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A Polylogarithmic Approximation for

Computing Non-Metric Terminal Steiner Trees

Iftah Gamzu∗

Danny Segev†

Abstract

The main contribution of this short note is to provide improved bounds on the approx-

imability of constructing terminal Steiner trees in arbitrary undirected graphs. Technically

speaking, our results are obtained by relating this computational task to that of computing

group Steiner trees. As a secondary objective, we make a concentrated effort to distinguish

between the factor by which constructed trees exceed the optimal backbone cost and between

the deviation from the optimal terminal linking cost.

∗Blavatnik

iftgam@tau.ac.il. Supported by the Israel Science Foundation, by the European Commission under the Inte-

grated Project QAP funded by the IST directorate as Contract Number 015848, by a European Research Council

(ERC) Starting Grant, and by the Wolfson Family Charitable Trust.

†Department of Statistics, University of Haifa, Haifa 31905, Israel. Email: segevd@stat.haifa.ac.il.

School ofComputerScience,Tel-AvivUniversity, Tel-Aviv69978,Israel.Email:

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1Introduction

An instance of the terminal Steiner tree problem consists of an undirected graph G = (V,E)

on n vertices, with non-negative edge costs specified by c : E → R+, and a subset of vertices

T = {t1,...,tk}, which we refer to as terminals. The objective is to identify a minimum cost

tree H ⊆ G spanning T ; however, some Steiner trees will not do the trick, since we also have to

take into account the following structural requirement: terminals cannot serve as intermediate

vertices in H, or in other words, each terminal is required to be a leaf. This structural constraint

is motivated by real-life applications in computational biology, VLSI design, and networking.

We refer the reader to directly related papers [9, 10, 3] and to the references therein for a

comprehensive review of these applications.

The metric case. There has been a growing line of work on metric terminal Steiner trees, in

which we are given a complete graph whose edge costs satisfy the triangle inequality. Lin and

Xue [9] seem to have been the first to consider this variant, for which they obtained a (2 + ρ)-

approximation; here, ρ denotes the best approximation ratio attainable in polynomial time for

the standard Steiner tree problem. Later on, several authors [2, 5, 3] independently established

an improved performance guarantee of 2ρ. Currently, the best known approximation ratio is

2ρ−ρ/(3ρ−2), which was achieved by Martinez, de Pina and Soares [11]. This factor evaluates

to roughly 2.14 by plugging in ρ = ln4 + ? < 1.39 [1].

The non-metric case. We study the approximability of computing terminal Steiner trees in

arbitrary undirected graphs. That is, the underlying graph is no longer required to be complete,

and its edge costs do not necessarily satisfy the triangle inequality. In spite of appearance, this

setting cannot be sensibly reduced to the metric case. To better understand the latter statement,

note that one cannot straightforwardly utilize the shortest-path metric induced by (G,c) since

replacing single edges by their corresponding shortest paths may quite possibly violate the

degree constraints of terminal vertices. Degree bounds are not intrinsic requirements in the

standard Steiner tree problem, for which such edge-to-path replacements are possible (see, for

instance, [12, Sec. 3]).

To our knowledge, the only non-trivial hardness result for the non-metric case is due to

Drake and Hougardy [3], who proved that this problem cannot be approximated within a factor

of (1 − ?)lnn, for any fixed ? > 0, unless NP ⊆ DTIME(nO(loglogn)). On the positive side,

Fukunaga [6, Sec. 6.5] obtained a performance guarantee of 2∆, where ∆ is the maximum

degree of any terminal. However, simple examples demonstrate that this factor may be Ω(n)

in the worst case. Consequently, approximating the terminal Steiner tree problem when one

makes no structural assumptions about edge costs is still an open question.

Main results. The primary contribution of this short note is to provide improved bounds on

the approximability of constructing terminal Steiner trees by relating this computational task

to that of computing group Steiner trees. As a secondary objective, we make a concentrated

effort to distinguish between the factor by which constructed trees exceed the optimal backbone

cost and between the deviation from the optimal terminal linking cost; both of these measures

will be defined later on. Our findings can be briefly summarized as follows:

• We present a randomized algorithm that constructs, with constant probability, a terminal

Steiner tree whose cost is within a factor of O(lognlogklog∆) of optimal1. The specifics

of our approach, along with a refined statement of the latter bound (in terms of backbone

and terminal linking costs), appears in Section 2.

1This improves on the upper bound of Fukunaga [6, Sec. 6.5] for large enough values of ∆, or more precisely,

whenever ∆/log∆ = ω(lognlogk).

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• We establish an Ω(log2−?k) hardness of approximation, for any fixed ? > 0, unless NP ⊆

ZTIME(npolylog(n)). Further details are given in Section 3.

Group Steiner trees: The bare necessities. Prior to delving into technicalities, some

background on the group Steiner tree problem is necessary. In this setting, we are given an

edge-weighted undirected graph G = (V,E) and a family of vertex groups g1,...,gk ⊆ V .

The goal is to identify a minimum cost subgraph spanning at least one representative from each

group. For our purposes, it would be sufficient to mention that Garg, Konjevod and Ravi [7] were

the first to achieve polylogarithmic approximability results, through randomized LP-rounding

on tree instances, which attains an approximation ratio of O(logklogN), where N = maxi|gi|.

Combining this approach with the probabilistic embedding method of Fakcharoenphol, Rao and

Talwar [4] implies a performance guarantee of O(lognlogklogN) for arbitrary graphs. From

an inapproximability point of view, Halperin and Krauthgamer [8] demonstrated that the group

Steiner tree problem cannot be approximated within a factor of O(log2−?k), for any fixed ? > 0,

unless NP ⊆ ZTIME(npolylog(n)).

2A Polylogarithmic Approximation

To provide an accurate description of our main algorithmic result, we note that the optimal

terminal Steiner tree consists of two separate components: Edges incident on terminals, whose

combined cost is denoted by OPTL(henceforth, the optimal terminal linking cost), and the tree

obtained by removing these edges, whose cost is denoted by OPTB(the optimal backbone cost).

The primary finding of this section can be formally stated as follows.

Theorem 2.1. There is a randomized algorithm that constructs, with constant probability, a

terminal Steiner tree whose backbone cost is O(lognlogklog∆)·OPTB. At the same time, the

terminal linking cost is only O(logk) · OPTL.

2.1Preliminaries

We assume without loss of generality that the collection of terminals T forms an independent

set in G. This assumption can be easily justified by observing that an edge joining two terminals

cannot be included in any feasible solution, unless |T | = 2. However, this special case can be

solved to optimality by applying a shortest path algorithm. We now introduce some notation

and terminology:

• Let H∗⊆ G denote the optimal terminal Steiner tree, of cost OPT = OPTB+ OPTL.

• We assume that an arbitrary root vertex r ∈ V (H∗) \ T in the backbone of H∗, and a

constant factor estimate EL∈ [OPTL,2 · OPTL] of the optimal terminal linking cost, are

known in advance. This assumption can be enforced by considering polynomially-many

potential candidates for r and EL.

• For every ti∈ T , let Gibe the group of this terminal, consisting of vertices reachable from

tivia a single edge of cost at most 2EL/k, that is, Gi= {v ∈ V : c(ti,v) ≤ 2EL/k}.

2.2 Overview

In the remainder of this section, we describe a randomized procedure that constructs, with

constant probability, a tree H ⊆ G satisfying the following properties:

1. H spans the root r, as well as Ω(k) terminals.

2. All terminals spanned by H are necessarily leaves.

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3. H has a backbone cost of O(lognlog∆)·OPTBand a terminal linking cost of O(1)·OPTL.

A procedure of this nature allows one to instantly validate our main result.

Proof of Theorem 2.1. Suppose we repeatedly construct O(logk) such trees (say, H1,H2,...),

focusing in each new iteration only on terminals that have not been spanned up until now. That

is, once a terminal tiis spanned by some tree, we immediately remove it from the current graph.

By amplifying the success probability of each iteration via O(loglogk) independent repetitions,

we can ensure that, with constant probability, each and every terminal is indeed spanned by the

union of all constructed trees,? H =?

that every terminal may appear in at most one tree, guarantee that this subgraph can be con-

verted into a terminal Steiner tree, since every possible cycle in? H consists solely of non-terminal

whereas its terminal linking cost is only O(logk) · OPTL.

jHj. Needless to say, property 1 implies that? H must be

connected, as each tree Hjspans r. In addition, property 2, in conjunction with the observation

vertices. Finally, property 3 ensures that the backbone cost of? H is O(lognlogklog∆)·OPTB,

2.3The randomized procedure

Tree embedding. Let?G be the subgraph of G created by removing the set of terminals T . We

tree T?= (V?,E?) using the method of Fakcharoenphol et al. [4], incurring an expected stretch

of O(logn). More specifically, the distance in T?between any pair of original vertices is at least

as large as the corresponding distance in?G, and the expected ratio between these distances is

Fractionally connecting Ω(k) groups. We now focus our attention on identifying a low-cost

subtree of T?that connects r to a fixed proportion of the groups G1,...,Gk. For this purpose,

consider the following linear program, which is an adaptation of the cut-based LP-relaxation of

group Steiner tree:

?

subject to(1)

begin by probabilistically embedding the shortest-path metric induced by (?G,c) into a random

O(logn), where the expectation is with respect to the distribution over tree metrics.

(LP1)minimize

e∈E?

c(e)xe

?

?

xe, yi∈ [0,1]

e∈δ(U)

xe≥ yi

∀U ⊆ V?, ∀i ∈ [k] :

r ∈ U, Gi∩ U = ∅

(2)

i∈[k]

yi≥ k/2

(3)

∀e ∈ E?, ∀i ∈ [k]

In an integral solution, the variable xeindicates whether the edge e is picked, and yiindicates

whether r is connected to some representative of Gi. Constraint (1) guarantees that, if r is

connected to Gi, at least one edge is picked from every cut (U,V?\U) separating r and Gi; here,

δ(U) denotes the set of edges crossing (U,V?\ U). Constraint (2) ensures that at least k/2 of

the groups are connected. We remark that although (LP1) has exponentially many constraints,

it admits a polynomial-time separation oracle based on a minimum cut procedure (see, for

instance, [7]).

Lemma 2.2. With constant probability, OPT(LP1) = O(logn) · OPTB.

Proof. One can easily argue that the backbone of H∗spans representatives from at least k/2 of

the groups G1,...,Gk. Otherwise, the individual linking cost of at least k/2 terminals is strictly

greater than 2EL/k, implying that the terminal linking cost of H∗is greater than EL≥ OPTL.

Consequently, when each backbone edge (u,v) is translated to the unique u-v path in T?, we

obtain a subtree that connects r to at least k/2 groups. Moreover, since the expected stretch

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is O(logn), a straightforward application of Markov’s inequality implies that the subtree cost

is O(logn)·OPTBwith constant probability. The lemma follows by observing that OPT(LP1)

provides a lower bound on the cost of any feasible integral solution.

Creating a group Steiner tree instance. Let (x∗,y∗) be an optimal fractional solution to

(LP1), and let I∗= {i : y∗

the extent of at least 1/4. Note that constraint (2) implies |I∗| ≥ k/3, since

k

2≤

i∈[k]

We proceed by setting up a group Steiner tree instance in which the groups are {Gi: i ∈ I∗};

the objective is to connect r to at least one representative of each and every group. Consider

the cut-based relaxation of this instance, formally defined as follows.

?

subject toxe≥ 1

i≥ 1/4} be the index set of groups that are fractionally connected to

?

y∗

i=

?

i∈I∗

y∗

i+

?

i/ ∈I∗

y∗

i≤ |I∗| +(k − |I∗|)

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.

(LP2)minimize

e∈E?

?

xe∈ [0,1]

c(e)xe

e∈δ(U)

∀U ⊆ V?, ∀i ∈ I∗:

r ∈ U, Gi∩ U = ∅

∀e ∈ E?

Notice that the main constraint in (LP2) is nearly identical to the one in (LP1), with an

additional restriction stating that yi= 1 if i ∈ I∗, and yi= 0 otherwise. With this observation

in mind, it is easy to verify that the vector ˆ x, defined by ˆ xe = min{4x∗

feasible solution to (LP2), since y∗

OPT(LP2) ≤ 4 · OPT(LP1).

Putting it all together. At this point in time, we remark that, given an optimal solution to

(LP2), the randomized rounding procedure of Garg et al. [7] generates an r-rooted subtree of

T?satisfying the following properties:

e,1}, constitutes a

i≥ 1/4 for every i ∈ I∗. As a result, we immediately have

1. For every i ∈ I∗, at least one representative of Giis spanned with probability Ω(1/log|Gi|).

2. The expected cost of this subtree is OPT(LP2) = O(1) · OPT(LP1).

Now let H ⊆ T?be a subtree created by unifying O(log∆) such random subtrees.

∆ ≥ maxi∈I∗ |Gi| and |I∗| ≥ k/3, it follows that the expected number of spanned groups is Ω(k).

In addition, the expected cost of H is O(log∆) · OPT(LP1). Therefore, based on Lemma 2.2

and simple probabilistic arguments, we can safely claim that, with constant probability, Ω(k)

groups will be spanned at a combined cost of O(lognlog∆)·OPTB. We proceed by translating

H to an analogous tree in the original graph G, creating a backbone of no greater cost. Finally,

we identify a representative of each group Gispanned by this backbone, and connect it to the

corresponding terminal ti. As each terminal can be linked to any representative of its group by

an edge of cost at most 2EL/k, we conclude that the terminal linking cost is O(k) · 2EL/k =

O(1) · OPTL.

Since

3Polylogarithmic Hardness

In what follows, we establish a polylogarithmic hardness of approximation for the terminal

Steiner tree problem on arbitrary undirected graphs.

reduction from the group Steiner tree problem.

This result is obtained via a simple

Theorem 3.1. The terminal Steiner tree problem cannot be approximated within a factor of

O(log2−?k), for any fixed ? > 0, unless NP ⊆ ZTIME(npolylog(n)).

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Proof. Given a group Steiner tree instance, as described in Section 1, we define an instance of

the terminal Steiner tree problem as follows. The underlying graph is augmented with k new

vertices, t1,...,tk, which constitute the set of terminals. In addition, zero-cost edges join each

terminal tito every vertex in its corresponding group gi. Figure 1 provides a concrete example

of this construction.

t1

t2

g1

g2

(a)(b)

Figure 1: (a) An instance of the group Steiner tree problem with two groups; (b) The resulting

terminal Steiner tree instance (edge costs are not indicated).

We begin by pointing out that a group Steiner tree H in the original instance can be

converted to a terminal Steiner tree of identical cost in the newly-created instance. For this

purpose, simply add an edge connecting each terminal tito some representative of gispanned by

H. Conversely, it is not difficult to verify that, given a terminal Steiner tree H, we can perform

a similar cost-preserving transformation in the opposite direction. Clearly, H must span at least

one representative from gi, as there is no other way to access ti; furthermore, noting that every

terminal is a leaf of H, connectivity is still maintained when each terminal is discarded along

with its incident edge.

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