1.5 Contributions
2.1.2 Approximation algorithms
As of today, the NP-hard problems can not be solved optimally in polynomial time unless P =N P. Since the ST problem is also an NP-hard, this also holds for it. Nevertheless, the ST problem has numerous practical applications. Therefore an efficient algorithm1 which computes a near-optimal ST is widely accepted. There are numerous ways to compute a near-optimal ST in polynomial time:– approximation, randomization, parameterization, heuristics etc. In this section we mainly focus on the approximation algorithms for the ST problem exist in literature. Note that the goal of an approximation algorithm [126]
is to compute a near-optimal solution of an NP-hard problem as fast as possible. An α- approximation algorithm is defined as follows.
Definition 2.1.1. An α-approximation algorithm for an optimization problem is a polyno- mial time algorithm such that for any instances of the problem it computes a solution whose value is within a factor of α of the value of an optimal solution.
To the best of our knowledge, the first sequential approximation algorithm for the ST problem was proposed by Edward Forrest Moore (1925-2003) which was published by Gilbert
1Here the efficient algorithm denotes an algorithm which solves a problem in polynomial time.
Sequential ST algorithms
and Pollak [57] in the year 1968 (by referring Moore). This algorithm, which is based on the MST heuristic, has an approximation ratio of 2. A Prim’s based shortest path heuristic (P-SPH) approximate ST algorithm and a Krushkal’s based shortest path heuristic (K- SPH) approximate ST algorithm were proposed by Takahashi and Matsuyama [125] in the year 1980 and Kou et al. [84] in the year 1981 respectively. The approximation ratio guaranteed by both the algorithms is 2 and remained the best known approximation ratio until Zelikovsky’s 11/6-approximation algorithm [132] proposed in 1993, which became the first algorithm to beat the 2-approximation algorithm for the ST problem. The total running time of this algorithm is O(nm+|Z|4). Berman and Ramaiyer [14] improved Zelikovsky’s 11/6 bound to 16/9 by showing that 3 element subset of Zelikovsky’s consideration can be further increased to 4 element subset. Based on the Zelikovsky’s central idea, sequence of improved approximation algorithms appeared in literature. The bound of 16/9 = 1.764 was improved to 1.693 by Zelikovsky in 1996 [133], then to 1.667 by Promel and Steger in 1997 [118], to 1.644 by Karpinski and Zelikovsky in 1997 [75], to 1.598 by Hougardy and Promel in 1999 [70] and to 1.550 (actually 1 + 12ln 3 + for any arbitrary small constant >0) by Robins and Zelikovsky in 2000 [121].
The sequence of improved approximation ratios for the ST problem in [14, 70, 75, 118, 121, 132, 133] are mainly based on the following idea. A full Steiner component (or full component for short) of an ST is a subtree whose internal vertices are Steiner vertices, and whose leaves are terminals. The edge set of any ST can be partitioned into full components by splitting the tree at terminals. An r-(restricted)-ST is defined to be an ST all of whose full components have at most r terminals. Borchers and Du [18] showed that in order to obtain a good approximation factor, it is sufficient to restrict our attention to r-restricted STs. Letc(opt) and c(optr) denote the costs of the optimal ST and an optimal r-restricted ST respectively. Then the following theorem holds.
Theorem 2.1.1. ( [18]) Let pr be the r-Steiner ratio, i.e. the supremum of the ratio c(optr)/c(opt). Then qr = (a+1)2a2a+ba+b ≤ 1 + [log1
2r], where a and b are non-negative integers such that r= 2a+b and b <2a.
Till date, the best known (polynomial time) approximation ratio for solving the ST problem in the sequential setting is ln(4) + ≈ 1.386 +, for any constant factor > 0 due to Byrka et al. [20]. Byrka et al. achieved this result by using theiterative randomized rounding technique. It is also known that in general graphs the ST problem can not be
solved in polynomial time with an approximation factor ≤ 9695 [28] unless P =N P.
Table 2.2: Summary of related work of sequential ST algorithms for general graphs. P-SPH: Prim’s based shortest path heuristic, K-SPH: Krushkal’s based SPH, LP: linear programming, >0is any constant, “*”
means particular result is not discussed in that work.
Authors Techniques Approx. Ratio Time Complexity
Gilbert and Pollak [57] Moore Theorem 2 O(nt2)
Takahashi and Matasuyama [125] P-SPH 2 *
Kou et al. [84] K-SPH 2 *
Goemans and Williamson [58] primal-dual 2 O(n2logn)
Jain [71] Iterative rounding 2 O(n10m7)
Zelikovsky [132] r-restricted ST 1.834 O(mn+t4)
Berman and Ramaiyer [14] r-restricted ST 1.734 O(n5)
Zelikovsky [133] r-restricted ST 1.694 O(n3)
Promel and Steger [118] r-restricted ST 1.67 O(n3)
Karpinski and Zelikovsky [75] r-restricted ST 1.644 O(n3)
Hougardy and Promel [70] r-restricted ST 1.598 *
Robins and Zelikovsky [121] r-restricted ST 1.55 O(mn2)
Byrka et al. [20] LP and Iterative randomization ln(4) + O(n˜ 6)
One of the known standard approaches to solve the ST problem is the use of proper LP relaxation. A natural formulation of the problem is undirected cut formulation [58].
In this formulation, a variable is assigned for each edge of the graph and a constraint is considered for each cut separating the set of terminals. Each constraint helps to pick at least one edge crossing the corresponding cut. Considering the LP relaxation, 2-approximation algorithms can be obtained by using the primal-dual framework developed by Goemans and Williamson [58] or the iterative rounding technique due to Jain [71].
To the best of our knowledge, the best known integrality gap of LP relaxation for the ST problem in general graphs is 2. However there exists integrality gap better than 2 for special class of graphs, known as quasi-bipartite graphs.1 For such graphs Rajagopalan and Vazirani [119] gave an upper bound on the integrality gap, which is 3/2. This was improved to 4/3 by Chakrabarty et al. [25]. Robins and Zelikovsky [121] showed that a 1.28- approximate ST can be computed if input graphs are restricted to quasi-bipartite graphs.
With a different LP formulation K¨onemann et al. [82] showed that for b-quasi-bipartite
1A graph is calledquasi-bipartite if all of its Steiner nodes form an independent set.
Distributed ST algorithms
graphs1 the integrality gap is upper-bounded by 2b+1b+1 . Moreover, the best known hardness result for the ST problem in this class of graphs is 128127 [28]. For planar graphs Borradaile et al. [19] proposed a PTAS for ST problem whose running time isO(nlogn). Recently Byrka et al. [21] presented a PTAS for the ST problem which holds for map graphs.2 A summary of performances of known sequential algorithms in special class of graphs for ST problem is listed in Table2.3.
Table 2.3: Summary of related work of sequential ST algorithms for special class of graphs. >0 and
“*” means particular result is not discussed in that work.
Authors Graphs Approx. Ratio Time Complexity
Rajagopalan and Vazirani [119] Quasi-bipartite 3/2 O(1mnlogn)
Chakrabarty et al. [25] Quasi-bipartite 4/3 *
Robins and Zelikovsky [121] Quasi-bipartite 1.28 O(mn2) K¨onemann et al. [82] b-quasi-bipartite 2b+1b+1 *
Borradaile et al. [19] Planar PTAS O(nlogn)
Byrka et al. [21] Map PTAS *