A Unifying Model for Locally Constrained Spanning Tree Problems
Luiz Alberto do Carmo Viana, Manoel Campêlo, Ignasi Sau, Ana Silva
aa r X i v : . [ c s . CC ] M a y A Unifying Model for Locally ConstrainedSpanning Tree Problems
Luiz Alberto do Carmo Viana , Manoel Campˆelo , Ignasi Sau , and Ana Silva Campus de Crate´us, Universidade Federal do Cear´a, Crate´us, Brazil Dep. de Estatstica e Matemtica Aplicada, Universidade Federal do Cear´a, Fortaleza, Brazil LIRMM, Universit´e de Montpellier, CNRS, Montpellier, France Departamento de Matem´atica, Universidade Federal do Cear´a, Fortaleza, Brazil [email protected], [email protected], [email protected],[email protected]
May 22, 2020
Abstract
Given a graph G and a digraph D whose vertices are the edges of G , we investigatethe problem of finding a spanning tree of G that satisfies the constraints imposed by D . The restrictions to add an edge in the tree depend on its neighborhood in D . Here,we generalize previously investigated problems by also considering as input functions ℓ and u on E ( G ) that give a lower and an upper bound, respectively, on the numberof constraints that must be satisfied by each edge. The produced feasibility problem isdenoted by G-DCST , while the optimization problem is denoted by
G-DCMST . We showthat
G-DCST is NP -complete even under strong assumptions on the structures of G and D , as well as on functions ℓ and u . On the positive side, we prove two polynomialresults, one for G-DCST and another for
G-DCMST , and also give a simple exponential-time algorithm along with a proof that it is asymptotically optimal under the
ETH .Finally, we prove that other previously studied constrained spanning tree (
CST ) prob-lems can be modeled within our framework, namely, the
Conflict CST , the
ForcingCST , the
At Least One/All Dependency CST , the
Maximum Degree CST ,the
Minimum Degree CST , and the
Fixed-Leaves Minimum Degree CST . Let G be a graph and D be a (directed or undirected) graph whose vertices are the edgesof G . In other terms, D defines a relation on the edge set of G . The dependencies of anedge e ∈ E ( G ) are given by its (in-)neighborhood in D , i.e., by the set dep D ( e ) = { e ′ ∈ ( G ) | ( e ′ , e ) ∈ E ( D ) } . We omit the subscript D in dep D ( e ) whenever the dependencygraph is clear from the context.Many problems have been investigated under the light of dependencies between pairs ofobjects, such as the knapsack problem [26], bin-packing [24], maximum flow [40], schedul-ing problems [9], maximum matchings [19], shortest paths [19], or maximum acyclic sub-graphs [36]. Generally, the dependency problems defined on graphs can be described asthe problem of finding a subgraph H of G satisfying the dependency constraints imposedby D .However, the notion of dependency itself may vary. For example, every ( e, e ′ ) in D could mean that, whenever e ′ is chosen (not chosen), we get that e cannot (must) be chosen,thus expressing a conflict constraint ( forcing constraint ), with D being the conflict graph ( forcing graph ). In this paper, we introduce a generalization of dependency constrainedproblems, and investigate this generalization for spanning trees. In particular, our modelgeneralizes many of the constrained spanning tree problems that have been investigated inthe literature. Our contribution.
For the generalized version of dependency constrained problems,together with graph G and (di)graph D , we also consider functions ℓ and u that assign,to each e ∈ E ( G ), a lower and an upper bound on the number of dependencies that mustbe ensured for e . This means that a subgraph H ⊆ G satisfies the imposed constraintsif and only if the number of edges in E ( H ) ∩ dep ( e ) is at least ℓ ( e ) and at most u ( e ),for every e ∈ E ( G ) ; we then say that H ( ℓ, u ) -satisfies D . When H is asked to bea spanning tree, we call the problem the Generalized Dependency ConstrainedSpanning Tree problem, and denote it by
G-DCST . Also, sometimes we deal with therelated optimization problem by considering weights on the edges of G ; this is called the Generalized Dependency Constrained Minimum Spanning Tree problem, andis denoted by
G-DCMST . Let us observe that when ℓ ( e ) = 0 and u ( e ) ≥ | dep ( e ) | , for all e ∈ E ( G ), then G-DCST is equivalent to deciding whether G is connected, and G-DCMST corresponds to the classical
Minimum Spanning Tree problem.Clearly, the feasibility problem
G-DCST is a particular case of the optimization problem
G-DCMST , where the weight of each edge is equal to one. This is why, whenever possible, wegive preference to prove NP -completeness results for the feasibility problem, and get poly-nomial results for the optimization problem. We use reductions from (3 , , -SAT to proveour main NP -completeness results, whereas our polynomial results arise as consequencesof the Matroid Intersection Theorem [23] (cf. Section 2).Considering the generalized version of the spanning tree problem, given a graph G ,a digraph D = ( E ( G ) , A ), and functions ℓ, u , we prove that deciding whether G has aspanning tree that ( ℓ, u )-satisfies D is NP -complete in the following cases: We can always assume that 0 ≤ ℓ ( e ) ≤ u ( e ) ≤ | dep ( e ) | , and so ℓ ( e ) = u ( e ) = 0 if dep ( e ) = ∅ . ℓ ( e ) = u ( e ) = | dep ( e ) | for every e ∈ E ( G ), D is a forest of oriented paths of lengthat most two where all components are directed paths, out-stars, or in-stars, and G is an outerplanar chordal graph with diameter at most two. Furthermore, thisproblem cannot be solved in time 2 o ( n + m ) unless the ETH fails, where n = | V ( G ) | and m = | E ( G ) | ;(ii) When ℓ, u are constant functions, for every pair of constant values such that ℓ ≤ u ;(iii) When ℓ ( e ) = 0, and u ( e ) = | dep ( e ) | −
1, for every e ∈ E ( G ).On the positive side, we prove the following:(a) G-DCST can be solved in polynomial time when D is an oriented matching, and ℓ ( e ) = u ( e ) = | dep ( e ) | for every e ∈ E ( G );(b) G-DCMST can be solved in polynomial time when ℓ = 0, D is a collection of symmetriccomplete digraphs D , . . . , D k , and u ( e ) = u ( e ′ ) whenever e, e ′ are within the samecomponent of D ;(c) G-DCST can be solved in time O (2 m · ( n + m )), where n = | V ( G ) | and m = | E ( G ) | .It is worth observing that (a) and (i) define a dichotomy between polynomial and hardcases for G-DCST when regarding D as a family of oriented paths. It can be solved inpolynomial time if length of the longest path in the underlying graph of D is at most one,and it is NP -Complete otherwise.We also prove that many of the constrained spanning tree (CST) problems that havebeen investigated in the literature can be modeled with our general problem, namely the Conflict CST [18], the
Forcing CST [19], the
At Least One/All DependencyCST [45], the
Maximum Degree CST [20], the
Minimum Degree CST [2], and the
Fixed-Leaves Minimum Degree CST [22]. All our reductions preserve the value ofthe solutions, which means that also the optimization version of these problems can bemodeled within our framework.Notice that the previously mentioned CST problems impose (vertex-wise or edge-wise)local constraints to describe their set of feasible spanning trees. This contrasts with
Max-imum Diameter CST [10,11],
Minimum Diameter CST [27,28] (with variations [7,32])and
Maximum Leaves CST [25, 35], examples of NP -hard problems that impose con-straints on global tree parameters. In [21], the authors propose an approach that includesalso these global constraints, but from a practical point of view. Related work.
In what follows, we talk sometimes about the feasibility version of theproblems, and sometimes about the optimization version, where also a weight function onthe edges of the input graph is given. Also, when ℓ and/or u are constant functions, wewrite directly the constant value inside the parenthesis when saying whether a spanningtree ( ℓ, u )-satisfies D . 3 onflict constraints: Recall that, in the
Conflict Constrained (Minimum) SpanningTree problem, we are given a pair of graphs G and D such that V ( D ) = E ( G ), and wewant to know whether there exists a spanning tree (find a minimum spanning tree) T of G such that E ( T ) ∩ dep ( e ) = ∅ for every e ∈ E ( T ). We denote the feasibility problem by CCST and the optimization problem by
CCMST . Note that, if we consider D ′ as an arbitraryorientation of D (i.e., each edge e e in D gives rise to either ( e , e ) or ( e , e ) in D ′ ),then we get that such a tree exists if and only if there exists a spanning tree T that (0 , D ′ . This means that our problem generalizes this one and therefore inherits the NP -complete results, as well as might help with some polynomial cases. Also, observe thatthe problems related to results (ii) when ℓ = 0, (iii) and (b) can be seen as generalizationsof the conflict constrained problems in the sense that ℓ = 0 (i.e., no lower bound constraintis imposed), but u = 0.Problems CCST and
CCMST have been introduced in [19], where
CCMST is proved to bepolynomial-time solvable if the conflict graph is a matching, and
CCST is proved to be NP -complete if the conflict graph is a forest of paths of length at most two. From what is saidpreviously, we then get that G-DCMST ( G, D, , , w ) is polynomial when D is an orientedmatching, and G-DCST ( G, D, ,
0) is NP -complete when D is an orientation of a forest ofpaths of length at most two. When D is one of these digraphs, observe that ∆ − ( D ) ≤ − ( D ) <
2, the other possibilities for constant values of ℓ, u are ℓ = 0 and u = 1,which is trivially polynomial, and ℓ ( e ) = u ( e ) = | dep ( e ) | for every e ∈ E ( G ). For the lattercase, we have results (i) and (a), which leaves open only the complexity of the optimizationproblem when D is an oriented matching. On the other hand, for ∆ − ( D ) = 2, i.e., D contains a forest of in-stars with at most two leaves, result (i) shows NP -completenesswhen ℓ ( e ) = u ( e ) = | dep ( e ) | for every e ∈ E ( G ), the other values of ℓ and u remainingopen.The CCST and
CCMST problems have also been investigated in [47], where the authorsprove that, if the input graph G is a cactus, then CCST is polynomial, while
CCMST is still NP -hard. They further show that the optimization problem is polynomial if the conflictgraph D can be turned into a collection of cliques by the removal of a constant numberof vertices, i.e., there exists a subset E ′ ⊆ E ( G ) = V ( D ) such that D − E ′ is a collectionof cliques, and | E ′ | is bounded by a constant. We prove something similar here for thegeneralized problem (result (b)).In [31], the authors investigate a conflict constrained problem where the conflict graphis only allowed to contain an edge ee ′ if e and e ′ share an endpoint in G (they called these forbidden transitions ). Among other results, they prove that the feasibility problem is NP -complete even if the input graph G is a complete graph. Practical approaches to theconflict constrained problem have been presented in [13, 42, 47].Another interesting, recently defined, problem that can be modeled as a conflict con- This means that ℓ ( e ) = k and u ( e ) = k ′ ≥ k , for all e ∈ E ( G ) such that dep ( e ) = ∅ , and ℓ ( e ) = u ( e ) = 0if dep ( e ) = ∅ . G-DCST ) is the so-called
Angular Constrained Spanning Tree problem [6]. In this problem, we are given aset V of points on the plane, a graph G = ( V, E ), and an angle α . A spanning tree T iscalled an α -spanning tree if, for every point v ∈ V , there is an angle on v of size smallerthan α containing all the edges (line segments) of T incident to v . Observe that, if we let D contain an arc ( vu, vw ) whenever the smaller angle formed by vu and vw is bigger than α , then an α -spanning tree T also (0 , D , and vice-versa. Besides, the conflictsin this case are forbidden transitions. In [6], one can find references on the decision versionof the problem, while the optimization version is investigated in [17]. Forcing constraints:
Recall that, in the
Forcing Constrained (Minimum) SpanningTree problem, we are given a pair of graphs G and D such that V ( D ) = E ( G ), and wewant to know whether there exists a spanning tree (find a minimum spanning tree) T of G such that E ( T ) ∩ { u, v } 6 = ∅ for every uv ∈ E ( D ). We denote the feasibility problem by FCST and the optimization problem by
FCMST .This problem was introduced in [19], where the authors prove that
FCST is NP -completeeven if the conflict graph is a forest of paths of length at most two. To the best of ourknowledge, this is the only existing paper that investigates this problem. Here, we show areduction from FCST ( G, D ) to
G-DCST ( G ′ , D ′ , ℓ, u ), where ℓ ( e ) ∈ { , } and u ( e ) = | dep ( e ) | for every e ∈ E ( G ′ ), and the maximum in-degree of D ′ is 2. If weights are being considered,such a reduction can be made to preserve the value of the solutions, and therefore it alsoapplies to the optimization problem. At least one/all dependency constraints:
The following two dependency constrained prob-lems are introduced in [45]. Given a graph G and a digraph D such that V ( D ) = E ( G ), onewants to know whether there exists a spanning tree T of G such that: E ( T ) ∩ dep ( e ) = ∅ forevery e ∈ E ( T ) with dep ( e ) = ∅ , called the At Least One Dependency ConstrainedSpanning Tree problem; or dep ( e ) ⊆ E ( T ) for every e ∈ E ( T ), called the All Depen-dency Constrained Spanning Tree problem. We denote these problems by
L-DCST and
A-DCST , and the related optimization problems by
L-DCMST and
A-DCMST , respectively.Note that these are special cases of our problem.In [45], it is proved that both
L-DCST and
A-DCST are NP complete, even if G is a planarchordal graph with diameter two or maximum degree three, and D is the disjoint union ofarborescences of height two. Here, we strengthen the constraints on D , while also getting alower bound on the running time of exponential algorithms for these problems (result (i)).Observe that this result comprises cases where the maximum in-degree of D is one, and sothe generalized problem coincides with both L-DCST and
A-DCST .Still in [45], the authors prove that
L-DCMST and
A-DCMST are W [2]-hard when parame-terized by the weight of a solution, and that, unless P = NP , they cannot be approximatedwith a ratio of ln | V ( G ) | even if: G is bipartite; the dependency relations occur only between5djacent edges of G ; and each weak component of D has diameter one. One can noticethat the weight of a solution in their W [2]-hardness reduction is O ( n ), where n = | V ( G ) | .This means that there is no FPT algorithm for
L-DCMST and
A-DCMST parameterized by n ,unless FPT = W [1]. This contrasts with the decision problem, which can be solved in time O (2 m · ( n + m )) = O (2 n · n ), where m = | E ( G ) | (result (c)). Maximum degree constraints:
Given a graph G = ( V, E ), and a positive integer k , the Maximum Degree Constrained Spanning Tree problem consists in deciding whether G has a spanning tree T such that d T ( v ) ≤ k for every v ∈ V ( G ), where d T ( v ) is thedegree of v in T . This problem was introduced in [20]. Observe that it is NP -complete,even for k = 2, since this case generalizes the Hamiltonian path problem [25]. In [39], itis proved to be NP -complete even for grid graphs of maximum degree three. Also, [39]tackles the Euclidean optimization version of the problem (i.e., vertices are points onthe plane, and edges are weighted according to the Euclidean distance). The Euclideanoptimization version remains NP -hard when k ≤
3, and is polynomial-time solvable when k ≥
5, remaining open for k = 4. Several heuristic, approximation, and exact approacheshave been proposed for the problem (see [8, 33, 43] and references therein).Here, we denote the feasibility version of this problem by MDST , and the optimizationversion by
MDMST . We present a reduction from
MDST ( G, k ) to
G-DCST ( G ′ , D, , u ) where u ( e ) ∈ { , k } for every e ∈ E ( G ′ ). The reduction also applies to the optimization problemsince it preserves the value of the solutions. Minimum degree constraints:
Given a graph G = ( V, E ), and a positive integer k , the Minimum Degree Constrained Spanning Tree problem consists in deciding whether G has a spanning tree T such that d T ( v ) ≥ k for every non-leaf vertex v of T . Here,we denote the feasibility version of this problem by mDST , and the optimization versionby mDMST . This problem was introduced in [2], where it is shown to be NP -hard for every k ∈ { , · · · , | V ( G ) | } . On the other hand, [2] proves that the problem can be solved (byinspection) for degree bounds between | V ( G ) | + 1 and | V ( G ) | −
1. In [3], the problem wasshown to be NP -hard for k = 3. The case k ≤ mDST is obtained when the set of leaves is fixed in the input.More formally, given a graph G , a subset C ⊆ V , and a positive integer k , it consists infinding a spanning tree T of G such that d T ( v ) ≥ k , for every v ∈ C , and d T ( v ) = 1,for every v ∈ V \ C . We denote the feasibility version of this problem by FmDST , and theoptimization version by
FmDMST . This problem was introduced in [22], where the authorsprove that
FmDST is NP -complete for k ≥
2, and
FmDMST is NP -hard even for completegraphs. Also, some necessary and sufficient conditions are given for feasibility.Here, we present a reduction from both mDST ( G, k ) and
FmDST ( G, C, k ) to
G-DCST ( G ′ , D, ℓ, u ) where ℓ ( e ) ∈ { , , k } and u ( e ) ∈ { , | dep ( e ) |} for every e ∈ E ( G ).6gain, our reduction preserves the values of the solutions and therefore works for theoptimization version as well. Applications. As G-DCST generalizes all these problems, it inherits their applications,such as design of wind farm networks [12], VLSI global routing [41], or low-traffic com-munication networks [37]. In particular, dependency relations can model communicationsystems with protocol conversion restrictions [44]. Besides, we can get unified results forall of them by considering
G-DCST . Organization.
In Section 2, we present the formal definitions and notation used through-out the paper; in Section 3 we present our NP -complete results; in Section 4, our positiveresults; in Section 5, we show how to model the many constrained spanning tree problemsas special cases of our problem; and in Section 6, we discuss our results and pose someopen questions. Graphs.
For missing basic definitions on graph theory, we refer the reader to [46]. Let G be a simple graph (henceforth called simply a graph), and D be a digraph. We denote by E ( G ) , E ( D ) the edge set of G and arc set of D , respectively. Also, we denote an edge { u, v } of G by uv , and arc with head v and tail u of D by ( u, v ). We say that D is symmetric if ( v, u ) ∈ E ( D ) whenever ( u, v ) ∈ E ( D ). A (di)graph G ( D ) is complete if uv ∈ E ( G )( { ( u, v ) , ( v, u ) } ⊆ E ( D )) for every pair of vertices u and v in G ( D ). It is empty if has noedges (no arcs).If C ⊆ V ( G ) is such that uv ∈ E ( G ) for every u, v ∈ V ( G ), u = v , then we call C a clique . And if there are no edges between vertices in C , we say that C is an independentset . A vertex v ∈ V ( G ) is called universal if N ( v ) = V ( G ) \ { v } , where N ( v ) stands for theset of neighbors of v . A tree T is called a star if it has a universal vertex v , called center .Similarly, an out-star ( in-star ) is a directed graph D with a vertex v such that any othervertex is an out-neighbor (in-neighbor) of v and V ( D ) \ { v } is an independent set. Definition of the problems.
Let G = ( V, E ) be a graph and D = ( E, A ) be a digraphwhose vertices are the edges of G . We say that e ∈ E is a D -dependency of e ∈ E if( e , e ) ∈ A . For each e ∈ E , we define its D -dependency set as dep D ( e ) = { e ′ ∈ E :( e ′ , e ) ∈ A } , and for E ′ ⊆ E , let dep D ( E ′ ) = ∪ e ∈ E ′ dep D ( e ); from now on we omit D fromthe subscript whenever it is clear from the context. Also, let ℓ, u : E → N be functionsthat assign a non-negative integer to each edge of G . We say that a subgraph H of G ( ℓ, u ) -satisfies D if ℓ ( e ) ≤ | dep ( e ) ∩ E ( H ) | ≤ u ( e ), for every e ∈ E ( H ).We introduce the Generalized Dependency Constrained Spanning Tree prob-lem as, given a graph G , a digraph D = ( E ( G ) , A ), and functions ℓ, u : E ( G ) → N , deciding7hether there exists a spanning tree T of G such that T ( ℓ, u )-satisfies D . We abbreviatethis with G-DCST ( G, D, ℓ, u ). Observe that it corresponds to the feasibility problem. Ifwe are also given a weight function w : E → R , then we define the Generalized De-pendency Constrained Minimum Spanning Tree problem as the problem of findinga spanning tree T ∗ of G that minimizes the weight sum and that ( ℓ, u )-satisfies D ; thisproblem is denoted by G-DCMST . Polynomial reductions and Exponential Time Hypothesis.
Given problems Π andΠ ′ , we write Π (cid:22) P Π ′ if there exists a polynomial reduction from Π to Π ′ . This means thatproblem Π ′ is at least as hard as problem Π. The Exponential Time Hypothesis (denotedby
ETH ) of Impagliazzo et al. [29, 30] states that the problem cannot be solved intime 2 o ( n + m ) , where n is the number of variables and m the number of clauses of the inputformula. In particular, if it is possible to reduce to problem Π and the producedinstance has size linear in the size of the input formula, then the ETH implies that problemΠ cannot be solved in time 2 | x | either, where | x | denotes the size of the input of Π. Werefer the reader to [5] for basic background on computational complexity. Parameterized complexity.
We refer to [16] for a recent monograph on parameterizedcomplexity. Here, we recall only some basic definitions. A parameterized problem is adecision problem whose instances are pairs ( x, k ) ∈ Σ ∗ × N , where k is called the parameter .A parameterized problem L is fixed-parameter tractable ( FPT ) if there exists an algorithm A , a computable function f , and a constant c such that, given an instance I = ( x, k ) of L ,we get that A (called an FPT algorithm ) correctly decides whether I ∈ L in time boundedby f ( k ) · | I | c . For instance, the Vertex Cover problem parameterized by the size of thesolution is
FPT .Within parameterized problems, the class W [1] may be seen as the parameterized equiv-alent to the class NP of classical optimization problems. Without entering into details(see [16] for the formal definitions), a parameterized problem being W [1]- hard can be seenas a strong evidence that this problem is not FPT . The canonical example of W [1]-hardproblem is Independent Set parameterized by the size of the solution. The class W [2] ofparameterized problems is a class that contains W [1], and such that the problems that are W [2]- hard are even more unlikely to be FPT than those that are W [1]-hard (again, see [16]for the formal definitions). The canonical example of W [2]-hard problem is DominatingSet parameterized by the size of the solution.
Matroids.
We state here some basic tools about matroids that we will use in the algo-rithms of Section 4, and we refer to [34, 38] for more background. A (finite) matroid M isa pair ( E, I ), where E is a finite set, called the ground set , and I is a family of subsets of E , called the independent sets , satisfying the following properties:1. The empty set is independent, that is, ∅ ∈ I .8. Every subset of an independent set is independent, that is, for each A ′ ⊆ A ⊆ E , if A ∈ I then A ′ ∈ I . This is called the hereditary property .3. If A, B ∈ I with | A | > | B | , then there exists x ∈ A \ B such that B ∪ { x } ∈ I . Thisis called the augmentation property .Every graph or multigraph G = ( V, E ) gives rise to a so-called graphic matroid having E as ground set, and a set F ⊆ E is independent if and only if G [ F ] is acyclic.Given a collection E = { E , E , . . . , E k } of pairwise disjoint sets, and integers { d , . . . , d k } such that 0 ≤ d i ≤ | E i | for every i ∈ [ k ] = { , . . . , k } , the partition matroid with ground set E = S ki =1 E i has S ⊆ E as an independent set if and only if | S ∩ E i | ≤ d i for every i ∈ [ k ].The Matroid Intersection Theorem, proved by Edmonds [23], states that the problemof finding a largest common independent set of two matroids over the same ground set canbe solved in polynomial time. NP -completeness results In this section, we present our NP -complete results. First, we impose in Section 3.1 con-straints on the structure of D (also getting constraints on G as a byproduct), and then wefocus in Section 3.2 on hardness results imposing constraints on functions ℓ and u . D We prove result (i). First, we consider the case where D is a forest of out-stars with at mostthree vertices. Later we show that the orientation of D in the reduction can be changed toget a forest of directed paths of length at most two. In addition, in Theorem 3, we modifythe reduction to obtain D as a forest of in-stars with at most three vertices, thus closingall the possible orientations of forest of paths of length at most two. Theorem 1.
G-DCST ( G, D, ℓ, u ) is NP -complete, even when ℓ ( e ) = u ( e ) = | dep ( e ) | forevery e ∈ E ( G ) , D is a forest of out-stars of maximum degree two, and G is an outerplanarchordal graph with diameter at most two. Furthermore, this problem cannot be solved intime o ( n + m ) unless the ETH fails, where n = | V ( G ) | and m = | E ( G ) | .Proof. Notice that, given a spanning tree T , one can check whether T ( ℓ, u )-satisfies D in polynomial time; hence, G-DCST ( G, D, ℓ, u ) is in NP . To prove NP -completeness, wepresent a reduction from (3 , , -SAT to G-DCST . In the (3 , , -SAT problem, we are givena CNF formula φ where each clause has at most three literals, and each variable appearsat most twice positively and at most twice negatively. This problem is well-known to be NP -complete [14, 25]. So consider a CNF formula φ on n variables and m clauses; we buildan instance ( G, D, ℓ, u ) of
G-DCST as follows (follow the construction in Figure 1):9
Add to G vertex v , and vertices v x , v ¯ x , and w x related to each variable x , and atmost three vertices { v c , v c , v c } related to each clause c (these vertices represent theliterals in c ). Then, make v adjacent to every other vertex; v x adjacent to v ¯ x for everyvariable x ; and add for each clause c a path ( v c , v c ) or ( v c , v c , v c ), depending on howmany literals c has (observe that we can suppose that c has at least two literals).Edge vv x will be interpreted as the true assignment of x , while edge vv ¯ x as the falseone. • For each variable x , add arc ( v x v ¯ x , vw x ) to D . For every variable x and each occur-rence of x in a clause c , say as the i -th literal in c , add to D arc ( vv x , vv ic ) if x appearspositively in c , or arc ( vv x , vv ic ) if x appears negatively in c . • Finally, let ℓ ( e ) = u ( e ) = | dep ( e ) | for every e ∈ E ( G ). v x vw x v ¯ x v c v c v c (a) Graph G . v x v ¯ x vw x vv x vv c vv c vv ¯ x vv c (b) Digraph D . Figure 1: Illustration of the reduction from (3 , , -SAT in Theorem 1. In Figure 1(a), werepresent a variable gadget together with a gadget of a clause containing three literals. InFigure 1(b), for a variable x , we represent the dependency between v x v ¯ x and vw x , and alsothe arcs leaving vv x and vv ¯ x when x appears positively in c and c , and negatively in c ,being related to the first literal in each of these clauses.One can see that G is an outerplanar chordal graph, and that each component (differentfrom the one containing v x v x and vw x ) of D is an out-star from vv x or vv ¯ x , for some variable x ; we get ∆ + ( D ) ≤ φ .We build a spanning tree T of G with the following edges: for each variable x , add to Tv x v ¯ x , vw x , and either vv x if x is true, or vv ¯ x , if x is false; for each clause c = ( ℓ ∨ ℓ ∨ ℓ ),add path ( v c , v c , v c ) and an edge vv ic for some i ∈ { , , } such that ℓ i is a true literal(analogously when c has only two literals). Because, for every variable x , vw x , v x v ¯ x andexactly one edge among { vv x , vv ¯ x } are chosen, and for every clause c exactly one edgeamong { vv c , vv c , vv c } is chosen, apart from the path ( v c , v c , v c ), one can see that T is10ndeed a spanning tree of G . The dependencies can also be seen to be satisfied since weonly choose an edge vv ic if the corresponding literal is true (hence the dependency is chosentoo).Conversely, let T be a solution for G-DCST ( G, D, ℓ, u ). For each variable x , because vw x is a cut edge and ( v x v ¯ x , vw x ) ∈ E ( D ), we get that { vw x , v x v ¯ x } ⊆ E ( T ). Besides, foreach variable x , since vv x and vv ¯ x form a cut and also form a cycle with v x v ¯ x , we get thatexactly one between vv x and vv ¯ x is in T . We then assign x to true if vv x ∈ E ( T ), and tofalse otherwise. Now, consider a clause c = ( ℓ ∨ ℓ ∨ ℓ ); since the edges { vv c , vv c , vv c } form a cut, at least one of them is in T , say vv c ∈ E ( T ) and say that x is the variablerelated to ℓ . If ℓ = x , then ( vv x , vv c ) ∈ E ( D ), which implies that vv x ∈ E ( T ) and that x is true. And if ℓ = x , then ( vv ¯ x , vv c ) ∈ E ( D ), which implies that vv ¯ x ∈ E ( T ) and that x is false (therefore ℓ is true). In any case, c is satisfied. The case where c has only twoliterals is analogous.Finally, for the lower bound 2 o ( n + m ) , just observe that the constructed instance hassize linear in the size of the given formula.Observe that when either ∆ − ( D ) = 0 or ∆ + ( D ) = 0, we get that D is the emptygraph, and that G-DCST ( G, D, ℓ, u ) reduces to deciding whether G is connected. Also, bythe previous theorem, we get that the problem is NP -complete if ∆ − ( D ) = 1, thus giving usa dichotomy with regard to the value of ∆ − ( D ). Concerning ∆ + ( D ), the previous theoremtells us that the problem becomes NP -complete for ∆ + ( D ) = 2. With a small modificationon the previous reduction, we can also get a dichotomy with regard to ∆ + ( D ). Theorem 2.
G-DCST ( G, D, ℓ, u ) is NP -complete, even when ℓ ( e ) = u ( e ) = | dep ( e ) | forevery e ∈ E ( G ) , D is a union of directed paths with length at most two, and G is a chordalouterplanar graph with diameter two. Furthermore, this problem cannot be solved in time o ( n + m ) unless the ETH fails, where n = | V ( G ) | and m = | E ( G ) | .Proof. Consider the same construction from the Theorem 1, except that each out-star withtwo leaves is turned into a directed path of length two. Observe that if ( vv ic , uv x , vv jc ′ ) is apath in D , then the previous arguments might not work simply because we might be forcedto pick edge vv ic when variable x is set to true (i.e., edge uv x is chosen). However, in thiscase we can remove some of the edges of the path ( v c , v c , v c ) in order to avoid cycles. Asimilar argument is made for out-stars containing a vertex of type vv x .Recall that L-DCST ( G, D ) and
A-DCST ( G, D ) denote the dependency constrained span-ning tree problem (
G-DCST ) where at least one dependency (if any exists) or all dependenciesare satisfied, respectively. Also, note that, if ∆ − ( D ) ≤
1, then we get that
L-DCST ( G, D )and
A-DCST ( G, D ) coincide with
G-DCST ( G, D, ℓ, u ) by assigning ℓ ( e ) = u ( e ) = | dep ( e ) | forevery e ∈ E ( G ). Thus, the following corollary, which strengthens the results in [45], is adirect consequence of the previous two theorems.11 v x v x w x (a) Graph G (variable gad-get). va ℓ i c b ℓ i c w ℓ i c v ℓ c v ℓ c v ℓ c (b) Graph G (clause gad-get). vw x v x v x a ℓ i c b ℓ i c vw ℓ i c va ℓ i c vv ℓ i c vb ℓ i c vv ℓ i (c) Digraph D (variable and literal gad-gets). Figure 2: Illustration of the reduction from (3 , , -SAT used in Theorem 3. Corollary 1.
L-DCST ( G, D ) and A-DCST ( G, D ) are NP -complete, even if G is an outer-planar chordal graph with diameter two, and D is the union of out-stars with ∆ + ( D ) = 2 ,or the union of paths of lenght at most two. Furthermore, these problems cannot be solvedin time o ( n + m ) unless the ETH fails, where n = | V ( G ) | and m = | E ( G ) | . In [19], it is shown that
CCMST is NP -complete if the conflict graph is a forest of paths oflength at most two. An orientation of such a forest may lead to directed paths, out-stars,or in-stars. The NP -completeness of G-DCST in the first two cases is proved in Theorems 1and 2. The case of in-stars is approached next.
Theorem 3.
G-DCST ( G, D, ℓ, u ) is NP -complete, even when ℓ ( e ) = u ( e ) = | dep ( e ) | for ev-ery e ∈ E ( G ) , D is a forest of in-stars of maximum in-degree two, and G is an outerplanarchordal graph with diameter at most two. Furthermore, this problem cannot be solved intime o ( n + m ) unless the ETH fails, where n = | V ( G ) | and m = | E ( G ) | .Proof. Notice that a spanning tree T of G can be checked to ( l, u )-satisfy D in polynomialtime, thus G-DCST ( G, D, ℓ, u ) is in NP . To prove NP -completeness, we again make a reduc-tion from (3 , , -SAT to G-DCST . This way, consider a CNF formula φ on n variables and m clauses. We build an instance ( G, D, ℓ, u ) of
G-DCST as follows (see Figure 2): • Start by adding a vertex v , which will be universal. For each variable x , add to G vertices v x , v x , and w x , making them adjacent to v , and add edge v x v x ; see Figure2(a). Selecting edge vv x will correspond to the true assignment for x , and vv x to thefalse one. • For each clause c = ( ℓ ∨ ℓ ∨ ℓ ) with three literals, add to G vertices { v ℓ i c , a ℓ i c , b ℓ i c , w ℓ i c | i ∈ [3] } , and make them adjacent to v . Then, add path ( v ℓ c , v ℓ c , v ℓ c ), and edges { a ℓ i c b ℓ i c | i ∈ [3] } ; see Figure 2(b). Proceed analogously if c has two literals. For eachliteral ℓ i , selecting edge va ℓ i c will indicate that c is satisfied by ℓ i , and selecting edge vb ℓ i c will indicate that c must be satisfied by some of its other literals.12 For each variable x , add arc ( v x v x , vw x ) to D . For each clause c and each literal ℓ i of c , add arcs ( a ℓ i c b ℓ i c , vw ℓ i c ) , ( va ℓ i c , vv ℓ i c ), and ( vb ℓ i c , vv ℓ i ) to D ; see Figure 2(c). • Finally, let ℓ ( e ) = u ( e ) = | dep ( e ) | for every e ∈ E ( G ).Observe that G is an outerplanar chordal graph, and that each component of D is anin-star. We get ∆ − ( D ) ≤ φ .We build a spanning tree T of G as follows. For each variable x , choose edges v x v x and vw x ; then choose edge vv x if x is true, and edge vv x otherwise. For each clause c withthree literals, add edges { a ℓ i c b ℓ i c , vw ℓ i c | i ∈ [3] } . Also, for each i ∈ [3], add { vv ℓ i c , va ℓ i c } if ℓ i is true; otherwise, add vb ℓ i c . Finally, use path ( v ℓ c , v ℓ c , v ℓ c ) to connect any possiblydisconnected vertex. Proceed analogously if c has two literals. Denote by X the set ofvariables of φ , and by C the set of clauses; also, write ℓ i ∈ c to denote the fact that literal ℓ i appears in c . We first show that T is a spanning tree of G . It is easy to see that T spans { v x , v x , v, w x | x ∈ X } , and also every vertex of degree 1 in G . Now, given a clause c ,because each literal ℓ i in c is either true or false, we get that either va ℓ i c or vb ℓ i c is in T , andsince a ℓ i c b ℓ i c ∈ E ( T ) we get that T also spans { a ℓ i c , b ℓ i c | c ∈ C, ℓ i ∈ c } . Finally, for each clause c , we know that at least one of its literals is true, which means that at least one of the edgeslinking the path ( v ℓ c , v ℓ c , v ℓ c ) to v is chosen, and since it is always possible to choose edgesfrom this path to connect any possible remaining disconnected vertex, we are done. Now,we prove that dependencies are satisfied. Dependencies in { ( v x v x , vw x ) | x ∈ X } , and in { ( a ℓ i c b ℓ i c , vw ℓ i c ) | c ∈ C, ℓ i ∈ c } are all satisfied since all the involved edges are contained in T . Dependencies in { ( va ℓ i c , vv ℓ i c ) | c ∈ C, ℓ i ∈ c } are also valid because we only add theseedges together. Finally, given a variable x , if x is true, then we choose edge vv x , and vb ℓ i c for each clause c such that x is the i -th literal of c ; and if x is false then we choose edge vv x , and vb ℓ i c for each clause c such that x is the i -th literal of c . This settles the last typeof dependencies.Conversely, let T be a solution for G-DCST ( G, D, ℓ, u ). For each variable x , because vw x is a cut edge and ( v x v x , vw x ) ∈ E ( D ), we get that { vw x , v x v x } ⊆ E ( T ). Besides, for eachvariable x , since vv x and vv x form a cut and also form a cycle with v x v x , we get that exactlyone between vv x and vv x is in T . We then assign x to true if vv x ∈ E ( T ), and to falseotherwise. Now, consider a clause c = ( ℓ ∨ ℓ ∨ ℓ ); since the edges { vv ℓ c , vv ℓ c , vv ℓ c } forma cut, at least one of them is in T , say vv ℓ c ∈ E ( T ). Hence, because ( va ℓ c , vv ℓ c ) ∈ A ( D ),we get that va ℓ c ∈ E ( T ). But then, since a ℓ c b ℓ c ∈ E ( T ), we have that vb ℓ c / ∈ E ( T ), whichin turn implies that vv ℓ / ∈ E ( T ) because of the dependency ( vb ℓ c , vv ℓ ). We then concludethat vv ℓ must be chosen, henceforth ℓ is a true literal in c . The case where c has onlytwo literals is analogous.Finally, since the constructed instance has size linear in the size of φ , we obtain theclaimed lower bound 2 o ( n + m ) under the ETH .Let us observe that, similarly to Corollary 1, the results in Theorem 3 can be stated to13 -DCST ( G, D ), since this problem is equivalent to
G-DCST ( G, D, dep , dep ). This also extendsthe achievements from [45]. ℓ and u In this subsection, we examine the complexity of
G-DCST ( G, D, ℓ, u ) by focusing on func-tions ℓ and u . Recall that, given constants c, c ′ , G-DCST ( G, D, c, c ′ ) denotes the problemrestricted to instances where ℓ ( e ) = c and u ( e ) = c ′ for every e ∈ E ( G ). Given a function f : E → N and a positive integer c , we denote by f + c the function obtained from f byadding c to f ( e ) for every e ∈ E . The following lemma will be useful. Lemma 1.
Let c, ℓ, u be positive integers with u ≥ ℓ . Then, instances ( G, D, ℓ, u ) and ( G ′ , D ′ , ℓ + c, u + c ) of G-DCST are equivalent.Proof.
Choose any v ∈ V ( G ), and let G ′ be obtained from G by adding ℓ + c + 1 vertices ofdegree one pending in v ; denote the new edges by e , . . . , e ℓ + c +1 . Now, add the symmetricclique on vertices { e , . . . , e ℓ + c +1 } to D , and add e i e to D , for every i ∈ [ c ] and every e ∈ E ( G ). Let G ′ and D ′ be the obtained graph and digraph, respectively. Finally, let ℓ ′ ( e ) = ℓ + c and u ′ ( e ) = u + c , for every e ∈ E ( G ′ ). We prove that ( G, D, ℓ, u ) is a yes -instance of
G-DCST if and only if ( G ′ , D ′ , ℓ ′ , u ′ ) is a yes -instance of G-DCST .First, let T be a spanning tree of G that ( ℓ, u )-satisfies D . Let T ′ be obtained from T byadding e , . . . , e ℓ + c +1 . Clearly T ′ is a spanning tree of G ′ ; we prove that T ′ ( ℓ ′ , u ′ )-satisfies D ′ . Let e ∈ E ( G ′ ). If e ∈ E ( G ), because at least ℓ and at most u dependencies of e arein T , and because E ( T ′ ) ∩ dep D ′ ( e ) = ( E ( T ) ∩ dep D ( e )) ∪ { e , . . . , e c } , we get that at least ℓ + c and at most u + c dependencies of e are in T ′ . And if e ∈ { e , . . . , e ℓ + c +1 } , we getthat E ( T ′ ) ∩ dep D ′ ( e ) = { e , . . . , e ℓ + c +1 } \ { e } and again the constraints hold.On the other hand, let T ′ be a spanning tree of G ′ ; we know that { e , . . . , e ℓ + c +1 } ⊆ E ( T ′ ). Let T be obtained from T ′ by removing these edges, and consider e ∈ E ( G ). Itfollows that | dep D ( e ) ∩ E ( T ) | = | ( dep D ′ ( e ) ∩ E ( T ′ )) \ { e , . . . , e c }| = | dep D ′ ( e ) ∩ E ( T ′ ) | − c ,and since ℓ + c ≤ | dep D ′ ( e ) ∩ E ( T ′ ) | ≤ u + c we get that T ( ℓ, u )-satisfies D .First, we analyze the cases where ℓ = 0. Recall that in CCST , whenever an edge e ischosen, no dependencies of e can be chosen; this translates to having ℓ ( e ) = u ( e ) = 0 forevery e ∈ E ( G ). So in a sense, when one considers instances ( G, D, ℓ, u ) of
G-DCST where ℓ ( e ) = 0 for each e ∈ E ( G ), one can think of the problem as a “weak” version of CCST .It thus makes sense to ask whether this version turns out to be polynomial. Indeed, wenotice that
G-DCST ( G, D, ℓ, u ) with ℓ = 0 and u ( e ) ≥ | dep D ( e ) | , for each e ∈ E ( G ), is aneasily solvable problem since every spanning tree of G trivially ( ℓ, u )-satisfies D . In thefollowing theorem, we see that this is not the case when u is a constant function. Theorem 4.
Let c be a positive integer. Then G-DCST ( G, D, , c ) is NP -complete. roof. Recall that
CCST ( G, D ) is NP -complete [19] and equivalent to G-DCST ( G, D ′ , , D ′ is an arbitrary orientation of D . Given an instance ( G = ( V, E ) , D ) of CCST ,where D is an undirected graph with V ( D ) = E , we construct and equivalent instance( G ′ , D ′ , , c ) of G-DCST as follows (cf. Figure 3) • Let G ′ be obtained from G by adding a new vertex p and, for each i ∈ [ c ] and eachedge e ∈ E ( G ), adding a new vertex p ie . Then, make p adjacent to every p ie , and toan arbitrary vertex q ∈ V ( G ). More formally, G ′ = ( V ∪ V ′ , E ∪ E ′ ), where V ′ = { p } ∪ { p ie : e ∈ E, i ∈ [ c ] } and E ′ = { pq } ∪ { pp ie : e ∈ E, i ∈ [ c ] } . • Let D ′ be obtained from an arbitrary orientation of D by adding an arc ( pp ie , e ) forevery e ∈ E ( G ) and every i ∈ [ c ].The constructed instance has size clearly polynomial on the size of ( G, D ) (recall that c isa constant). Note that G ′ [ V ′ ∪ { q } ] is a star. Now, we show that ( G, D ) is a yes -instanceof
CCST if and only if ( G ′ , D ′ , , c ) is a yes -instance of G-DCST .First, let T be a solution for CCST ( G, D ), and let T ′ be obtained from T by adding E ′ .Clearly T ′ is a spanning tree of G ′ ; hence it remains to show that T ′ (0 , c )-satisfies D ′ .For this, consider e ∈ E ( T ′ ). If e ∈ E ( T ), then because T (0 , D , we have that E ( T ′ ) ∩ dep D ′ ( e ) = { pp e , . . . , pp ce } and therefore 0 ≤ | E ( T ′ ) ∩ dep D ′ ( e ) | = c . And if e ∈ E ′ ,we have that dep D ′ ( e ) = ∅ and trivially 0 = | dep D ′ ( e ) ∩ E ( T ′ ) | ≤ c .Conversely, let T ′ be a spanning tree of G ′ that (0 , c )-satisfies D ′ , and let T = T ′ [ V ].Because p separates V ′ \ { p } from V , we know that T is connected and, therefore, it isa spanning tree of G ; so it remains to show that T (0 , D . For this consider e ∈ E ( T ). Since each edge in E ′ is a cut edge in G ′ , we get that E ′ ⊆ E ( T ′ ). Therefore,since | E ( T ′ ) ∩ dep D ′ ( e ) | ≤ c and { pp c , . . . , pp ce } ⊆ E ( T ′ ) ∩ dep D ′ ( e ), we get that E ( T ′ ) ∩ ( E ( G ) \ E ′ ) = ∅ , i.e., | E ( T ) ∩ dep D ( e ) | = 0, as we wanted to show.Combining Lemma 1 and Theorem 4, we get result (ii), that is, G-DCST ( G, D, ℓ, u ) is NP -complete for every combination of constant values ℓ and u . Corollary 2.
For every pair of positive integers ℓ, u with ℓ ≤ u , we have that G-DCST ( G, D, ℓ, u ) is NP -complete.Proof. Let c = u − ℓ . By Theorem 4, we have that G-DCST ( G, D, , c ) is NP -complete, andby Lemma 1, we have that G-DCST ( G, D, ℓ, c + ℓ = u ) also is.As we have already mentioned, if u ( e ) = | dep ( e ) | for every e ∈ E ( G ), then G-DCST ( G, D, , u ) is easy since any spanning tree (0 , u )-satisfies D . Hence, it is natu-ral to ask whether the problem continues to be easy when u ( e ) is just slightly smaller than | dep ( e ) | . The following corollary is trivially obtained from previous results. It answers the15 v upp e p ce p ce p e G e e (a) G ′ . e e (b) G c . e e pp e pp ce pp e pp ce (c) D . Figure 3: Illustration of the reduction for
CCMST .aforementioned question negatively and yields result (iii). Given a positive integer c , wedenote by dep − c the function u : E ( G ) → N defined by u ( e ) = max {| dep ( e ) | − c, } . Corollary 3.
G-DCST ( G, D, , dep − is NP -complete, even when D is a collection ofdirected paths of length at most two.Proof. Assume that D is a collection of directed paths of length at most two. Then,∆ − ( D ) = 1 and therefore dep − G-DCST ( G, D, , CCST ( G, D ′ ), where D ′ isthe (undirected) underlying graph of D . Then D ′ is a collection of paths of length at mosttwo, and it is known that CCST ( G, D ′ ) is NP -complete [19]. In this section, we present some polynomial cases of the
G-DCST problem. Wheneverpossible, we deal with the optimization version instead, i.e., we consider that we are alsogiven a weight function w and that the objective is to find a spanning tree that ( ℓ, u )-satisfies D having minimum weighted sum-weight. This is denoted by G-DCMST ( G, D, ℓ, u, w ).Similarly to the previous section, we first focus in Section 4.1 on constraints on thedependency graph D , and then in Section 4.2 on constraints on the functions ℓ and u .Finally, we present in Section 4.3 a simple exponential-time algorithm to solve the problem. D We start by investigating the case where D is a collection of directed paths of length atmost one (note that there might be some edges of G that are isolated vertices in D ). So,given a graph G = ( V, E ) and a digraph D = ( E, A ), write E as { e , . . . , e m } and assumethat A = { ( e i , e i + t ) : i ∈ { , . . . , t }} , for some t ≤ ⌊ m ⌋ . Let S = { e i : i ∈ { t + 1 , . . . , t }} ,16.e., S is the set of the edges e ∈ E ( G ) with dep ( e ) = ∅ . The idea is to find a subset ofedges S ′ ⊆ S that connects the components of G − S and such that the dependencies of S ′ do not form a cycle in G . Let C = { C , . . . , C k } be the set of connected components of G − S , and let H be the multigraph obtained from G by contracting each C i to a singlevertex (loops are removed). Observe that there is an injection from the multiedges of H tothe edges in S . Hence, given S ′ ⊆ S , we can pick H ( S ′ ) to be the subgraph of H inducedby S ′ , i.e., H ( S ′ ) has C as vertex set, and there is an edge between C i , C j , i = j , for eachedge e ∈ S ′ with one endpoint in C i and the other one in C j . Observe that some of theedges of S might appear inside a component. However, as we will see, these edges canactually be ignored, and this is why we do not need to add the loops in H . Lemma 2.
Let G = ( V, E ) be a graph, D = ( E, A ) be a digraph, before, and ℓ, u be suchthat ℓ ( e ) = u ( e ) = | dep ( e ) | ∈ { , } for every e ∈ E ( G ) . If there exists S ′ ⊆ S suchthat H ( S ′ ) is a spanning tree of H and ( V, dep ( S ′ )) is acyclic, then G-DCST ( G, D, ℓ, u ) isfeasible. Conversely, any feasible solution of G-DCST ( G, D, ℓ, u ) contains such a subset S ′ .Proof. First, consider S ′ ⊆ S such that H ( S ′ ) is a spanning tree of H and ( V, dep ( S ′ )) isacyclic. Observe that, because ( V, dep ( S ′ )) is acyclic, S ′ ⊆ S , and S ∩ dep ( S ′ ) = ∅ , we getthat each C i is a connected component of G − S . Thus, we can add edges of G − S to( V, dep ( S ′ )) so as to obtain a spanning forest F of G having connected components withthe same vertex sets as C , . . . , C k . After this, just add edges of S ′ to F ; because H ( S ′ )is a spanning tree of H , we get that the obtained graph T connects all components of F without forming a cycle (i.e., T is a spanning tree of G ). Finally, since dep ( S ′ ) ⊆ E ( T )and S ∩ E ( T ) = S ′ , we get that T satisfies D .Conversely, let T = ( V, E T ) be a feasible solution of DCST ( G, D ). As T is a spanningtree of G , we get that the edges in E T ∩ S must connect the components of G − S , i.e., H ( E T ∩ S ) is connected. Thus, choose S ′ as the edge set of any spanning tree of H ( E T ∩ S ).We get that S ′ also forms a spanning tree of H , and since S ′ ⊆ E T and T satisfies D , weget that dep ( S ′ ) ⊆ E T and therefore ( V, dep ( S ′ )) cannot contain a cycle.In the following theorem we use the Matroid Intersection Theorem [23] to get result (a). Theorem 5.
Let G = ( V, E ) be a graph, D = ( E, A ) be a digraph such that each componentis a directed path of length at most , and ℓ, u be such that ℓ ( e ) = u ( e ) = | dep ( e ) | ∈ { , } for every e ∈ E ( G ) . Then DCST ( G, D ) can be solved in polynomial time.Proof. Given E ′ ⊆ E ( G ), denote by G ( E ′ ) the subgraph ( V ( G ) , E ′ ). Let H be ob-tained as before. Also, let I = { S ′ ⊆ S | G ( dep ( S ′ )) is acyclic } and I = { S ′ ⊆ S | H ( S ′ ) is acyclic } . We have that ( S, I ) and ( S, I ) are matroids (on a common groundset S ), since they are equivalent to the graphic matroids of the graph ( V, dep ( S )) and ofthe multigraph H , respectively. According to Lemma 2, DCST ( G, D ) is feasible if and onlyif there is S ′ ∈ I ∩ I such that | S ′ | = k −
1, where k is the number of components of G ′ = ( V, E \ S ). The existence of such S ′ can be checked in polynomial time using theMatroid Intersection Theorem [23]. 17 .2 Constraints on ℓ and u Recall that
CCMST ( G, G c , w ) is equivalent to G-DCMST ( G, D, , , w ) when D is a symmetricdigraph. From a result in [47] for the CCMST problem, we get that
G-DCMST ( G, D, , , w ) issolvable in polynomial time when D is the union of complete digraphs. We generalize thisresult for upper bound functions that have the same value on each clique (result (b)). Theorem 6.
Let ( G, D, , u, w ) be an instance of G-DCMST such that D = D ∪ D ∪· · ·∪ D k is the union of k disjoint complete digraphs and, for each i ∈ [ k ] , there exists u i ∈ [ | V ( D i ) | ] such that u ( e ) = u i − for every e ∈ V ( D i ) . Then, G-DCMST ( G, D, , u, w ) can be solved inpolynomial time.Proof. Note that, in this case, a solution for
G-DCMST ( G, D, , u, w ) can have at most u i edges from D i , for each i ∈ [ k ]. We show that solving such an instance can be formulatedas a matroid intersection problem.Observe that a subgraph T of G (0 , u )-satisfies D if and only if | E ( T ) ∩ D i | ≤ u i for every i ∈ [ k ]. Therefore, if M is the partition matroid on E ( G ) defined by { V ( D ) , . . . , V ( D k ) } and { u , . . . , u k } , we get that T ⊆ G is a solution for our problem if and only if T is aspanning tree of G and E ( T ) is an independent set of M . Hence, if M ′ is the graphicmatroid associated with G (where the independent sets are the sets of edges inducing aspanning tree of G ), we can solve our problem in polynomial time by applying the MatroidIntersection Theorem [23] to the matroids M and M ′ .Observe that the same approach used in the previous theorem does not work when thevalues of u can differ inside the same clique. For instance, suppose that D is the completedigraph on edges { e , . . . , e } and that u ( e ) = 1 and u ( e i ) = 2 for every i ∈ { , , } .Then S = { e , e } and S = { e , e , e } are acceptable subsets within a solution, howeverbecause of e , there does not exist e ∈ S \ S such that S ∪ { e } is an acceptable subset.This means that the augmentation property (cf. Section 2) is not satisfied and the subsetsthat define feasible solutions do not form a matroid. In this section, we present an exponential exact algorithm for
G-DCST ( G, D, ℓ, u ), as statesresult (c). Recall that, as a consequence of Theorem 3 in [45] for the optimization problems
L-DCMST and
A-DCMST , we get that the optimization problem
G-DCMST ( G, D, ℓ, u, w ) is W [2]-hard when parameterized by n = | V ( G ) | . The importance of the algorithm below, despiteits simplicity, is that it separates the complexity of the feasibility and the optimizationproblems. Theorem 7.
G-DCST ( G, D, ℓ, u ) can be solved in time O (2 m · ( n + m )) , where n = | V ( G ) | and m = | E ( G ) | . roof. It suffices to observe that, given a subset S ⊆ E ( G ), one can test in time O ( n + m )whether S forms a spanning tree of G , and whether the constraints imposed by D, ℓ, u aresatisfied by S . Because there are 2 m possible subsets to be tested, the theorem follows. As discussed in the introduction, the spanning tree problem has been investigated underthe most various constraints. In this section, we show that some of them can be modeledas special cases of our problem. We have already remarked that this is the case for the
Conflict Constrained Spanning Tree problem. One should also notice that, in allthe reductions presented below, when the feasibility version reduces to our problem, so doesthe optimization version. It is a matter of observing that the graph in the source instanceis a subgraph of the graph in the
G-DCST instance, and that the reduction preserves thesolution value when keeping the weights of the initial edges and setting to zero the weightsof the new edges.In Subsection 5.1, we present a reduction from
Forcing CST (denoted by
FCST ), inSubsection 5.2, a reduction from
Maximum Degree CST (denoted by
MDST ), and inSection 5.3, from
Minimum Degree CST and
Fixed-Leaves Minimum Degree CST (denoted by mDST and
FmDST , respectively).
Recall that, given graphs G and D such that V ( D ) = E ( G ), FCST consists in decidingwhether G has a spanning tree T such that | E ( T ) ∩ { e, f }| ≥ ef ∈ E ( D ). Theorem 8.
Denote by
G-DCST ∗ the problem G-DCST restricted to instances ( G ′ , D ′ , ℓ, such that ℓ ( e ) ∈ { , } for every e ∈ E ( G ′ ) and ∆ − ( D ′ ) = 2 . Then, FCST (cid:22) P G-DCST ∗ .Proof. Let (
G, D ) be an instance of
FCST , and construct G ′ , D ′ as follows (cf. Figure 4).Choose any v ∈ V ( G ), and let G ′ be obtained from G by adding, for each ee ′ ∈ E ( D ), apendant degree one vertex in v ; denote the new edge by p ee ′ . Then, let D ′ be the digraphwith vertex set E ( G ′ ) and arcs ( e, p ee ′ ) and ( e ′ , p ee ′ ) for every ee ′ ∈ E ( D ). Finally, let ℓ ( e ) = 0 if e ∈ E ( G ), and ℓ ( e ) = 1 otherwise. We prove that ( G, D ) is a yes -instance of
FCST if and only (
G, D, ℓ,
2) is a yes -instance of
G-DCST .First, let T be a solution for FCST , and let T ′ be obtained from T by adding p ee ′ forevery ee ′ ∈ E ( D ). Clearly T ′ is a spanning tree of G ′ , and since | E ( T ) ∩ { e, e ′ }| ≥ ee ′ ∈ E ( D ), we get that | E ( T ′ ) ∩ dep D ′ ( p ee ′ ) | ≥
1. The reverse implication can beproved similarly. 19 Gp ee ′ (a) G ′ . p ee ′ e e ′ V ( D ) (b) D ′ . Figure 4: Illustration of the reduction for
FCST . Given a graph G = ( V, E ) and a positive integer k , recall that in the MDST ( G, k ) problemwe want to find a spanning tree T such that d T ( v ) ≤ k , for every v ∈ V ( G ). We prove thata generalized version of this problem reduces to ours. In MDST ( G, d ∗ ), instead of being givenan integer k , we are given a function d ∗ : V ( G ) → N that separately sets upper bounds tothe degrees of the vertices. Theorem 9.
Denote by
G-DCST ∗∗ the problem G-DCST restricted to instances ( G, D, , u ) .Then, MDST (cid:22) P G-DCST ∗∗ .Proof. Let (
G, d ∗ ) be an instance of MDST . We build an equivalent instance ( G ′ , D, , u ) of G-DCST as follows (cf. Figure 5). • G ′ = ( V ∪ V ′ , E ∪ E ′ ), where V ′ = { v ′ | v ∈ V } has a copy of each vertex, and E ′ = { vv ′ | v ∈ V } connects each vertex v ∈ V to its copy v ′ ∈ V ′ . • D = ( E ∪ E ′ , A ), where A = { ( uv, uu ′ ) , ( uv, vv ′ ) | uv ∈ E } . • u ( e ) = 0, for each e ∈ E , while u ( vv ′ ) = d ∗ ( v ), for each v ∈ V .Observe that dep ( vv ′ ) is the set of edges incident to v in G for every v ∈ V , and that dep ( e ) = ∅ for every e ∈ E . Also, note that because ℓ ( e ) = u ( e ) = 0 = | dep ( e ) | for every e ∈ E , and ℓ ( e ) = 0 for every e ∈ E ′ , the only real constraints being imposed by D areupper bounds on the chosen dependencies for edges in E ′ . More specifically, we get that aspanning tree T ′ of G ′ ( ℓ, u )-satisfies D if and only if | E ( T ′ ) ∩ dep ( vv ′ ) | ≤ d ∗ ( v ) for every vv ′ ∈ E ′ ∩ E ( T ′ ). Note that each v ′ has degree one in G ′ , which implies that every edge in E ′ must be part of every spanning tree of G ′ . Thus, we get that ( V, E T ) ⊆ G is a tree thatsatisfies the maximum degree constraints if and only if ( V ∪ V ′ , E T ∪ E ′ ) ( ℓ, u )-satisfies D . 20 vu ′ v ′ (a) G ′ . uvuu ′ vv ′ (b) D . Figure 5: Illustration of the reduction for
MDST . Given a graph G and a positive integer k , recall that the mDST ( G, k ) problem consists infinding a spanning tree T of G such that d T ( v ) ≥ k for every nonleaf vertex v ∈ V ( T ).We introduce a generalized version of this problem, denoted by G-mDST ( G, ℓ, u ), where wereplace the integer k by functions ℓ, u : V → N and require that each nonleaf vertex v of T satisfies ℓ ( v ) ≤ d T ( v ) ≤ u ( v ). Clearly, mDST ( G, k ) is a special case of
G-mDST ( G, ℓ, u ) where ℓ ( v ) = k and u ( v ) = d ( v ) for every v ∈ V ( G ). Theorem 10.
G-mDST (cid:22) P G-DCST .Proof.
Given an instance ( G = ( V, E ) , ℓ, u ) of G-mDST , we build an instance ( G ′ , D, ℓ ′ , u ′ )of G-DCST as follows (cf. Figure 6): • G ′ = ( V ∪ V ′ , E ∪ E ′ ), where V ′ = { v , v , v | v ∈ V } and E ′ = { vv , vv , v v , v v | v ∈ V } ; • D = ( E ∪ E ′ , A ∪ A ), where A = { ( uv, vv ) , ( uv, vv ) | uv ∈ E } and A = { ( v v , v v ) , ( v v , v v ) | v ∈ V } ; • For each e ∈ E , let ℓ ′ ( e ) = u ′ ( e ) = 0; and for each v ∈ V , let ℓ ′ ( vv ) = ℓ ( v ), u ′ ( vv ) = u ( v ), and ℓ ′ ( e ) = u ′ ( e ) = 1, for each e ∈ { vv , v v , v v } .Observe that, for each v ∈ V and i ∈ { , } , we have that dep ( vv i ) is the set ofedges incident to v in G . We show that ( G, ℓ, u ) is a yes -instance of
G-mDST if and only( G ′ , D, ℓ ′ , u ′ ) is a yes -instance of G-DCST .First, let T = ( V, E T ) be a solution for G-mDST ( G, ℓ, u ). We expand T into T ′ =( V ∪ V ′ , E T ′ ) ⊆ G ′ , where E T ′ is equal to E T together with the following edges. Foreach v ∈ V , add v v and v v , and if v is a leaf in T then add vv , otherwise add vv .Observe that T ′ is a spanning tree of G ′ such that T = T ′ [ V ]. It remains to show that the D -dependencies are satisfied. Every edge e ∈ E T has dep ( e ) = ∅ and ℓ ′ ( e ) = u ′ ( e ) = 0, so ℓ ′ ( e ) ≤ | dep ( e ) ∩ E ( T ′ ) | ≤ u ′ ( e ) trivially follows. The remaining types of edges in E T ′ \ E T are analyzed below: 21 vv v v u u u (a) G ′ . uvuu uu vv vv u u u u v v v v (b) D . Figure 6: Illustration of the reduction for
G-mDST .(i) v i v , for v ∈ V and i ∈ { , } : recall that v v is the unique dependency of v v ,and vice-versa, and they are both in T ′ . Hence 1 = ℓ ′ ( v i v ) ≤ | dep ( v i v ) ∩ E ( T ′ ) | ≤ u ′ ( v i v ) = 1.(ii) vv : by construction of T ′ , we get that v is necessarily a leaf in T . Since exactlyone edge of dep ( vv ) is in E T ′ , namely the edge incident to v in T , we have that1 = ℓ ′ ( vv ) ≤ | dep ( vv ) ∩ E ( T ′ ) | ≤ u ′ ( vv ) = 1.(iii) vv : by construction of T ′ , we get that v in T is not a leaf in T . From the feasibilityof T , we know that ℓ ( v ) ≤ d T ( v ) ≤ u ( v ), i.e., at least ℓ ( v ) and at most u ( v ) edges of dep ( e v ) are in E T ⊆ E T ′ , which implies that ℓ ( v ) = ℓ ′ ( vv ) ≤ | dep ( vv ) ∩ E ( T ′ ) | ≤ u ′ ( vv ) = u ( v ).Conversely, suppose that T ′ = ( V ∪ V ′ , E T ′ ) is a solution for G-DCST ( G ′ , D, ℓ ′ , u ′ ). Weshow that T = T ′ [ V ] is a solution for G-mDST ( G, ℓ, u ). Due to the dependency constraints,we get that v v and v v are in T ′ , for each v ∈ V . From this, and since vv and vv area cut in G ′ , exactly one of vv and vv is in T ′ , for each v ∈ V . Take v ∈ V . If vv is in T ′ , then there are at least ℓ ( v ) and at most u ( v ) edges incident to v in T ′ . And if vv isin T ′ , then there is exactly one edge uv ∈ E in E T ′ . Therefore, either ℓ ( v ) ≤ d T ( v ) ≤ u ( v )or d T ( v ) = 1. Since T ′ is a spanning tree of G ′ , T is a spanning tree of G , and thus T is asolution of GD-MST ( G, ℓ, u, w ).Finally, given a graph G , a subset C ⊆ V , and a function ℓ : C → Z + , recall that FmDST ( G, C, ℓ ) denotes the problem of finding a spanning tree T of G such that d T ( u ) ≥ ℓ ( u )for every u ∈ C , and d T ( v ) = 1 for every v ∈ V \ C (i.e., the set of leaves is prefixed).Observe that the same reduction of Theorem 10 can be applied to this problem by removingedge e v for each v ∈ C , and edge e v for each v ∈ V \ C . We then get the following: Theorem 11.
FmDST (cid:22) P G-DCST . In this paper, we investigated a dependency constrained spanning tree problem that gener-alizes many previously studied spanning tree problems with local constraints, as for instance22egree constraints. We then inherit all of the NP -completeness results for these problems,as well as polynomial results and practical approaches to our problem will therefore holdfor these other problems. Interestingly, there are other spanning tree problems that imposeglobal constraints on the tree, as for instance, a bound on the diameter of the producedtree [10, 11], or on the number of leaves [25, 35]. A good question therefore is whetherproblems with this kind of constraints can be modeled within our framework. Question 1.
Can instances of CST problems with global constraints be modeled as
G-DCST instances?
Concerning NP -completeness results, we have investigated the problem under restric-tions on the dependency digraph D , and on the lower and upper bound functions ℓ and u .Among other restrictions, we have proved that G-DCST is NP -complete when D is either aforest of directed paths, a forest of out-stars, or a forest of in-stars, and each componenthas at most three vertices. In the first two cases, we have considered all possible values forconstant functions ℓ, u . The following cases for in-stars remain open. Question 2.
What is the complexity of
G-DCST ( G, D, ℓ, u ) when D is a forest of in-starswith at most three vertices and ( ℓ, u ) = (0 , or, for every e ∈ E ( G ) such that dep ( e ) = ∅ , ℓ ( e ) = 1 and u ( e ) ∈ { , } ? Concerning positive results, we have proved that
G-DCST ( G, D, dep , dep ) can be solvedin polynomial time when D is an oriented matching. We ask whether this also holds forthe optimization problem (recall that G-DCMST ( G, D, ,
0) is polynomial in this case [18]).
Question 3.
What is the complexity of
G-DCMST ( G, D, dep , dep ) when D is an orientedmatching? Finally, we have proved that
G-DCMST can be solved in polynomial time when ℓ = 0, D is a collection of symmetric graphs D , . . . , D k , and u ( e ) = u ( e ′ ) whenever e, e ′ are withinthe same component, and that G-DCST ( G, D, ℓ, u ) can be solved in time O (2 m · ( n + m ))by a naive brute-force algorithm, where n = | V ( G ) | and m = | E ( G ) | . The latter resultis important in the face of the fact that, as a byproduct of a result in [45], we get thatno algorithm running in time 2 O ( n ) exists for the optimization problem, unless ETH fails.Also, the results presented in Section 3.1 imply that no algorithm that runs in time 2 o ( n + m ) exists for the feasibility problem under the ETH , which means that the algorithm presentedin Section 4.3 is asymptotically optimal. We mention that our algorithm can also be seenas an FPT algorithm parameterized by m . We ask whether the problem is FPT underother parameters. Question 4.
Under which parameters is
G-DCST or G-DCMST
FPT ? In order to identify parameters for the above question, note that the maximum degreeof the input graph G is not enough, since a particular case of the problem is already NP -complete restricted to graphs with maximum degree at most three [45]. Similarly, the23aximum degree of the dependency graph D is not enough either, as another particularcase of the problem is NP -complete even if D is a forest of paths of length at most two (seee.g. [19], as well as our results presented in Section 3.1).A promising candidate parameter for obtaining FPT algorithms is the treewidth ofthe input graph G (see [16] for the definition); note that the treewidth of the underlyinggraph of D is not enough by the last sentence of the above paragraph, since forests havetreewidth one. Suppose that, in order to use Courcelle’s Theorem [15] or any of its op-timization variants [4], one tries to express the G-DCST problem in monadic second-order ( MSO ) logic. In order to guarantee that the dependencies of D are satisfied for every edge e of the desired spanning tree of the input graph G , one would probably need to evaluate the functions ℓ ( e ) and u ( e ) inside the eventual MSO formula, and this seems to be a funda-mental hurdle since these values are a priori unrelated to the treewidth of G . Nevertheless,for the particular case of G-DCST (or
G-DCMST ) where both functions ℓ and u are constants (or even equal to some constant value that depends on the treewidth of G ), it is indeedpossible, using standard techniques, to write such an MSO formula expressing the problem,and therefore it is
FPT parameterized by the treewidth of the input graph. Note that thisrestriction of the
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