most k swaps leads to Lemma 18, which is an analogue o Lemma 6. Corollary 19 is analogous to Corollary 7. Lemma 18. A label-p cascade contains at most k p q label-q swaps. Corollary 19. A label-p cascade contains at most (k p q+1 1) involuntary losses to nodes with labels at least q. Proo. From Lemma 18, the total number o swaps with label at least q in a label-p cascade is at most kp q+1 1. The corollary k 1 ollows rom the act that each swap causes at most (k 1) involuntary losses. Lemma 20. The number o involuntary losses to nodes in W p +1 is at most k W p +1 ( µ µ k ). The proo o Lemma 20 is analogous to the proo o Lemma 9. We conclude with an approximation guarantee or the pr-mdmcb algorithm: Theorem 21. Given a k-ary hypergraph G = (V, E), a weighted matroid M = (E, I, c), and a real-valued parameter µ > k, the pr-mdmcb algorithm obtains in polynomial time an MCB o degree, where kµ opt (G, M) kµ µ k. Proo. On input(g, M, µ), the pr-mdmcb algorithm terminates with a maximum independent subset T and a pair W p = (F, X). We now compute the number ϑ T (W p +1) o edges in T incident on Wp +1. Each node in W p +1 has degree at least ( 1) ater its last voluntary loss, and it may then suer some involuntary losses. Using the bound rom Lemma 20, the total loss in degree to W p +1 rom involuntary losses is kµ Wp µ k +1. The sum o degrees in T o nodes in Wp +1 is thereore at least ( 1 kµ ) Wp µ k +1. Since each edge is incident on at most k nodes, the number ϑ T (W p +1) o edges in T incident on W p +1 is at least ( 1 ) times this sum o degrees. Thus rom the generalization o Lemma 4, k ( opt (M) 1 kµ ) 1 µ k kµ. Rearranging, we get kµ opt (M) kµ µ k. Remark: The bound above does not strictly generalize the bound in Theorem 10 since we do not have the structure necessary to bound the number o edges internal to W p +1 as we did in the proo o Theorem 10. Instead we use a cruder argument to lower bound the cardinality o ϑ T (W p +1). Choosing µ = k + k, we get the ollowing bound. opt (G,M) Corollary 22. Given a k-ary hypergraph G = (V, E) and a matroid M on E, there is a polynomial-time algorithm that outputs an MCB o degree, where k 2 opt (G, M) + 2k 3 2 opt (G, M) + k A density-based bound In this section, we generalize the result o Section 5.1. We irst deine a notion o density or a hypergraph. Given a e ϑ hypergraph G = (V, E) and a subset U V, we deine the density o U to be the ratio E (U) ( e U 1) U. In other words, each edge e incident on U contributes e U 1 to the numerator. The local density s(g) o a hypergraph G is then deined to be e ϑ the maximum density o any subset o V, i.e. s(g) = max E (U) ( e U 1) U V U. Let T be the tree and W p the witness output by the pr-mdmcb algorithm on input (G, M) and µ = 1. Let be the degree o T. Since µ = 1, the sets W p and W p +1 must be equal, and hence, level p is empty when the algorithm terminates. The ollowing lemma is an analogue o Lemma 12.

13 K. Chaudhuri et al. / Theoretical Computer Science 410 (2009) Lemma 23. Let T be the MCB, l the labeling, and p the empty level when the algorithm pr-mdmcb terminates. Let ϑ T (W p +1) be the set o edges in T that are incident on W p +1. Then ϑ T (W p +1) ( 1) Wp u ϑ E (W p +1 ) ( e Wp +1 1). Proo. For a node u W p +1, let δ T (u) T be the set o edges in T incident on u, and let δ L (u) E be the set o edges that node u loses involuntarily ater its last voluntary loss. Since each node has degree at least 1 ater its last voluntary loss, it ollows that δ T (u) 1 δ L (u) \ δ T (u). Moreover, i v is the last node to be relabeled, δ T (v). Thus the sum o degrees o nodes in W p +1 in T is at least ( 1) Wp u W p +1 δ L (u) \ δ T (u). An edge e may be in δ L (u) or at most e Wp +1 1 nodes u e. It ollows that δ L (u) \ δ T (u) ( e Wp +1 1). u W p +1 e u Wp (δ +1 L (u)\δ T (u)) Moreover, u Wp +1 (δ L (u) \ δ T (u)) is a subset o ϑ E (W p +1) \ ϑ T (W p +1). An edge e in ϑ T (W p +1) contributes to δ T (u) or each endpoint u e W p +1, and thus is counted e Wp +1 times in the sum o degrees o nodes in W p +1 in T. Thus the number ϑ T (W p +1) o edges in T incident on Wp +1 is δ T (u) ( e Wp +1 1) u W p +1 e ϑ T (W p +1 ) ( 1) Wp δ L (u) \ δ T (u) ( e Wp +1 1) u W p +1 e ϑ T (W p +1 ) ( 1) Wp ( e Wp +1 1) ( e Wp +1 1) = ( 1) Wp u ϑ E (W p +1 )\ϑ T (W p +1 ) u ϑ E (W p +1 ) ( e Wp +1 1). e ϑ T (W p +1 ) We now use the above bound on ϑ T (W p +1) to prove a lower bound on opt (G, M). Theorem 24. Let T and W p be the output o the pr-mdmcb algorithm given a hypergraph G, a matroid M, and µ = 1. Then the degree o T is at most opt (G, M) + s(g). Proo. From Lemma 23, ϑt (W p +1) ( 1) Wp u ϑ E (W p +1 ) ( e Wp +1 1). By the deinition o local density, the ratio u ϑ E (W p +1 ) ( e Wp 1) +1 W p +1 is at most s(g). An analogue o Lemma 4 then implies that opt (G, M) s(g) Since opt (G, M) is an integer, we conclude that opt (G, M) s(g). W p +1 We note that we are not aware o an analogue o Theorem 14 or the MDMCB problem. Such an analogue would imply an additive approximation via Theorem Bounded-degree minimum spanning trees BDMST problem: Given a weighted graph G = (V, E, c), c : E R +, and a positive integer B 2, ind a minimum-cost spanning tree in the set {T : v V, deg T (v) B}. Könemann and Ravi [15] show that an MDMST or a certain cost unction is also a BDMST with related guarantees on degree and with cost within a constant actor o the optimum. For the sake o completeness, we present this argument below Linear programs or BDMST An integer linear program or the BDMST problem is given by opt B = min c e x e e E such that x e B v V e δ(v) (1) x sp G x e {0, 1}

14 4502 K. Chaudhuri et al. / Theoretical Computer Science 410 (2009) where δ(v) is the set o edges o E that are incident to v, and sp G is the convex hull o edge-incidence vectors o spanning trees o G. A tree deined by a vector x sp G, the entries o which are not necessarily all integer, is called a ractional spanning tree and can be written as a convex combination o spanning trees o G. For a ractional tree T with edge incidence vector x sp G, let deg T (v) = e δ(v) x e. Now we can write the linear program relaxation o (1) as min{c(t ) : T sp G, v V deg T (v) B}. (2) The approach used by Könemann and Ravi [15] is to take the Lagrangean dual o (2), given by max λ 0 min {c(t ) + λ v (deg T (v) B)}. T sp G v V (3) We can think o the optimal solution to the dual as a vector λ B o Lagrangean multipliers on the nodes and a set O B o optimal trees, such that every tree T B O B minimizes c(t B ) + v V λ B v (deg T B (v) B). The optimal multipliers λ B and set O B o trees can be computed in polynomial time. The optimal value opt LD(B) o this dual program (3) is a lower bound on opt B and a tight lower bound on the optimal value o the LP relaxation (2). Following the analysis o [15], we deine a new cost unction c λb, where c λb (e) = c e + λ B + u λb v or an edge e = (u, v). Since B v V λb v is constant or a ixed choice o λb, every tree in O B is an MST o G under the cost unction c λb. An optimal solution to the linear program (2) is a ractional tree T B opt = T O B α T T that is also an MST under the cost unction c λb. Let B = B(1 + β) or some β > 0, and let T B = T O B α T T be an optimal solution to the LP (2) or degree bound B. Since λ B is a easible solution or the dual LP (3), it is clear that c(t B ) + λ B v (deg T B (v) B) opt LD(B). (4) v V Recall that we are guaranteed by complementary slackness conditions that i λ B v along with (4), we get opt LD(B) v V λb v (deg T B (v) B) = v V λb v (B B) = β v V λb v B. > 0 then deg T B (v) = B. Using this act Now let T be any MST o G under the cost unction c λb. Then we arrive at the cost bound o T in [15] as ollows: c(t) = c λb (T) v V c λb (T B ) λ B v opt LD(B) + B v V deg T (v) λ B v by (4) ( ) opt LD(B) by (5), β rom which we conclude Theorem 25. Theorem 25 ([15]). Let T be an MST o G under the cost unction c λb with B = B(1 + β). Then c(t) From Theorem 25 and Theorem 11, we derive the ollowing theorem. ( ) opt β LD(B). Theorem 26. For any β > 0, there is a polynomial-time algorithm that, given a graph G and degree bound B, computes a spanning tree T with maximum degree at most 2 (1 + β) B + O( ( ) (1 + β) B) and cost at most opt β LD(B). For example, or β = 1, we produce a spanning tree o cost at most twice the optimum and o degree at most 4B + 4 2B + O(1). (5) Acknowledgement We would like to thank the anonymous reerees or several helpul comments.

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