1
Approximating Minimum Bounded Degree Spanning Trees
to within One of Optimal
MOHIT SINGH, Microsoft Research, Redmond
LAP CHI LAU, The Chinese University of Hong Kong
In the Minimum Bounded Degree Spanning Tree problem, we are given an undirected graph G = (V, E)
with a degree upper bound B
v
on each vertex v ∈ V , and the task is to find a spanning tree of minimum
cost that satisfies all the degree bounds. Let
OPT be the cost of an optimal solution to this problem. In this
article we present a polynomial-time algorithm which returns a spanning tree T of cost at most
OPT and
d
T
(v) ≤ B
v
+ 1forallv,whered
T
(v) denotes the degree of v in T . This generalizes a result of F
¨
urer and
Raghavachari [1994] to weighted graphs, and settles a conjecture of Goemans [2006] affirmatively. The
algorithm generalizes when each vertex v has a degree lower bound A
v
and a degree upper bound B
v
,and
returns a spanning tree with cost at most
OPT and A
v
− 1 ≤ d
T
(v) ≤ B
v
+ 1forallv ∈ V . This is essentially
the best possible. The main technique used is an extension of the iterative rounding method introduced by
Jain [2001] for the design of approximation algorithms.
Categories and Subject Descriptors: F.2.2 [Analysis of Algorithms and Problem Complexity]: Non-
numerical Algorithms and Problems—Computations on discrete structures; G.2.2 [Discrete Mathematics]:
Graph Theory—Network problems, Trees
General Terms: Algorithms, Theory
Additional Key Words and Phrases: Approximation algorithms, spanning trees, bounded degree, iterative
rounding
ACM Reference Format:
Mohit Singh and Lap Chi Lau. 2015. Approximating minimum bounded degree spanning trees to within one
of optimal. J. ACM 62, 1, Article 1 (February 2015), 19 pages.
DOI: http://dx.doi.org/10.1145/2629366
1. INTRODUCTION
The Minimum Bounded Degree Spanning Tree (MBDST) problem is defined as follows:
Given an undirected graph G = (V, E), a cost function c : E → R on the edges, and a
degree upper bound B
v
on each vertex v ∈ V , find a spanning tree of minimum cost
that satisfies all the degree bounds. When all degree bounds are two (i.e., B
v
= 2 for all
v), the MBDST problem specializes to the Minimum Cost Hamiltonian Path problem,
and thus even the problem of checking whether there exists a feasible solution is
NP-complete. In unweighted graphs, F
¨
urer and Raghavachari [1994] gave an elegant
algorithm that returns a spanning tree in which the degree of each vertex is at most
B
v
+ 1, or returns a witness certifying that the degree bounds are infeasible. It was
conjectured in Goemans [2006] that this result can be generalized to weighted graphs.
A preliminary version of this article appeared in Proceedings of the 39th ACM Symposium on Theory of
Computing (STOC 2007).
Authors’ addresses: M. Singh, Microsoft Research, 1 Microsoft Way, Redmond, WA 98004; L. C. Lau, Com-
puter Science and Engineering, The Chinese University of Hong Kong, Shatin, Hong Kong. Correspondence
email: mohits@microsoft.com.
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DOI: http://dx.doi.org/10.1145/2629366
Journal of the ACM, Vol. 62, No. 1, Article 1, Publication date: February 2015.
1:2 M. Singh and L. C. Lau
CONJECTURE 1.1. In polynomial time, one can find a spanning tree of maximum
degree at most k + 1 whose cost is no more than the cost of a minimum cost tree with
maximum degree at most k.
Note that this conjecture is formulated in the special case where B
v
= k for each
vertex v ∈ V . Goemans [2006] made a major step towards proving this conjecture by
giving a polynomial-time algorithm t hat returns a tree with maximum degree k + 2,
whose cost is at most the cost of a minimum cost tree with maximum degree at most
k. The algorithm also generalizes when there are different degree bounds on different
vertices, where the algorithm returns a spanning tree violating the degree bounds by
at most two. In this article, we settle Conjecture 1.1 positively by proving the following
result.
T
HEOREM 1.2. There exists a polynomial-time algorithm for the Minimum Bounded
Degree Spanning Tree problem that returns a tree T in which each vertex v ∈ V has
degree at most B
v
+ 1 and the cost of the tree T is at most OPT, where OPT is the minimum
cost of a spanning tree which satisfies all degree bounds.
Theorem 1.2 also generalizes to the setting when there is a degree lower bound A
v
and a degree upper bound B
v
for each vertex v ∈ V . In this case, the algorithm returns
a spanning tree T such that A
v
− 1 ≤ d
T
(v) ≤ B
v
+ 1 for each vertex v ∈ V and the cost
of T is at most
OPT, where OPT is the minimum cost of a spanning tree which satisfies
all degree (upper and lower) bounds. Note that we do not assume that the cost function
satisfies triangle inequalities (or even non-negativity). With this general cost function,
it is not possible to obtain any approximation algorithm if we insist on satisfying all
the degree upper bounds [Garey and Johnson 1979].
1
Thus, Theorem 1.2 is essentially
the best possible.
1.1. Techniques
Polyhedral combinatorics has proved to be a powerful, coherent, and unifying tool
in combinatorial optimization [ Schrijver 2003]. In the last two decades, polyhedral
methods have also been applied very successfully to the design of approximation algo-
rithms [Vazirani 2004]. A standard approach to design approximation algorithms is to
first formulate the problem as an integer program, and then use the linear relaxation
of this program as a way to lower-bound the cost of an optimal solution to a minimiza-
tion problem. We shall also use this approach. Given an undirected graph G = (V, E)
and a subset S of vertices, we use E(S) ={e ∈ E : |e ∩ S|=2} to denote the set of
edges which have both endpoints in S ⊆ V . For any subset of edges F ⊆ E and vertices
S ⊆ V , we denote δ
F
(S) to be the set of edges in F which have exactly one endpoint
in S.WhenF = E, we drop the subscript F from the notation. For any vertex v ∈ V ,
we let δ
F
(v) denote δ
F
({v})andd
F
(v) =|δ
F
(v)|.Forx : E → R
+
and F ⊆ E, we de-
note x(F):=
e∈F
x(e). Following Goemans [2006], we use the following natural linear
programming relaxation for the Minimum Bounded Degree Spanning Tree problem.
minimize c(x) =
e∈E
c
e
x
e
(1)
subject to x(E(V )) =|V |−1(2)
x(E(S)) ≤|S|−1 ∀ S ⊂ V (3)
1
Assuming P = NP, there is no p(n)-approximation which satisfies the degree bound exactly for any polyno-
mial p(n)ofn where n is the number of vertices.
Journal of the ACM, Vol. 62, No. 1, Article 1, Publication date: February 2015.
Approximating Minimum Bounded Degree Spanning Trees to within One of Optimal 1:3
x(δ(v)) ≤ B
v
∀ v ∈ V (4)
x
e
≥ 0 ∀ e ∈ E (5)
The constraints (2) state that every tree must have exactly |V |−1 edges. Con-
straints (3) state that for any subset S ⊂ V , the number of edges picked with both
endpoints in S must be at most |S|−1. These constraints are valid since any spanning
tree induces an acyclic subgraph on any subset S ⊆ V and the maximum number of
edges in an acyclic subgraph on S is at most |S|−1. The constraints (4) state that the
number of edges included in the solution is at most B
v
for each v ∈ V .
Using a polyhedral approach, a general strategy is to construct a spanning tree of
cost no more than the optimal value of this linear program, and in which the degree of
each vertex v ∈ V is at most B
v
+ 1. This would prove Theorem 1.2. In fact, this general
strategy has been used in previous work, and different techniques have been proposed
to “round” a fractional s olution to this linear program. An important observation of
Goemans is that an extreme point solution of this linear program is characterized by
a laminar family (definitions will be provided later) of tight constraints, inequalities
that are satisfied as equalities. This fact was exploited cleverly to obtain the result in
Goemans [2006].
We note that a very similar observation was made by Jain [2001] in his breakthrough
work on the Survivable Network Design problem, where he first introduced the idea of
iterative rounding to the design of approximation algorithms. This potential connection
was initiated in Lau et al. [2009], where Jain’s iterative rounding method was extended
to give the first constant factor (bi-criteria) approximation algorithm for bounded de-
gree network design problems including the Minimum Bounded Degree Steiner Tree
problem and the Bounded Degree Survivable Network Design problem. Inspired by
these results, we attempted Conjecture 1.1 using the iterative rounding method.
The basic setting of the iterative rounding method for network design problems goes
as follows. First, we solve the linear program to obtain an optimal extreme point solu-
tion x
∗
. We proceed by adding the edges with the highest fractional value to the integral
solution. Then we construct the residual problem where the edges added previously
are fixed, and update the linear program appropriately. A key feature of the iterative
rounding method is to repeat this procedure: solve again the linear program for the
residual problem to obtain an optimal extreme point solution (instead of using x
∗
), and
add the edges with the highest fractional value in this new fractional solution to t he
integral solution. This procedure is iterated until the integral solution constructed is
a feasible solution. In the Survivable Network Design problem, the crucial theorem in
Jain’s approach is that the edges picked in each iteration have fractional value at least
1/2, which ensures that the above algorithm has an approximation ratio of two. This
theorem relies heavily on the properties of an extreme point solution, as in the work of
Goemans [2006].
Our first contribution is to extend the iterative rounding method and show that it
can be used to solve problems optimally. This is achieved by setting integral variables;
an variable e such that x
e
= 0 is removed and a variable e with x
e
= 1 is picked in the
solution. One then formulates the residual problem and iterates. The technical claim
is to show that one can always find an integral variable in each iteration. We apply this
approach to the minimum spanning tree problem and give two different proofs of the
integrality of the spanning tree polyhedron in Section 2.
For the Minimum Bounded Degree Spanning Tree problem, however, the direct ap-
proach of iterative rounding would not work. The standard iterative rounding frame-
work of picking an edge e with x
∗
e
≥ 1/2 would not work because we can not guarantee
the optimality (with respect to the cost of the linear program) of the solution. The latter
approach of picking integral edges would not work since we do not expect the linear
Journal of the ACM, Vol. 62, No. 1, Article 1, Publication date: February 2015.
1:4 M. Singh and L. C. Lau
programming solution to only have integral values. Our algorithm uses the iterative
relaxation framework as introduced in Lau et al. [2009]. The key insight is that i f
the algorithm cannot find integral edges to pick, then it can remove/relax one of the
degree constraints. In particular, we find a vertex with degree upper bound B
v
and
with at most B
v
+ 1 edges incident on it in the support of an extreme point solution.
The algorithm then removes the degree constraint of v and proceeds to re-solve the
linear program. The heart of our analysis is to show that one can always find such a
vertex. This is proved by a counting argument which relies heavily on the fact that an
extreme point solution is characterized by a laminar family of tight constraints [ Jain
2001; Goemans 2006]. Since the algorithm only picks integral edges, the optimality
of the cost follows naturally. The condition when the degree constraint of a vertex is
removed ensures that the degree constraint can be violated by at most one, giving us
Theorem 1.2.
1.2. Related Work
The Minimum Bounded Degree Spanning Tree problem is a well-studied problem
and has been attacked using a variety of techniques. Initial efforts on the problem
were concentrated on obtaining bi-criteria approximation algorithms. Let
OPT be the
cost of an optimal solution to the MBDST problem. An (α, f (B
v
))-approximation algo-
rithm
2
is an algorithm which returns a spanning tree T with cost at most α · OPT and
d
T
(v) ≤ f (B
v
) for all v, where d
T
(v) denotes the degree of v in T . Ravi et al. [1993] gave
an (O(log n), O(B
v
log n))-approximation for the MBDST problem using a matching-
based augmentation technique. K
¨
onemann and Ravi [2002, 2005] used a Lagrangian-
relaxation based approach to obtain an (O(1), O(B
v
+ log n))-approximation algorithm.
Chaudhuri et al [2009b, 2009a] presented an (1, O(B
v
+ log n))-approximation algo-
rithm, and an (O(1), O(B
v
))-approximation algorithm based on the push-relabel frame-
work developed for the maximum flow problem. Ravi and Singh [2006] considered a
variant of the problem in which the tree returned must be a minimum spanning tree,
and gave an algorithm that returns an MST in which the degree of any vertex v is at
most B
v
+ p, where p is the number of distinct costs in any MST. Recently, Goemans
[2006] presented an (1, B
v
+ 2)-approximation algorithm using matroid intersection
techniques. This was the previous best guarantee for the MBDST problem. I n the spe-
cial case where the graph is unweighted, F
¨
urer and Raghavachari [1994], building on
the work of Win [1989], gave an algorithm that returns a spanning tree in which the
degree of each vertex v is at most B
v
+ 1 or returns a witness certifying infeasibility of
the degree bounds.
The iterative rounding technique that we use in our algorithm was developed in Jain
[2001] for the Survivable Network Design problem and has later been successfully
applied to various problems [Cheriyan et al. 2006; Fleischer et al. 2006]. Recently,
this technique has been extended t o give constant factor bi-criteria approximation
algorithm for the Bounded Degree Survivable Network Design problem [Lau et al.
2009; Lau and Singh 2008].
1.3. Organization
The rest of this article is organized as follows. In Section 2, we give two iterative
algorithms which shows the integrality of the spanning tree polyhedron. Then, in
Section 3, we present a new proof of the result of Goemans [2006] that gives an (1, B
v
+
2)-approximation algorithm for the MBDST problem. In Section 4, we present the main
algorithm and the proof of Theorem 1.2. Both the results for the MBDST problem build
2
Notice that the first parameter is used to specify the ratio, while the second parameter is used to specify
the actual bound.
Journal of the ACM, Vol. 62, No. 1, Article 1, Publication date: February 2015.
Approximating Minimum Bounded Degree Spanning Trees to within One of Optimal 1:5
on the iterative algorithms for the spanning tree problem given in Section 2. Finally,
in Section 5, we extend the algorithm to deal with degree lower bounds.
2. SPANNING TREE POLYHEDRON
In this section, we present two iterative arguments to show that the minimum spanning
tree polytope is integral. This motivates the main result of this article and illustrates
the basic proof techniques.
2.1. Linear Program
Let G = (V, E) be a graph with a cost function c on edges. A classical result of Edmonds
[1971] states that the following linear program LP-MST(G) is integral, and an optimal
extreme point solution is always a minimum spanning tree.
minimize c(x) =
e∈E
c
e
x
e
(6)
subject to x(E(V )) =|V |−1(7)
x(E(S)) ≤|S|−1 ∀ S ⊂ V (8)
x
e
≥ 0 ∀ e ∈ E (9)
2.2. Characterization of the Extreme Point Solutions
An extreme point solution is defined as the unique solution of m linearly independent
tight constraints (constraints which achieve equality), where m denotes the number
of variables in the linear program. We focus on extreme point solutions x
∗
such that
x
∗
e
> 0 for each e ∈ E. We first give some basic definitions. For a set F ⊆ E, we define
the characteristic vector of F in R
|E|
, χ(F), as the vector that has a 1 corresponding
to each edge e ∈ F, and 0 otherwise. For any set family F ⊆ 2
V
,letspan(F)bethe
vector space generated by the set of vectors {χ(E(S)) | S ∈ F}. A family of sets L ⊆ 2
V
is laminar if for any two sets in the family, either one contains the other or they are
disjoint.
L
EMMA 2.1 [CORNUEJOLS ET AL. 1985]. Let x
∗
be any extreme point solution to LP-
MST(G) such that x
∗
e
> 0 for each edge e ∈ E and let F ={S ⊆ V | x
∗
(E(S)) =|S|−1}
be the collection of sets corresponding to tight constraints. Then there exists a laminar
family L ⊆ F such that
(1) The set of vectors {χ (E(S)) : S ∈ L} are linearly independent and span(L) =
span(F).
(2) |L|=|E|.
(3) x
∗
is the unique solution to the set of equations {x(E(S)) =|S|−1:S ∈ L}.
The proof of Lemma 2.1 is quite standard [Cornu
´
ejols et al. 1985], but we include it
for completeness. It also illustrates the uncrossing technique.
P
ROOF. First, we need the following uncrossing lemma on intersecting sets among
tight sets.
LEMMA 2.2 [CORNUEJOLS ET AL. 1985]. If S, T ∈ F and S ∩ T = ∅, then both S ∩ T
and S ∪ T are in F. Furthermore, χ (E(S)) + χ (E(T )) = χ(E(S ∩ T )) + χ (E(S ∪ T )).
Journal of the ACM, Vol. 62, No. 1, Article 1, Publication date: February 2015.
