This note extends a result of Frank, Jordan, and Szigeti [3] on parity constrained orientations with connectivity requirements. Given a hypergraph H, a non-negative intersecting supermodular set function p, and a preferred in-degree parity for every node, a formula is given on the minimum number of nodes with wrong in-degree parity in an orientation of H covering p

J, Szabo

Király, Tamás

ELTE Digital Institutional Repository (EDIT)

Egerv´ary Research Groupon Combinatorial OptimizationTechnical reportSTR-2003-11. Published by the Egrerva´ry Research Group, Pa´zma´ny P. se´ta´ny 1/C,H–1117, Budapest, Hungary. Web site: www.cs.elte.hu/egres . ISSN 1587–4451.A note on parity constrainedorientationsTama´s Kira´ly and Ja´cint Szabo´December 2003EGRES Technical Report No. 2003-11 1A note on parity constrained orientationsTama´s Kira´ly⋆ and Ja´cint Szabo´⋆⋆AbstractThis note extends the results of Frank, Jorda´n, and Szigeti [1] on parityconstrained orientations with connectivity requirements. Given a hypergraphH, a non-negative intersecting supermodular set function p, and a preferredin-degree parity for every node, a formula is given on the minimum number ofnodes with wrong in-degree parity in an orientation ofH covering p. It is shownthat the minimum number of nodes with wrong in-degree parity in a stronglyconnected orientation cannot be characterized by a similar formula.1 IntroductionIn [1], Frank, Jorda´n, and Szigeti proved that the existence of a parity-constrainedk-rooted-connected orientation of a graph can be characterized by a partition-typecondition. In this note it is shown that the requirement of k-rooted-connectivity canbe replaced by any requirement given by a non-negative intersecting supermodular setfunction. We also extend the characterization to hypergraphs, and show a min-maxformula on the minimal number of nodes violating the parity condition. The proof isbased on the ideas in [1]. In the last chapter we show that it is not possible to give asimilar characterization for parity-constrained strongly connected orientations.For a hypergraph H = (V, E) and a set X ⊆ V , iH(X) denotes the number ofhyperedges of H spanned by X. For a partition F , eH(F) denotes the number ofcross-hyperedges of H; in other words,eH(F) = |E| −∑X∈FiH(X). (1)A directed hypergraph consists of hyperarcs: hyperedges that have one node des-ignated as head node. An orientation of a hypergraph H is a directed hypergraphobtained by selecting a node from each hyperedge of H as head node. For a directed⋆MTA-ELTE Egerva´ry Research Group, Department of Operations Research, Eo¨tvo¨s University,Pa´zma´ny Pe´ter se´ta´ny 1/C, Budapest, Hungary, H-1117. e-mail: tkiraly@cs.elte.hu . Supportedby the Hungarian National Foundation for Scientific Research, OTKA T037547 and OTKA N034040.⋆⋆Department of Operations Research, Eo¨tvo¨s University, Pa´zma´ny Pe´ter se´ta´ny 1/C, Budapest,Hungary, H-1117. e-mail: jacint@cs.elte.hu . The author is a member of the MTA-ELTEEgerva´ry Research Group (EGRES). Supported by the Hungarian National Foundation for ScientificResearch, OTKA T037547 and OTKA N034040.December 2003Section 2. Main result 2hypergraph D = (V,A) and a set X ⊆ V , ̺D(X) denotes the number of hyperarcs inD which have their head node inside X and have at least one node outside of X.A set function p : 2V → Z is called intersecting supermodular ifp(X) + p(Y ) ≤ p(X ∩ Y ) + p(X ∪ Y )holds whenever X ∩ Y 6= ∅. A set function p : 2V → Z is monotone decreasing ifp(X) ≥ p(Y ) whenever ∅ 6= X ⊆ Y . We always assume that p(∅) = 0. Clearly,if p(V ) = 0 and p is monotone decreasing, then p is non-negative. For intersectingsupermodular functions the converse is also true:Claim 1.1. If p is intersecting supermodular, non-negative, and p(V ) = 0, then p ismonotone decreasing.Proof. Let ∅ 6= X ( Y ⊆ V , and let Z := (V − Y ) ∪ X. By the intersectingsupermodularity and non-negativity of p, p(Y ) ≤ p(Y )+p(Z) ≤ p(Y ∩Z)+p(Y ∪Z) =p(X) + p(V ) = p(X).An orientation D = (V,A) of a hypergraph H = (V, E) covers a set function p if̺D(X) ≥ p(X) for every X ⊆ V . If the in-degrees of the orientation are specified,then the following is true (see e.g. [2]):Lemma 1.2. Let H = (V, E) be a hypergraph, p : 2V → Z+ a non-negative setfunction, and m : V → Z+ an in-degree specification such that m(V ) = |E|. Then Hhas an orientation covering p such that the in-degree of each node v ∈ V is m(v) ifand only ifm(X) ≥ iH(X) + p(X) for every X ⊆ V .For non-negative intersecting supermodular set functions, the following can beproved using basic properties of polymatroids:Theorem 1.3. Let H = (V, E) be a hypergraph, and p : 2V → Z+ an intersectingsupermodular and non-negative set function. Then H has an orientation covering pif and only ifeH(F) ≥∑X∈Fp(X) for every partition F . (2)2 Main resultLet H = (V, E) be a hypergraph, T ⊆ V a fixed set, and p : 2V → Z a set functionsuch that p(V ) = 0. An orientation of H is called (p, T )-feasible if it covers p andthe in-degree of v ∈ V is odd if and only if v ∈ T . A set X ⊆ V is called evenif |X ∩ T | + iH(X) + p(X) is even; X is called odd if |X ∩ T | + iH(X) + p(X) isodd (the notion of odd and even sets will be used with respect to different H, T ,and p values, but the actual meaning will always be clear from the context). Clearly,EGRES Technical Report No. 2003-11Section 2. Main result 3̺D(X) ≥ p(X) + 1 must hold for an odd set X in a (p, T )-feasible orientation of H.We define the following set function:pT (X) :={p(X) if X is even,p(X) + 1 if X is odd.(3)Note that pT depends on H too. The definition implies thatpT (X) ≡ |X ∩ T |+ i(X) mod 2 (4)for every X ⊆ V . Given a partition F , the valueµT (F) :=∑Z∈FpT (Z)− eH(F)is called the deficiency of F , which depends also on H and p.Claim 2.1. For given H, T , and p, the deficiency of every partition has the sameparity.Proof. According to (1), the deficiency of a partition F has the same parity as |E|+∑Z∈FiH(Z) +∑Z∈FpT (Z), which by (4) has the same parity as |E|+ |T |.It is easy to see that if an orientation D of H is (p, U)-feasible for some U ⊆ V ,then |T∆U | ≥ max{µT (F) : F is a partition}. The main result of this note is that ifp is non-negative, intersecting supermodular, and there exists an orientation coveringp, then equality can be attained.Theorem 2.2. Let H = (V, E) be a hypergraph, T ⊆ V a fixed set, and p : 2V →Z+ an intersecting supermodular and non-negative set function for which p(V ) = 0.Suppose that H has an orientation covering p, i.e. (2) holds. Then there exists a setU ⊆ V such that|T∆U | = max{µT (F) : F is a partition} (5)and H has a (p, U)-feasible orientation.Proof. Indirectly, let us consider a counterexample where |V | + |E| is minimal. Apartition F is called non-trivial if F 6= {V }. Let µ := max{µT (F) : F is a non-trivialpartition}. If µ is negative and odd, then the deficiency of the trivial partition is 1.Let us delete an arbitrary hyperedge fromH. By induction, the remaining hypergraphhas a (p, T )-feasible orientation. By adding the deleted hyperedge oriented arbitrarily,we get an orientation satisfying (5).If µ is negative and even, then we delete an arbitrary hyperedge e from H, andlet T ′ := T∆{v} for some v ∈ e. By induction, this hypergraph has a (p, T ′)-feasibleorientation. By adding the hyperarc e with head v, we get a (p, T )-feasible orientation.In the following we assume that µ is non-negative. Let F∗ = {V1, . . . , Vt} be a non-trivial partition of maximal cardinality for which µT (F∗) = µ holds. Let H∗ = (V∗, E∗)EGRES Technical Report No. 2003-11Section 2. Main result 4denote the hypergraph obtained by contracting each partition member Vi to a nodevi, let p∗ denote the contracted set function, and let T∗ ⊆ V∗ consist of the nodes vifor which |Vi ∩ T | + iH(Vi) is odd (thus pT∗∗ (vi) = pT (Vi) for every i). It follows fromthe choice of F∗ that∑v∈XpT∗∗ (v) − pT∗∗ (X) − iH∗(X) is non-negative and even forevery X ⊆ V∗.First we transform T into a set U such that |T∆U | = µ,∑ti=1 pU∗∗ (vi) = |E∗|, and∑v∈XpU∗∗ (v) − pU∗∗ (X) − iH∗(X) ≥ 0 for every X ⊆ V∗. If µ = 0 then U := Tis appropriate, so suppose that µ > 0. An even set X ⊆ V∗ is called critical if∑v∈XpT∗∗ (v) − pT∗∗ (X) − iH∗(X) = 0; thus every even singleton is critical. By theintersecting supermodularity of p, the intersection and union of intersecting criticalsets are critical. If every node of V∗ is covered by a critical even set, then there is apartition of V∗ consisting of critical even sets, which induces a partition on V thatviolates (2). Thus there is an odd singleton vi ∈ V∗ that is not covered by a criticaleven set. Let T ′ := T∆{v} for an arbitrary v ∈ Vi. Then∑ti=1 pT ′∗∗ (vi) = |E∗|+ µ− 1,and∑v∈XpT ′∗∗ (v) − pT ′∗∗ (X) − iH∗(X) ≥ 0 holds for every X ⊆ V∗. If we repeat theabove procedure µ times, we obtain the required U .Claim 2.3. There exists a (p∗, U∗)-feasible orientation D∗ of H∗ for which the in-degree of vi is pU∗∗ (vi) (i = 1, . . . , t).Proof. We know that∑ti=1 pU∗∗ (vi) = |E∗|. Lemma 1.2 implies that a good orientationexists if and only if∑v∈XpU∗∗ (v) ≥ pU∗∗ (X)+iH∗(X) for everyX ⊆ V∗. This is satisfieddue to the way U was constructed.Let D∗ be a fixed (p∗, U∗)-feasible orientation of H∗, and let D0 denote the directedhypergraph on V corresponding to the hyperarcs of D∗. From now on, the parity ofsets is determined with respect to U or U∗. The next step is to show that it is possibleto obtain a directed hypergraph D′∗ by deleting exactly one hyperarc entering eachodd singleton {vi}, such that ̺D′∗(X) ≥ p∗(X) still holds for every X ⊆ V∗. Let {vi}be an odd singleton. If a hyperarc a with head vi cannot be deleted, then there existsan even set Xa ⊆ V∗ such that a enters Xa and ̺D∗(Xa) = p∗(Xa). We call such aset tight – notice that every tight set is even. Since D∗ is a feasible orientation andp∗ is intersecting supermodular, the intersection and union of intersecting tight setsare also tight sets. Thus if no hyperarc with head vi can be deleted, then there existsa tight set X ⊆ V∗ such that every hyperarc of D∗ with head vi enters X. But thisis impossible, since ̺D∗(X) = p∗(X) ≤ p(Vi) < pU(Vi) = ̺D∗(vi) by the monotonedecreasing property of p and the fact that Vi is an odd set. Therefore we can delete ahyperarc a with head vi, and change U∗ by adding/deleting vi, so that {vi} becomesan even set.By repeating the above operation for every odd singleton {vi} (always consideringthe updated parameters when deciding the parity of sets), we get a directed hyper-graphD′∗. LetD′0 denote the directed hypergraph on V corresponding to the hyperarcsof D′∗, let H′ denote the hypergraph obtained from H by deleting the hyperedges cor-responding to hyperarcs in D0 − D′0, and let U′ be the new parity requirement set,i.e. U ′ := U∆{v : ̺D0−D′0(v) = 1}. It is easy to see that ̺D′0(X) ≥ p(X) holds ifX is the union of some members of F∗, and ̺D′0(Vi) = p(Vi) = pU ′(Vi) for every i.EGRES Technical Report No. 2003-11Section 2. Main result 5Furthermore, if D′0 can be extended to a (p, U′)-feasible orientation of H ′, then D0can be extended similarly to a (p, U)-feasible orientation of H.In the following we construct an orientation of H[Vi] for every i, which togetherwith D′0 will give a (p, U′)-feasible orientation of H ′. Let pi : 2Vi → Z be defined aspi(X) := p(X)− ̺D′0(X) (X ⊆ Vi).Then pi is intersecting supermodular, monotone decreasing, and pi(Vi) = 0 since̺D′0(Vi) = p(Vi). We define Ui ⊆ Vi byUi := (U′ ∩ Vi)∆{v ∈ Vi : ̺D′0(v) ≡ 1 mod 2}.Let pUii: 2Vi → Z be the set function defined similarly to (3) but with respect toH[Vi], pi, and Ui.Claim 2.4. The following holds for each Vi and for every partition F of Vi:eH[Vi](F) ≥∑Z∈FpUii(Z). (6)Proof. Suppose that there is a partition F for which the inequality does not hold.Then eH[Vi](F) ≤∑Z∈FpUii(Z)− 2 by Claim 2.1. We define the following partition ofV : F i := F∪F∗−{Vi}. We consider the original deficiency of Fi: µT (Fi) = µT (F∗)−pT (Vi)+∑Z∈FpT (Z)−eH[Vi](F) ≥ µT (F∗)+∑Z∈FpUii(Z)−eH[Vi](F)−2 ≥ µT (F∗) =µ, since∑Z∈FpUii(Z) ≤∑Z∈FpT (Z)−̺D′0(Vi)+1 ≤∑Z∈FpT (Z)−pT (Vi)+2. ThusF i would be a partition of deficiency µ with more elements than F∗, in contradictionwith the way F∗ was chosen.By induction, Theorem 2.2 is true for H[Vi], pi, and Ui. Thus Claim 2.4 impliesthat there is an orientation Di of H[Vi] such that ̺Di(X) ≥ pi(X) for every X ⊆ Vi,and ̺Di(v) is odd if and only if v ∈ Ui.Let D′ be the directed hypergraph obtained as the union of D′0 and D1, . . . , Dt.The above property means that ̺D′(X) ≥ p(X) if X ⊆ Vi for some i, and ̺D′(v) isodd if and only if v ∈ U ′. The construction method of D′0 implies that ̺D′(X) ≥ p(X)also holds if X is the union of some members of F∗.Suppose that there are sets for which ̺D′(X) < p(X); let X be such a set, withthe property that X ⊆ Vi or Vi ⊆ X or X ∩ Vi = ∅ holds for a maximum number ofmembers of F∗. There must be a member Vi of F∗ for which none of those relationsare true, sinceX is neither a subset of a member of F∗, nor the union of some membersof F∗. Since ̺D′(Vi) = p(Vi), the intersecting supermodularity of p implies that either̺D′(X ∩ Vi) < p(X ∩ Vi) or ̺D′(X ∪ Vi) < p(X ∪ Vi). But both cases are impossibledue to the way X was chosen.We obtained that D′ is a (p, U ′)-feasible orientation of H ′. This means that if Dis the directed hypergraph obtained as the union of D0 and D1, . . . , Dt, then D is a(p, U)-feasible orientation of H. This completes the proof of Theorem 2.2EGRES Technical Report No. 2003-11Section 3. Remarks 63 RemarksThe Berge-Tutte formula on the size of a maximum matching in a graph G = (V,E)easily follows from Theorem 2.2. To see this, we define the graph G′ = (V ′, E ′) byadding one node ve to V for every e ∈ E, and by replacing every edge e = uv in Eby edges uve and vve. For v ∈ V , let p({v}) := dG(v) − 1, and let p(X) := 0 onevery other set. Let T consist of the nodes in V for which dG(v)− 1 is odd. It is easyto see that every orientation of G′ covering p determines a matching of G (an edgee = uv is in the matching if the orientation contains the directed edges uve and vve),and the number of nodes not covered by the matching equals the number of nodesthat do not match the parity-specification T . Therefore Theorem 2.2 implies that ifF is a partition of V ′ for which µT (F) :=∑Z∈FpT (Z) − eG′(F) is maximal, then2ν(G) ≥ |V | − µT (F). The following Claim proves the Berge-Tutte formula.Claim 3.1. Let W := {v ∈ V : {v} ∈ F}. thenµT (F) ≤ oddG(W )− |W |, (7)where oddG(W ) denotes the number of components of G[V −W ] having an odd numberof nodes. Thus 2ν(G) ≥ |V |+ |W | − oddG(W ).Proof. Let F ′ consist of the members of F which are not singletons in W . Thenp(X) = 0 for every X ∈ F ′. We can assume that there is no edge between twomembers of F ′, otherwise we can replace them by their union. By this assumption,the number of sets X ∈ F ′ for which pT (X) = 1 is at most oddG(W ). Thus µT (F) =∑Z∈FpT (Z) − eG′(F) =∑v∈W(dG(v) − 1) +∑Z∈F ′pT (Z) − eG′(F) ≤ oddG(W ) −|W |.It the next paragraphs we describe a few negative results concerning possible gen-eralizations of Theorem 2.2. First, let us state a corollary which follows easily fromTheorem 2.2.Corollary 3.2. Let H = (V, E) be a hypergraph, T ⊆ V a fixed set, and p : 2V →Z+ an intersecting supermodular and non-negative set function for which p(V ) = 0.Suppose that there exists an orientation of H covering pT (as defined in (3)). Thenthere exists a (p, T )-feasible orientation of H.One may try to extend this corollary to more general set functions. A possibility isto include upper bounds on the in-degrees of the nodes (which may violate intersectingsupermodularity). However, Frank, Sebo˝, and Tardos [3] showed that if p consists oflower and upper bounds on the in-degrees of nodes, then the equivalent of Corollary3.2 is not necessarily true.Another problem that is not contained in the intersecting supermodular case isto find a strongly connected orientation of a graph. In this case p(X) equals 1 forevery ∅ 6= X ( V . In the following we describe an example where the equivalent ofCorollary 3.2 for strongly connected orientations does not hold.Let G be the graph on the left side of Figure 1, let T be the set of black nodes. ThenG has no (p, T )-feasible orientation (i.e. it has no strongly connected orientation whereEGRES Technical Report No. 2003-11References 7XY YZtsFigure 1T is the set of nodes with odd in-degree). To see this, observe that in a (p, T )-feasibleorientation every node of X must have at least 2 in-edges, every node of Z must haveat least 2 out-edges, and every node of Y must either have an in-edge coming from X,or an out-edge going to Z. Thus the graph must have at least 2|X|+2|Z|+ |Y | = 38edges, but it has only 36.On the other hand, G has an orientation covering pT , as shown on the right sideof Figure 1. It is easy to check that the orientation is strongly connected, and thein-degree parity is incorrect only at the nodes of Y . Thus it suffices to show that thein-degree of every set separating Y is at least 2. This can be seen by checking thatthere are 2 edge-disjoint paths from s to any v ∈ Y , there are 2 edge-disjoint pathsfrom any v ∈ Y to t, and there are 2 edge-disjoint paths from t to s.References[1] A. Frank, T. Jorda´n, Z. Szigeti, An orientation theorem with parity conditions,Discrete Applied Mathematics 115 (2001), 37–45.[2] A. Frank, T. Kira´ly, Z. Kira´ly, On the orientation of graphs and hypergraphs,Discrete Applied Mathemathics 131 No. 2 (2003), 385–400.[3] A. Frank, A. Sebo˝, E´. Tardos, Covering directed and odd cuts, Mathematical Pro-gramming Study 22 (1984), 99–112. 385–400.EGRES Technical Report No. 2003-11

A NOTE ON PARITY CONSTRAINED ORIENTATIONS

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A NOTE ON PARITY CONSTRAINED ORIENTATIONS

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