Matroidal Degree-Bounded Minimum Spanning Trees

Matroidal Degree-Bounded Minimum Spanning T rees Rico Zenklusen ∗ August 2, 2018 Abstract W e consider the minimum spanning tree (MST) problem u nder the restriction that for e very vertex v , the edges of the tree th at are adjacent to v satisfy a given family o f constraints. A famous example thereof is the c lassical degree-boun ded MST p roblem, where for e very vertex v , a simple upper bound on the degree is imposed. Iter ativ e roun ding/relax ation algorithm s b ecame the tool of choice for degree- constrained network design pro blems. A cor nerstone f or this development was the work of Sing h and Lau [18], who showed that fo r the degree -boun ded MST pr oblem, one can find a spanning tree violating each degree bou nd by at most on e unit and with cost at most the cost of an optimal solution that respects the degree bounds. Howe ver, current itera ti ve rou nding app roaches face several limits whe n d ealing with more gene ral degree constraints. In p articular, when se veral constraints are imposed o n the edg es adjacent to a vertex v , as for example when a partition of the edges adjacent to v is gi ven and only a fixed number of elements can b e cho sen out of each set of the p artition, cu rrent app roaches might v iolate ea ch of the c onstraints by a constant, instead of violating the whole f am ily o f constraints by at most a con stant nu mber of edges. Furthermo re, it is also not clear how previous iterativ e roundin g appr oaches can be used for degre e constraints where some edges are in a super-constant number of constraints. W e extend iterative roundin g/relaxation approach es b oth o n a concep tual level as well as aspects in volving their analysis to ad dress these limitation s. Based on these extensions, we pr esent an algorithm for the degree-constra ined MST pro blem where for e very vertex v , the ed ges adjacent to v have to be indepen dent in a given ma troid. The alg orithm retu rns a spann ing tr ee o f co st at most OPT such that for every vertex v , it suffices to rem ove at m ost 8 ed ges fr om the spannin g tr ee to satisfy the matroid al degree constraint at v . 1 Introd uction Recently , much ef fort has been put on designing approximatio n algori thms for deg ree-constrai ned netw ork design problems. This dev elopment was motiv ated by vari ous applic ations as for example VLSI design, veh icle routing, and applicat ions in communicati on networks [7, 3, 17]. One of the most prominent and elementa ry probl ems here, which attracted lots of atten tion in recen t years, are degr ee-constrain ed (MST) proble ms. In the most classical s etting, kno wn as the de gr ee-bounde d MST p r oblem , the problem is t o find a spann ing tree T ⊆ E of minimum cost in a graph G = ( V , E ) under the restriction that the degr ee of each verte x v with respect to T is at most some gi ven value B v . Since checkin g feasibility of a degree - bound ed MST problem is already NP-hard, interest arose in fi nding lo w -cost spann ing trees that viola te the gi ven degree constrai nts sligh tly . A long chain of papers (see [7, 12, 13, 4 , 5] and reference s therein) led ∗ Dept. of Mathematics, MIT , Cambridge. E-mail: r icoz@math.mit. edu . Supported by Swiss National Scienc e Foundation grant P BEZP2-129524, by NSF grants C CF-1115849 and CCF-0829878, and by ONR grants N00014-11-1-00 53 and N00014-09- 1-0326. 1 to algorit hms with var ious trade-of fs between cost of the spanni ng tree and violation of the degr ee bounds. In recen t years, important progress was achie ved for the degre e-boun ded MST problem, which also led to a v ariety of ne w techn iques. G oemans [8] sh o w ed ho w to find a spanning tree vi olating each de gree const raint by at most two u nits, and whose cost is bo unded by the cost OPT of an opti m um spanning tree that satisfies the de gree constraints . Enhancing the iterat i ve roun ding framewo rk introdu ced by Jain [9] with a relax ation step, Singh and Lau [18] obtained a strong er versio n of the abov e result, which is essen tially best possi ble, where degr ee constrai nts are only violated by at most one unit. The y work with an LP relaxation of the proble m, and iterati vely drop degre e constrain ts from the LP that cannot be violated by more than one unit in later itera tions. The ad apted LP is th en solv ed again to obtain a possibly spar ser basic sol ution that allo ws for furth er degre e relaxatio ns. Edges not used in the current op timal so lution to the LP are remo ved from the graph, and edges that hav e a weight of one are fixed, w hile updating degr ee bounds accordingl y . A degre e bound at a verte x v is remov ed whenev er it is at most one unit lower than the current number of edges adjace nt to v . W e are intere sted in obtaining results of similar strength for more genera l degree bounds. Consider for exa mple the follo wing type of degree con straint s: for ev ery vertex v , a partition E v 1 , . . . , E v n v of the set δ ( v ) of edge s adjace nt to v is gi ven, and withi n each set E v i of the pa rtition , only a gi ven number of edge s can be chosen . The al gorithm of S ingh and Lau [1 8 ] as well as the one of Goemans [8] ca n easily be adap ted to thi s setting . (In particular , the algorit hm of Singh and Lau was ev en presented in this precise setting .) Ho w e ver , with both of these approache s, the constraint imposed by each set E v i can be violated by a constant. W e are interes ted in ha ving at most a con stant violatio n over all d egree constra ints at v , i.e., for e very verte x v ∈ V , at most a constan t number of edges ha ve to be remo ved from the spanning tree to satisfy all constra ints at v . Ano ther m ore general example that will be useful to illustrat e limits of current methods is obtained when imposing constra ints for each verte x v on a laminar family on the edges adjacent to v , instead of only consid ering a partition. Adapting G oemans’ algorithm to these st ricter boun ds on the de gree v iolation se ems to be difficul t, si nce a crucial step of this algorit hm is to cov er the support E ∗ of a basic soluti on to the natural LP relax ation by a constant number of spanning trees (for the degree-b ounded MS T problem, G oemans sho wed [8] that two spann ing trees suf fice). This resul t allo ws for orient ing the edges in E ∗ such that ev ery v erte x has at most a const ant number of incoming arcs, at m ost one in each spann ing tree. Dropp ing for ev ery verte x all incoming arcs from its degr ee constraint then leads to a matroid intersection proble m, whose solu tion violat es each deg ree constraint by at most a constant. T o be able to decompose E ∗ into a constant number of sp anning trees, one ne eds to sho w that fo r any sub set of the v ertices S ⊆ V , on ly a linea r number (in | S | ) of edges ha ve both endpoints in S . In the classic al degree- bound ed MST problem, this sparsen ess property follo ws from the fact that when consider ing only edges with both endpoi nts in S , there are at most a linear number (in | S | ) of linea rly independe nt and tight spann ing tree constrai nts due to combin atorial uncrossing , and only a linear number of degree constr aints within S . Howe ver , in more general settings as highlighted abo ve, the number of deg ree constra ints within S can be super -linear . Iterati ve relaxation looks more promising for a possi ble exte nsion to generalized degr ee bounds. Ho w- e ver , cur rent iterati ve roun ding app roaches fa ce se veral limits when t rying to adapt them. In par ticular , when dealin g with the partitio n bound s as explai ned abov e, a simple adaptatio n of the relaxation rule, where for a vertex v all constrain ts at v wou ld be droppe d as soon as it is safe to do so due to a small suppor t E ∗ , risks to get stuck because there might be no verte x whose degree constrain t can be relaxed. Furthermor e, pre vious appro aches (as used in [18, 2 ]) to sho w that the suppor t is sparse fai l in our settin g because of a possib le super -linearity of the tota l number of degree c onstra ints. Additiona lly , previ ous iterativ e relaxation approa ches cruciall y rely on the pro perty that any edge is in a t most a cons tant number of d egree cons traints to obtain violation s that are bounded by a consta nt. H o w e ver , this does not hold when dealing for example with de gree constrain ts gi ven by a laminar family . In this pape r we show ho w to extend iterati ve relaxat ion approa ches, both from a concept ual point of 2 vie w as well as aspects in v olving their analysi s, to tackle a wide class of MS T problems with generaliz ed deg ree bounds, namely when the degre e bounds for e very verte x are gi ven by a matroid. In particul ar , this includ es the partitio n bounds and the m ore gene ral laminar bounds mentioned abov e. Our re sults. W e present an it erativ e ro unding/rela xation algorit hm for finding a matr oidal d e gr ee-boun ded MST . T he degree bound s are giv en as follo ws: for e very ver tex v , a matroid M v = ( δ ( v ) , I v ) ov er the groun d set δ ( v ) is gi ven wit h ind epende nt sets denoted by I v ⊆ 2 δ ( v ) . The prob lem (without rela xed de gree constr aints) is to find a span ning tree T in G satisfyin g T ∩ δ ( v ) ∈ I v ∀ v ∈ V , and minimizing a linea r cost function c : E → R + . W e say that a gi ven spanni ng tree T v iolates a deg ree constraint M v by at m ost k ∈ N units, if it suffices to remove at most k edges R ⊆ δ ( v ) ∩ T from T to satisfy the constrain t M v , i.e., ( T \ R ) ∩ δ ( v ) ∈ I v . Hence, the partiti on and laminar bounds mentione d abov e correspond to the case where all matroid s M v are partitio n or laminar matroids, respecti vely . W e sh ow the follo wing. Theor em 1. Ther e is an effic ient algori thm for the matr oida l de gr ee-bou nded MST pr oblem that r eturns a spann ing tr ee of cost at most the cost of an optimal solution , and violate s each de gr ee bound by at most 8 units. T o o vercome pro blems faced by pre vious itera ti ve relaxati on approach es, w e enh ance the iterati ve relax - ation step, and exploit polyh edral struct ures to prov e stronge r spars eness result s. The polyt ope used as a relaxa tion of the m atroid al degr ee-bou nded MS T asks to fi nd a point x ∈ R E in the spanning tree polytope such that for e very verte x v ∈ V , the restri ction of x to δ ( v ) is in the matroid polytope P M v of M v . T o be able to alw ays find possible relaxat ion steps, our iterati ve round ing procedure tries to achie ve a somewhat weak er goal than pre vious approach es. The algor ithm of Singh and Lau [18] relaxes de gree constr aints with the goal to ap proach the s pannin g tree polyt ope, which is inte gral. In ou r a pproac h, the goal we pursue is to remov e eve ry edge { u, v } from at least one of the two degree constrain ts at u or v . As soon as no edge is part of both degr ee constra ints at its endpoints, the problem is a matroid intersectio n problem, since all degree constr aints together can be descri bed by a single matroid ov er the suppo rt of the current LP solu tion. Thus , once we are in this situa tion, the curre nt LP will be inte gral and no furth er rounding steps are needed . Hence, in our relaxat ion step, we try to fi nd a vert ex v such that we can remov e all edges adjace nt to v that a re sti ll in both deg ree constraint s from the de gree constrai nt at v . Edges ad jacent to v that are only contain ed in the degre e constrai nt at v will not be removed from the constraint M v . Our approach has t hus s ome similarities with Goeman s’ method, bu t instead of remo ving righ t at the start e very edg e from one degree constraint, w e do this itera ti vely and hereby profit from additi onal sparsen ess that is obtained by solving the LP relaxation after each degree adaptatio n step. As we w ill see in Section 2, the way how we remov e edges from a constrain t is strictly speaking not a relaxation , and we therefor e prefer to use the term de gr ee adaptation instead of deg ree relaxation . T he abo ve degree adaptatio n step alone sho ws not to be suf ficient for our approac h, since one m ight still end up in a situatio n were no further degree adaptation can be perfo rmed because the graph is too den se. T o ob tain greater sparsity , we use a secon d type of degree adapta tion, w here for some v ertex v we remov e (almost) the full degree cons traint at v if this cannot lead to a lar ge violatio n of the de gree constra int at v . The main step in the analys is is to pro ve that it is alway s possibl e to apply at least one of two sug gested deg ree adaptations. A first ste p in this proof is to sh ow that the sup port of a b asic solution to the LP relaxati on is sparse. W e obtain sparsi ty by showin g that if there are k ∈ N linearly indep endent and tight constraints (with respect to the current L P soluti on x ) of the polytope P M v , then x ( δ ( v )) ≥ k . Since summing x ( δ ( v )) ov er all vertices is equal to 2( | V | − 1) , be cause x ( E ) = | V | − 1 as x is in the spanning tree polytope , there are at most 2( | V | − 1 ) linearly indepen dent and tight degr ee constra ints. The crucial part in the analysis is to sho w that ver tices to which no further degree adaptat ion can be perfor med do not hav e very lo w degrees in a verage, implying that some of the other verti ces are likely to 3 ha ve low degree s and therefore admit a degr ee constra int adaptat ion. T o pro ve this property , we explo it the interp lay between degr ee bounds and spanning tree constrai nts to show that an y degree two no de can either be treated separately and allo w s for reduc ing the proble m, or implies a reduction in the maximum number of linea rly indepe ndent and tight spanning tree constraint s. Related w ork. The s tudy of spanning trees with deg ree constrain ts can be tra ced bac k to F ¨ urer and Ragha vach ari [ 7], who presented an approximat ion a lgorithm for the degree -bounded S teiner Tree prob- lem which violates each degree bound by at most one, b ut does not cons ider costs. This resul t genera ted much interest in the study of deg ree-bo unded network design problems, leading to numerous results and ne w techniques in recent years for a v ariety of problems, including degree-b ounded arboresce nce problems, deg ree-bo unded k -edge -conne cted subgra phs, deg ree-bou nded submodular flows, de gree-bounde d bases in matroids (see [16, 17, 11, 14, 15, 10, 2, 6, 1] and referen ces therein ). Spanning tree problems with a somewhat differ ent notion of general ized degree bound s hav e been con- sidere d in [2] and [1]. In these papers , the term “general ized degre e bounds” is used as follows: gi ven is a family of sets E 1 , . . . , E k ⊆ E , and the number of edges that can be chosen out of each set E i is bounde d by some giv en va lue B i ∈ N . In [2], using an iterati ve relaxatio n algorithm, whose analy sis is based on a fractional token countin g ar gument, the authors show ho w to ef fi cientl y obtain a spanning tree of cost at most OPT a nd violatin g each degr ee bo und by a t most max e ∈ E |{ i ∈ [ k ] | e ∈ E i }| , the m aximum c ov erage of an y edg e by the sets E i . In [1], a n e w iterat i ve roun ding app roach was presented for the p roblem when the sets E 1 , . . . , E k corres pond to the edges E i = δ ( C i ) of a family of cuts C i ⊆ V for i ∈ [ k ] that is laminar . Contrary to pre vious settings w here iterati ve rounding appro aches were applie d, here, it is possible that an edge lies in a super -consta nt number of de gree const raints. At each iterat ion, the alg orithm reduces th e num- ber of degree constraints by a constant factor , repla cing some constraints with ne w ones if necessary . This is done in such a w ay that degree constra ints are violat ed by at most a constant in e very iteratio n, leading to a spann ing tree of cost at most OPT , that violate s each de gree constrain t by at most O (log ( | V | )) . Organiz ation. In Section 2 we present our algor ithm for the matroidal degr ee-bounded MST problem. The analys is of the algo rithm is presented in Section 3. 2 The algorithm Since durin g the exe cution of our algorithm the unde rlying graph will be m odified, we denote by H = ( W , F ) the current state of the graph, whereas G = ( V , E ) alw ays denotes the original graph . For bre vity , terminol ogy and notation is with respect to the current graph H when not specified further . T o distin guish between initial deg ree constra ints and current degr ee const raints, we denot e by N w the current constrai nts for w ∈ W —which will as well be of m atroid al type—whereas M v denote s the initial degree constra ints at v ∈ V . The vertice s of H are called nodes since they might contain sev eral ver tices of G due to edge contra ctions . The algorith m start s with H = G and N v = M v for v ∈ V , and the LP relaxation we use is the follo wing, ( LP 1 ) min c T x x ∈ P st x   δ ( w ) ∈ P N w ∀ w ∈ W where P st denote s the spanning tree polyt ope of H , P N w denote s the matroid polytope that correspon ds to N w , and x   δ ( w ) denote s the vect or obtained from x ∈ R E by consider ing only the components that corres pond to δ ( w ) . 4 Algorithm fo r Matroidal Degr ee-Bounded Minimum Spanning T rees 1. Initiali zation: H = ( W , F ) ← G = ( V , E ) , N v ← M v for v ∈ V . 2. While | W | > 1 do a) Determine basic optimal solu tion x to ( LP 1) . Delete all edges f ∈ F with x ( f ) = 0 . b) C ontrac t all edges f ∈ F with x ( f ) = 1 . c) Fix a maximal family of li nearly independ ent and tight spannin g tree const raints. d) T ype A de gr ee ada ptatio n : for e ach node w ∈ W such that the set of a ll edges U ⊆ δ ( w ) that are still in both degr ee constraints is non-empty and satisfies | U | − x ( U ) ≤ 4 , remov e U from the degr ee constraint N w . e) T ype B de gr ee adaptatio n : for each no de w ∈ W such th at the set of all edg es U ⊆ δ ( w ) contai ned in the degr ee constraint N w b ut not adjacent to a node in Q is non-empty and satisfies | U | − x ( U ) ≤ 4 , remov e U from the de gree constraint N w . 3. Return all contr acted edge s. There is a set of nodes Q = Q ( H , x ) ⊆ W that has a special role in our algor ithm due to its relation with tight spanning tree constr aints. The node set Q is defined and used after havin g contracted edges of weight one. Hence, assume that H does not contai n any edge f ∈ F wit h x ( f ) = 1 . Then Q is defined as follo ws: we start with Q = ∅ and as long as there is a node w ∈ W such that x ( δ ( w ) ∩ F [ W \ Q ]) = 1 , where F [ W \ Q ] is the set of all edg es with bot h end points in W \ Q , we add w to Q . One ca n easily o bserve that Q does not depend ent on the order in which n odes are added to it 1 . As we will see l ater , edges adjace nt to these nodes can often be ignored from degree constr aints due to strong restrictio ns that are imposed by the spanni ng tree constr aints. The box on top of the page gi ves a desc ription of our algor ithm, omitting deta ils of how to deal w ith the matroidal degree boun ds when remov ing or contr acting edges. W e discuss these missing poin ts in the follo wing. Notice, that a basic solution to ( LP 1) can be determined in polyn omial time by the ellip soid method, e ven if th e in volv ed matroid s are only accessib le trough an ind ependence oracl e. Dependin g o n the matroidal deg ree bounds in vol ved, ( LP 1) can be solve d more ef fi ciently by using a polyno m ially-s ized extended formulat ion. A tight spanning tree constr aint, as considered in step (2c), corresp onds to a set L ⊆ W , L 6 = ∅ such that x ( F [ L ]) = | L | − 1 . F ixing a tight spanning tree constr aint m eans that this constraint has to be fulfilled with equality in all linear programs of type ( LP 1) solv ed in future iteratio ns. It is well-kno w n that if supp ( x ) = F , then an y maximal family of linea rly indepen dent and tight spanning tree cons traints with respec t to x defines the minimal face of the spanning tree poly tope on which x lies (see e.g. [8]). Hence, due to step (2c), w e ha ve that if the LP solution at some iteration of the algorith m is on a gi ven face of the spann ing tree polytope, then all future solu tions to ( LP 1) will be as well on this face . Fixing tight spann ing tree constra ints sho ws to be useful since the y often imply strong condi tions on the edges, w hich can be exploit ed when ha ving to make sure that degre e constrain ts are not violat ed too 1 The fact that H does not contain 1 -ed ges is needed here t o mak e sure that the order is unimportant in the definition of Q . With 1 -edges it might be t hat during the it erativ e construction of Q , one ends up wit h two nodes connected by a single edge of weight one, in which case any one of t he two re maining nodes can be included in Q , but not bo th. This is actua lly the only bad constellation that leads to a dependenc y on the order in the definition of Q . 5 much. In particu lar , con sider a node w ∈ Q which, in the itera ti ve construc tion of Q , could hav e been added as the first node, i.e., x ( δ ( w )) = 1 . When fixing tight spanning tree constraint s, one can observe that any spann ing tree satisfyi ng those tight constr aints with equality contains precisely one edge adjacent to w . Furthermor e, the fixing of tight spannin g tree constrain ts guarantees that a node w ∈ Q will stay in Q in later iterat ions until an edge adja cent to w is contrac ted. Hence, all edge s being in some iter ation adjac ent to a node w ∈ Q , will be ad jacent to a no de in Q in all later iterat ions unt il they are eit her d eleted o r contra cted. This property is important in our approach since a type B deg ree adap tation ignores edges adjacen t to Q , and we wan t to make s ure that an edge which is on ce ignored will ne ver be consid ered during a later type B deg ree adaptation . Contracting and remov ing edges. T o fill in the remaining details of our algori thm, it remains to discuss ho w edges are cont racted and removed . Throu ghout the algorithm, any deg ree constraint N w of a node w contai ning the vertices v 1 , . . . , v k ∈ V can al ways be written as a disjoint union of matroid al constrai nts N v 1 , . . . , N v k , wher e N v i corres ponds to the “ remaining” degr ee bound at v i and i s a matroid o ver the edge s δ ( w ) that are adjacent to v i . When e ver an edge f = { w 1 , w 2 } of weight one is contracted in step (2b) of the algorit hm to obtain a new node w , the ne w degree const raint N w at w is obtained by taking a disjoint union of the matroid s N w 1 /f and N w 2 /f , where N w 1 /f and N w 2 /f c orresp ond to the matroids obtained from N w 1 and N w 2 , respecti vely , by contrac ting f . T his operation simply translates the degree constrain ts on w 1 and w 2 to the mer ged node w . The proper ty that a degree bound on w is a disjoint union of de gree bound s of the vert ices represent ed by w , is clearly maintained by this contracti on. As highl ighted in the box, we adapt cons traints by r emovin g for some node w ∈ W a set of edges U ⊆ δ ( w ) from the constraint N w . When remo ving U from N w , we construct a ne w de gree constr aint gi ven by a matroid N ′ w ov er the elements δ ( w ) such that the follo wing propert ies hold. Pro p erty 2. i) N ′ w is a di sjoint union of matr oidal constr aints N ′ v 1 , . . . , N ′ v k corr espon ding to vertic es contain ed in w , ii) edges of U are f ree elements of N ′ w , i.e., if I is indepen dent in N ′ w then I ∪ U is independen t in N ′ w , iii) any inde penden t set of N ′ v i can b e tr ansformed into on e o f N v i by r emovin g at most ⌈| U | − x ( U ) ⌉ edg es, iv) the pr evio us LP solu tion x is still feasible w ith r espect to N ′ w , i.e., x   δ ( w ) ∈ P N ′ w . Any remov al operat ion satisfying the abov e propertie s can be used in our algorithm. B efore presenti ng such a remo val o peratio n, we first mention a fe w impor tant poi nts. T o av oid confusio n, we want to hig hlight that remo ving U from N w does not simp ly corres pond to de leting the elemen ts U from the matroid N w . For any edge f ∈ δ ( w ) that is free in N w , we say that f is not contained in the degree constrain t N w , and it is contai ned otherwise. When all edges adjacent to a giv en node w are not contained in its degr ee constraint , which corresp onds to N w being a free matroid, we say that the node w has no degr ee constraint . W e now discuss how to remove a set of edges U ⊆ δ ( w ) from N w to obtain an adapte d degree bound N ′ w satisfy ing Property 2. Let v 1 , . . . , v p ∈ V be all vertices contain ed in the node w , and w e consider the decompo sition of N w into a disjoint unio n of matroids N v 1 , . . . , N v k , where N v i for i ∈ [ k ] corresp onds to the “remaini ng” de gree bound at v i . T o remo ve U from N w , we adapt each matroid N v i as follo ws to obtain a new matroid N ′ v i . Let S i be the ground set of N v i , i.e., all edges in δ ( w ) being adjacen t to v i . Let M 1 = ( S i , I 1 ) be the matroid with indepen dent sets I 1 = { I ⊆ S i ∩ U | | I | ≤ | S i ∩ U | − ⌊ x ( S i ∩ U ) ⌋} . Hence, M 1 is a specia l case of a partit ion matroid. Let M 2 = M 1 ∨ N v i be the union of the matroid s M 1 and N v i , and let M 3 = M 2 / ( S i ∩ U ) be the matroid obtained from M 2 by contractin g S i ∩ U . The degree bound N ′ v i is obtaine d by a disjo int union of M 3 and a free matroid ov er the elements in S i ∩ U . The new deg ree constra int N ′ w , that results by r emoving U from N w , is gi ven by the disjoin t union of the matroids N ′ v 1 , . . . , N ′ v k . 6 Lemma 3. The above pr ocedu r e to remo ve elements fr om a de gr ee constr aint satisfies P r operty 2. Pr oof. By construct ion, when remo ving a set U ⊆ δ ( w ) from a degree bound N w , which can be written as a disjoint unions of N v 1 , . . . , N v k , a matroidal bound N ′ w is determin ed which is a disjoi nt union of N ′ v 1 , . . . , N ′ v k . Hence point (i) of Property 2 holds. Let S i be the grou nd set of the matroid s N ′ v i , N v i for i ∈ [ k ] . Since N ′ w is a disj oint union of N ′ v 1 , . . . , N ′ v k , it suf fi ces for point (ii) to pro ve that if I ′ is independ ent in N ′ v i then I ′ ∪ ( S i ∩ U ) is in- depen dent in N ′ v i . This follo w s since N ′ v i was obtained by a disjoi nt union of the matroid M 3 , as defined abo ve, and a free matroid o ver S i ∩ U . For point (iii), consider an indepe ndent set I ′ in N ′ v i . Since all edges in U ∩ S i are free in N ′ v i , we can assume ( U ∩ S i ) ⊆ I ′ . Consider ho w the matroid N ′ v i was construct ed by the use of the matroids M 1 , M 2 , M 3 . W e s tart by obse rving that U ∩ S i is an ind epende nt set in M 2 = M 1 ∨ N v i . Let r i be th e ran k functi on of N v i , and r 2 be the rank function of M 2 . Since x ∈ P N v i , we hav e that r i ( S i ∩ U ) ≥ x ( S i ∩ U ) . Furthermor e, since M 2 = M 1 ∨ N v i and any | S i ∩ U | − ⌊ x ( S i ∩ U ) ⌋ elements of S i ∩ U are independen t in M 1 , we ha ve r 2 ( S i ∩ U ) = min {| S i ∩ U | , r i ( S i ∩ U ) + | S i ∩ U | − ⌊ x ( S i ∩ U ) ⌋} = | S i ∩ U | , sho wing indep endence of S i ∩ U in M 2 . B ecause N ′ v i was obtained by a disjoint union of the matroid M 3 and a free m atroid over the elements S i ∩ U , we can w rite I ′ = I 3 ∪ ( S i ∩ U ) w ith I 3 indepe ndent in M 3 . Furthermor e, as M 3 = M 2 / ( S i ∩ U ) and S i ∩ U is independen t in M 2 , the set I ′ is indep enden t in M 2 . As M 2 = M 1 ∨ N v i , we ha ve I ′ = I 1 ∪ I , with I 1 indepe ndent in M 1 and I independe nt in N v i . S ince M 1 is a matroid of rank | S i ∩ U | − ⌊ x ( S i ∩ U ) ⌋ , we hav e that I is obtained from I ′ by removing at most | I 1 | ≤ | S i ∩ U | − ⌊ x ( S i ∩ U ) ⌋ ≤ | U | − ⌊ x ( U ) ⌋ elements as desired. Let x i = x   S i for i ∈ [ k ] . T o sho w point (iv), it suf fices to prov e that x i ∈ P N ′ v i ∀ i ∈ [ k ] , since N ′ w is a disjoi nt union of N ′ v 1 , . . . , N ′ v k . Let z i ∈ [0 , 1] S i be gi ven by z i ( f ) = ( 1 if f ∈ S i ∩ U, x i ( f ) if f ∈ S i \ U. Observ e that z i − x i ∈ P M 1 becaus e the support of z i − x i is a subset of S i ∩ U , k z i − x i k 1 = | S i ∩ U | − x ( S i ∩ U ) and any | S i ∩ U | − ⌊ x ( S i ∩ U ) ⌋ elements of S i ∩ U are indepen dent in M 1 . Hence z i ∈ P M 2 , since M 2 = M 1 ∨ N v i , x i ∈ P N v i and z i − x i ∈ P M 1 . As M 3 = M 2 / ( S i ∩ U ) , we hav e that the rest riction of z i on S i \ U , which is equal to x i   S i \ U , is in P M 3 . Since N ′ v i is the union of M 3 and a free matroid over S i ∩ U , this finally implies that x i ∈ P N ′ v i . 3 Analysis of the algorithm Lemma 4. During t he e xecution of the al gorithm, for eve ry verte x v ∈ V , a t mos t one con stra int adaptatio n of type A and one of type B is perfo rmed that r emoves edg es of δ ( v ) fr om de gr ee constr aints containi ng v . Pr oof. When a type A degre e adaptatio n is applied to a node w ∈ W that conta ins v , no furthe r type A deg ree adaptat ion can remov e any edg es in δ ( v ) ∩ F from the con straint contain ing v , sinc e those edges are not any more contain ed in both degr ee constraints at their endpoints . Similarly , when a type B deg ree adaptat ion is appli ed to a node w that contains v , all edges in δ ( w ) ∩ F [ W \ Q ] are remove d from the degre e constr aint at w and thus cannot be remove d again at a later type B degre e adap tation. Hence, the only possibility to remov e furthe r edges adjacent to v in a later type B 7 deg ree adaptation is that some edge f ∈ F which was—at some iteration of the algorith m —not consider ed for a possib le remov al by a type B adaptat ion because of being adjacent to a node in Q , can be remove d by a type B adaptation at a later stage. Howe ver as alrea dy discu ssed, since we fi x all tight spanning tree constr aints, an edge tha t is adjacent to a nod e w ∈ Q in some iteration, wil l remain so until it is ei ther deleted or contr acted in step (2a) or (2b) of the algo rithm. Hence, this “bad const ellatio n” can nev er occur . W e explo it that our remov al operation satisfies point (iii) of P ropert y 2 to bound the m aximum possibl e deg ree violation. In particular , for each vert ex v ∈ V , ev ery time edges U w ith U ∩ δ ( v ) 6 = ∅ are removed from the curren t degree constrai nt N w at the node w th at contains v , the degr ee constraint at v can be violat ed at most by an additional ⌈| U | − x ( U ) ⌉ units. Since w e only perform degree adaptations for sets U with | U | − x ( U ) ≤ 4 , and L emma 4 guaran tees that at most two adaptati ons are performed that in v olve the deg ree constraint at v , w e obtain the follo wing result. Cor ollary 5. If the algorithm terminates, then the r eturne d tr ee violates each de gr ee constr aint by at most 8 units. A main step for pro ving that we can alway s appl y one of the two suggested degre e adaptat ions, is to pro ve that a basic solutio n to ( LP 1) is suf fi ciently spars e. A fi rst important build ing block for prov ing sparsi ty is the follo wing result. Lemma 6. L et x be any solution to ( LP 1) whose suppo rt equals F . Then for ever y node w ∈ W , the maximum number of linearly ind ependent con stra ints of th e matr oid polyt ope P N w that ar e tight with r espect to x , is bound ed by x ( δ ( w )) . Pr oof. Let C ⊆ 2 δ ( w ) be a family with a maximum number of sets that correspond to linearly independen t constr aints of the matroid polytope P N w that are tight w ith resp ect to x . By standard uncrossing ar guments, C can be chosen to be a chain, i.e. C = { C 1 , . . . , C p } with C 1 ( C 2 ( · · · ( C p (see [9] for m ore details). W e hav e to show that p ≤ x ( δ ( w )) . Let r be the rank functio n of N w . Define C 0 = ∅ and for i ∈ [ p ] let R i = C i \ C i − 1 . Since C is a family of tigh t constraints , we hav e x ( R i ) = r ( C i ) − r ( C i − 1 ) ∀ i ∈ [ p ] . (1) Because R i ⊆ sup p ( x ) , the left-hand side of (1) is strictly larg er than zero. Furthermore, the right-hand side is integral and must therefore be at least one. Hence x ( R i ) ≥ 1 for i ∈ [ p ] , which implies x ( δ ( v )) ≥ P p i =1 x ( R i ) ≥ p . Notice, that the abov e lemma implies that a basic solution x to ( LP 1) has a support of size at most 3( | W | − 1) , becaus e of the follo wing. W e can assume that all edges that are not in the supp ort of x are delete d from the graph. Due to Lemma 6 , at most P w ∈ W x ( δ ( w )) linear ly independe nt constraints of the polyto pes { P N w | w ∈ W } can be tight with respe ct to x , and since x is in the spann ing tree polyto pe of H , this bound equals P w ∈ W x ( δ ( w )) = 2( | W | − 1) . F urthermor e, at most | W | − 1 linear ly indepen dent constr aints of P st are tight with respec t to x due to uncro ssing. This sho ws in parti cular that in the first iterati on of the algorit hm, we can find a nod e w ∈ W to which a ty pe A de gree constrain t adaptati on can be applie d, because X w ∈ W ( | δ ( w ) | − x ( δ ( w )) = 2 | F | − 2( | W | − 1) ≤ 4( | W | − 1) , and hence there must be a node w ∈ W with | δ ( w ) | − x ( δ ( w )) ≤ 4 . Ho wev er , in later iteratio ns, the abov e reaso ning alone is not anymore sufficien t because m any vertice s do not ha ve degree constraints anymore. Still, by assuming that no type B constrain t adaptation is possible, 8 and u sing se veral ideas to ob tain stro nger sparsity , we s how that the ab ov e approac h of findi ng a g ood v ertex for a type A de gree adaptatio n by an av eraging argumen t can be extended to a general iteratio n. For the rest of this section, we consid er an iteration of the algorithm at step (2d) with a curren t basic soluti on x to ( LP 1) , and assume that | W | > 1 , and that no type B degre e adapta tion can be applied 2 . W e then show that there is a type A constraint adaptation that can be performed under these assumptions. This implies that our algor ithm ne ver gets stuck, and hence prov es its correctness. Since we often deal with the spar e 1 − x ( f ) of an edge f ∈ F , we use the notatio n z = 1 − x . Furthermor e, we partition F into the sets F 2 , F 1 and F 0 of ed ges that are conta ined in 2 , 1 an d 0 de- gree constraint s, respe cti vely . Hence, at the first iteratio n we hav e F 2 = F . Our goal is to sho w that P w ∈ W z ( δ ( w ) ∩ F 2 ) = 2 z ( F 2 ) ≤ 4 | Y | , w here Y ⊆ W is the set of all nodes w with δ ( w ) ∩ F 2 6 = ∅ . By an a veraging arg ument this then implies that there is at least one node w ∈ Y to which a type A constraint adapta tion can be applied. Notice that the set F 2 canno t be empty (and hence also Y 6 = ∅ ): if F 2 = ∅ , then the curr ent LP 1 corresp onds to a matro id interse ction problem since ev ery edge is conta ined in at most one deg ree constra ints, and hence all de gree constr aints form together a sin gle matroid ov er F ; in this case LP 1 is integral and a full spanning tree would hav e been contracted after step (2b), which leads to | W | = 1 and contra dicts our assumptio n | W | > 1 . Lemma 7. Let L be a maximum f amily of lin early indepen dent spanning tr ee constr aints that ar e tight with r espect to x . Then 2 z ( F 2 ) ≤ 2 |L| + 2( | W | − 1) − 2( | F 0 | + | F 1 | ) − 2 x ( F 0 ) . Pr oof. W e ca n re write 2 z ( F 2 ) as follo ws b y using t he f act that x ( F ) = | W | − 1 (because x is in the spann ing tree polyto pe of H ). 2 z ( F 2 ) = 2 z ( F ) − 2 z ( F 0 ) − 2 z ( F 1 ) = 2( | F | − x ( F )) − 2 z ( F 0 ) − 2 z ( F 1 ) = 2 | F | − 2( | W | − 1) − 2 z ( F 0 ) − 2 z ( F 1 ) (2) Using classical ar guments w e can bound the size of the suppor t of x , which is by assumptio n equal to | F | , by the number of linear ly indepen dent tig ht constrai nts from the spannin g tree polytope and the deg ree polyto pes P N w for w ∈ W . In particula r x is uniqu ely defined by the tight spanning tree constrai nts L complete d with some set D of linearly independen t degree constrai nts, and we hav e | F | = |L| + |D | . The deg ree constra ints D can be partit ioned into D w for w ∈ W , where D w are line arly independ ent constrain ts of th e matroid po lytope P N w . By Lemma 6 , |D w | is boun ded by the sum o f x ov er all edges i n δ ( w ) that are contai ned in the degre e constr aint at w . When summing these bound s up over all w ∈ W , each edge in F 2 is coun ted exa ctly twice, and each edge in F 1 exa ctly once. Hence, |D | ≤ 2 x ( F 2 ) + x ( F 1 ) = 2 x ( F ) − x ( F 1 ) − 2 x ( F 0 ) = 2( | W | − 1) − x ( F 1 ) − 2 x ( F 0 ) . Using | F | = |L| + |D | and the abov e bound, w e obt ain from (2) 2 z ( F 2 ) ≤ 2 |L| + 2( | W | − 1) − 2 ( z ( F 0 ) + z ( F 1 ) + 2 x ( F 0 ) + x ( F 1 )) = 2 |L| + 2( | W | − 1) − 2( | F 0 | + | F 1 | ) − 2 x ( F 0 ) , where the last ineq uality follows from z ( U ) + x ( U ) = | U | for any U ⊆ F . The size of a family L of linearly independent tight spannin g tree constrain ts can easily be bounded by | W | − 1 us ing t he fact th at on e c an assu me L to be laminar by standard u ncross ing a r guments (and L contains 2 Notice t hat the assumption | W | > 1 is not redundant. Whereas we kno w that at t he beginning of the iteration | W | > 1 did hold, this could hav e changed after contracting edges in step (2b). 9 no single ton sets). H o w e ver , this result sho w s not to be strong enough for our purpose s. T o strength en this bound we e xploit the fa ct tha t if L cont ains close to | W | − 1 sets, then t here are m any nod es w ∈ W that are “sandwic hed” between two set s of L , i.e., t here are two sets L 1 , L 2 ∈ L w ith L 2 = L 1 ∪ { w } , w hich in tu rn implies x ( δ ( w ) ∩ E [ L 2 ]) = 1 . Notice that for any de gree two node w which is not in Q , we hav e x ( U ) 6 = 1 for all U ⊆ δ ( w ) . Hence, such a node cannot be “sandwiched ” between two tight spanning tree constr aints, and w e expect that the more such nodes we hav e, the smaller is |L| . T he follo wing result quantifies this observ ation. It is s tated in the gene ral context of a sp anning tree poly tope of a general conne cted graph (not being link ed to our degree -constr ained problem). Lemma 8. Let y be a point in the span ning tr ee polyto pe for a given graph G = ( V , E ) , and let S ( G, y ) = { v ∈ V | | δ ( v ) | = 2 , y ( U ) 6 = 1 ∀ U ⊆ δ ( v ) } . Let L ⊆ 2 V be any linear ly indepen dent family of spanning tr ee constr aints that ar e tight w ith r espect to y . Then |L| ≤ | V | − 1 −  1 2 | S ( G, y ) |  . Pr oof. T o simplify notation let S = S ( G, y ) . By standa rd uncro ssing ar guments (see for example [8 ]), w e can assume that L is laminar . W e first consider the case that there is a set L ∈ L with L ⊆ S . Let L be a minimal set in L with this property . Since L is a tight spanni ng tree constraint, we hav e that y   E [ L ] is in the spann ing tree polytope of G [ L ] , and hence y ( δ ( v ) ∩ E [ L ]) ≥ 1 for v ∈ L . As L ⊆ S , we hav e | δ ( v ) | = 2 and y ( e ) < 1 for v ∈ L and e ∈ δ ( v ) . T his implie s that e very vert ex in L must ha ve both of its ne ighbors in L to satisfy y ( δ ( v ) ∩ E [ L ]) ≥ 1 . Since G is connected , as we assumed that there is a point in the spann ing tree p olytope of G , we must ha ve L = V = S . Furthermore | V | ≥ 3 , b ecause v ertices in L ha ve d egree two. Hence the claim tri vially follo ws since |L | = 1 . No w assume that there is no set L ∈ L with L ⊆ S . W e sho w that there exist s a set R ⊆ S of size at least | R | ≥ 1 2 | S | , such that the laminar family L R = { L \ R | L ∈ L} over th e elements V \ R satisfies the follo wing: i) L R has no sing leton sets, ii) |L R | = |L| , i.e., any two set s L 1 , L 2 ∈ L with L 1 ( L 2 , satis fy L 2 \ L 1 6⊆ R . Notice that this will imply the claim since |L R | ≤ | V \ R | − 1 , because L R is laminar without singleto n sets, and hen ce |L| = |L R | ≤ | V \ R | − 1 ≤ | V | − 1 − 1 2 | S | . It remains to define the se t R with the des ired proper ties. For L ∈ L , let V L ⊆ L be all vertices in L that are not contained in any set P ∈ L w ith P ⊆ L . For each set L ∈ L , inclu de an arbitrary set of ⌈ 1 2 | S ∩ V L |⌉ elements of S ∩ V L in R . Since the sets V L for L ∈ L are a partiti on of all verti ces V , we clearly hav e | R | ≥ 1 2 | S | . F urther more R satisfies the desired proper ties as we sho w below . i) A ssume by sake of contra diction that L R contai ns a singleton set, i.e., there is a set L ∈ L with | L \ R | = 1 . W e can assume that L is a minimal se t in L . By ass umption we ha ve L 6⊆ S , and since R ⊆ S , the e lement in L \ R is not in S . Hence, R co ntains all elemen ts L ∩ S , which is only po ssible if | L ∩ S | = 1 and therefo re | L | = 2 . Howe ver , this implies that there m ust be an edges of weight one between the two ver tices in L , which contrad icts the fact that one of those v ertices is in S . ii) Assume by contradictio n that there are two sets L 1 , L 2 ∈ L with L 1 ( L 2 that satisfy L 2 \ L 1 ⊆ R . W e can choo se L 1 and L 2 such that there is no set L ∈ L with L 1 ( L ( L 2 . By choic e of R , this can only happe n if L 2 \ L 1 contai ns exac tly one vertex v ∈ S . This implies y ( δ ( v ) ∩ E [ L 2 ]) = 1 , w hich contr adicts the fact that v ∈ S . Lemma 8 ca n easily be generali zed to the subgrap h of a gi ven graph G obtaine d by deleting the ver tices Q ( G, y ) . This form of th e lemma is more usefu l for our a nalysis bec ause of o ur sp ecial tre atment of v ertices in Q . 10 Lemma 9. Let y be a p oint in the spanning tr ee polyto pe of a given gr aph G = ( V , E ) with y ( e ) 6 = 1 ∀ e ∈ E , let G ′ = G [ V \ Q ( G, y )] , and let y ′ be the pr oject ion of y to the edg es i n G ′ . Let L be any linearl y indepe ndent family of spann ing tree constr aints of G that ar e tight w ith r espect to y . T hen |L| ≤ | V | − 1 −  1 2 | S ( G ′ , y ′ ) |  . Pr oof. By standar d uncrossing argumen ts, w e can assume that L is a maximal laminar f amily of tight spann ing tree constraint s. W e prov e the result by inductio n on the number of elements in Q = Q ( G, y ) . If Q = ∅ , then the result follo ws from Lemma 8. Let q ∈ Q be a possible first element added to Q during the iterati ve const ruction of Q , i.e., y ( δ ( q )) = 1 . This implies that V \ { q } is a tight spanning tree const raint. Let H = G [ V \ { q } ] , y H = y   E [ V \{ q } ] and Q H = Q ( H , y H ) . Since Q H = Q \ { q } , we can apply the induct ion hypo thesis to the graph H to obtain that any m aximal family L H of linear ly independ ent tight spann ing tree cons traints in H with respec t to y H satisfies |L H | ≤ | V \ { q }| − 1 −  1 2 | S ( G ′ , y ′ ) |  . The claim follo ws by observin g that L = L H ∪ { V } is a maximal family of tigh t spannin g tree constraints in G , and hence |L| = |L H | + 1 ≤ | V | − 1 −  1 2 | S ( G ′ , y ′ ) |  . Combining Lemma 9 w ith Lemm a 7 we obtain the follo wing bound , where we use S = S ( H [ W \ Q ] , x   F [ W \ Q ] ) to simplify the notation. T o get rid of the roun ding on 1 2 | S | we use 2 ⌊ 1 2 | S |⌋ ≥ | S | − 1 . Cor ollary 10. 2 z ( F 2 ) ≤ 4( | W | − 1) − 2 ( | F 0 | + | F 1 | ) − 2 x ( F 0 ) − | S | + 1 . The follo wing lemma implies the correct ness of our algori thm. W e recall that Y ⊆ W i s the set of all nodes w ∈ W s uch that δ ( w ) ∩ F 2 6 = ∅ . Lemma 11. T her e is a node w ∈ Y such that a typ e A constr aint adaptatio n can be applied to w . Pr oof. Let Y = W \ Y . W e will p rove that 4 | Y | ≤ 2( | F 0 | + | F 1 | ) + 2 x ( F 0 ) + | S | . (3) T ogether with Corollary 10 this then implies 2 z ( F 2 ) ≤ 4 | Y | − 3 , which in turn implies by an av eraging ar gument that there is at least one node in Y to which a type A constraint adaptati on can be appl ied. T o pro ve (3) we apply a fractional tok en countin g argumen t: we show that if we interpret the right-h and side of (3) as a (frac tional) amount of tok ens, then we can ass ign those tok ens to the v ertices in Y suc h that each ver tex in Y gets at least 4 tokens. W e think of t he tok ens corres ponding to 2( | F 0 | + | F 1 | ) + 2 x ( F 0 ) as residing at the endpoints o f the edges in F 0 ∪ F 1 . Each edge f ∈ F 0 gets 2 + 2 x ( f ) tokens, 1 + x ( f ) at each endp oint. Each edge f ∈ F 1 gets 1 + x ( f ) token s at the end point which d oes not con tain f in it s de gree constra int, and 1 − x ( f ) token s at the other endpoin t. The tok ens assigne d to the endpoi nts of the edges thus sum up to 2( | F 0 | + | F 1 | ) + 2 x ( F 0 ) . W e start by assigning tokens to ver tices in Q . By definitio n of the vertic es in Q , we can order the elements in Q = { q 1 , . . . , q p } such that for i ∈ [ p ] , we ha ve x ( F q i ) = 1 where F q i = {{ q i , v } ∈ F | v ∈ W \ { q 1 , . . . , q i − 1 }} . Sinc e x ( F q i ) = 1 and no edge f ∈ F sati sfies x ( f ) = 1 (such an edge would ha ve been contracted) , we hav e | F q i | ≥ 2 . Each vertex q i ∈ Q gets all the tokens at both endp oints of the edges in F q i . Since | F q i | ≥ 2 , q i recei ves indeed at least four token s. 11 Let H ′ = ( W ′ , F ′ ) = H [ W \ Q ] be the induced subgrap h ov er the vertices W \ Q , and let x ′ = x   F ′ . Notice that x ′ is in the spannin g tree polyto pe of H ′ since the set of edges U ⊆ F that ha ve at least one endpo int in Q satisfy x ( U ) = | U | , and hence x ′ ( F ′ ) = | F ′ | − 1 . The remai ning token s are allocat ed as follo ws. Each nod e w ∈ Y ∩ W ′ gets for e very edg e f ∈ δ H ′ ( w ) , the token s of f at the endpoint at w . Further more, ev ery node in S gets an additional token from the term | S | . The attrib uted tokens clearly do not e xceed the right- hand side of (3). It remains to sho w that each node w ∈ Y ∩ W ′ gets at least 4 tok ens. W e distingu ish the follo w ing three cases: (i) w ∈ S , (ii) w 6∈ S and none of the edges δ H ′ ( w ) is contained in the degre e constraint at v , and (iii) w 6∈ S and at least one edge of δ H ′ ( w ) is cont ained in the de gree cons traint at w . Notice that th e ver tices consi dered in cas e ( i) are pre cisely all ver tices in H ′ of degree two, because if there was a degree two vertex w ∈ W ′ \ S , then w would hav e been inclu ded in Q . Hence, all vertic es considere d in case (ii) or case (iii) ha ve degree at least 3 in H ′ . Case (i): w ∈ S . Because | δ H ′ ( w ) | = 2 , we hav e that both edges in δ H ′ ( w ) are not contained in the deg ree constraint at w , since otherwis e a type B deg ree adaptation co uld hav e bee n per formed at w . Hence, w recei ves 2 + x ( f 1 ) + x ( f 2 ) token s f rom those two edges plus one token from | S | , resu lting in 3 + x ( f 1 ) + x ( f 2 ) tok ens. Since x ′ is in the spannin g tree polyt ope of H ′ , we hav e x ( f 1 ) + x ( f 2 ) = x ( δ H ′ ( w )) ≥ 1 , and thus w recei ves at least 4 token s. Case (ii): w 6∈ S and none of the edg es δ H ′ ( w ) is contain ed in the de gr ee constr aint at w . The total number of toke ns recei ved by w thus equals | δ H ′ ( w ) | + x ( δ H ′ ( w )) ≥ 3 + x ( δ H ′ ( w )) , since | δ H ′ ( w ) | ≥ 3 . The claim follo ws again by observing that x ′ is in the spanni ng tree polytop e of H ′ , which implies x ( δ H ′ ( w )) ≥ 1 . Case (iii): w 6∈ S and at least one edge of δ H ′ ( w ) is contained in the de gr ee const raint at w . Let U be the set of all edges in δ H ′ ( w ) that are contai ned in the degree constrai nt at w . Since no type B degre e adapta tion can be perfo rmed at w , w e ha ve | U | − x ( U ) > 4 . H o w e ver , | U | − x ( U ) is exac tly the number o f tok ens that w receiv es from the edges in U . Hence, at least 4 token s are assigned to w . Refer ences [1] N. Bansal, R. Khandekar , J. K ¨ onema nn, V . Nagaraja n, and B. Peis. On generalizat ions of network design problems with degree bounds. In Pr oceedin gs of Inte ger P r ogr amming and C ombinato rial Optimizatio n (IPCO) , pages 110–123, 2010. [2] N. Bans al, R. Khandek ar , and V . Nagaraj an. Additi ve g uarante es for de gree-bound ed directe d netwo rk design . SIAM Jou rnal on Computing , 39(4):141 3–1431, 2009. [3] F . Bauer and A. V arma. Degre e-constraine d multicasti ng in point-to-po int networks. In Pr oceedi ngs of the F our teenth Annual Join t Confer ence of the IEE E Computer and Communicati on Societies (IN- FOCOM) , pages 369–37 6, 1995. [4] K. C haudh uri, S. Rao, S . Riesenfe ld, and K. T alwar . A push-rela bel a pproxi m ation algorithm for approx imating the minimum-deg ree MST problem and its generaliz ation to matroid s. Theor etical Computer Scien ce , 410:4489– 4503, October 200 9. [5] K. Chau dhuri, S. Rao, S. Riesenf eld, and K. T al war . What woul d Edmo nds d o? Augmenti ng paths and witnesse s for deg ree-bounded MSTs. A lgorith m ica , 55:157 –189, May 2009. [6] C. Chekuri, J. V ondr ´ ak, and R. Zenkluse n. Dependen t randomized roundi ng via exchan ge prop erties of combin atorial structures . In P r oceedin gs of the 51st IEEE Symposi um on F oun dation s of Computer Scienc e (FO CS) , page s 575–584, 2010. 12 [7] M. F ¨ ure r and B. Raghav achari. Approximating the minimum-deg ree Steiner Tree to within one of optimal. Journ al of A lgorith m s , 17(3):409 –423, 1994. [8] M. X . Goemans. Minimum bounde d deg ree spanni ng trees. In P r oceedi ngs of the 47th IEE E Sympo- sium on F oundatio ns of Computer Science (FOCS) , pages 273–2 82, 2006. [9] K. Jain. A facto r 2 approximati on algorith m for the generaliz ed Steiner N etwork Problem. Combina - torica , 21:39– 60, 2001. [10] T . Kir ´ aly , L . C. Lau, and M. Singh. De gree bound ed matroids and submodular flows. In P r oceedi ngs of Inte ger Pr ogr amming and Combinatoria l Optimizati on (IPCO ) , pag es 259–272, 2008. [11] P . N. Klein, R. Krishnan, B . Ragha vacha ri, and R. Ra vi. A pproxi m ation algor ithms for finding lo w - deg ree subgraphs . Networ ks , 44:203 –215, O ctober 2004. [12] J. K ¨ oneman n and R . R a vi. A matter of de gree: Impro ved approximatio n algorithms for d egre e-bounded minimum spann ing trees. SIAM Jou rnal on Computing , 31:1783–1 793, June 2002. [13] J. K ¨ one m ann and R. Rav i. Primal-dual meets local search: approx imating MS T’ s with nonun iform deg ree bounds . In Pr oceedings of the 35th Annual ACM Sympos ium on Theory of Computing (STOC) , pages 389–39 5, 2003. [14] L. C. Lau, J. Naor , M. R. Salav atipou r , and M. Singh. Survi v able network des ign with degree or order constr aints. In Pr oceeding s of the 39th Annual A CM Symposi um on Theory of Computing (STOC) , pages 651–66 0, 2007. [15] L. C. Lau and M. Singh. Additi ve approx imation for bounded degre e survi vabl e network design. In Pr oceedin gs of the 40th Annual A CM Symposi um on T heory of Computing (STOC) , pages 759–7 68, 2008. [16] B. R agha vac hari. A lgorith m s for finding low de gr ee structur es , pages 266–295. PWS Publishing Co., Boston, MA, USA, 1997 . [17] R. Ravi, M. V . Marathe, S. S. R a vi, D. J. Rosenkran tz, and H. B. H unt III. Approximatio n algorithms for degr ee-con straine d minimum-cos t network-des ign proble ms. A lgorith m ica , 31(1):58–7 8, 2001. [18] M. Singh and L. C . Lau. Appro ximating minimum bounded degree spanning trees to w ithin one of optimal. In Pr oceedings of the 39th A nnual A CM Symposium on Theory of Computing (STOC) , pag es 661–6 70, 2007. 13