Bicriterial Approximation for the Incremental Prize-Collecting Steiner-Tree Problem

TL;DR

The paper studies incremental, budget-constrained prize-collecting Steiner-tree problems, introducing a bicriteria notion where a solution with budget slack $B+α$ achieves a multiplicative approximation to the optimal profit at budget $B$. It shows a tight, tree-specific result: the density-greedy algorithm attains a $(χ,1)$-competitive guarantee, where $χ$ is the root eccentricity, and proves this bound is best possible. For general graphs, the authors extend the method to obtain a $(γ,2)$-competitive density-greedy variant, with a polynomial-time $(γ,3)$-competitive version, and they propose a capacity-scaling algorithm achieving $((4ℓ-1)χ, 2^{ℓ+2}/(2^{ℓ}-1))$-competitiveness for all natural numbers $ℓ$, approaching an $O(χ)$-type bound as the slack grows. They also establish fundamental lower bounds showing that no algorithm with α depending only on $γ$ can beat a μ of at least 17/16, highlighting intrinsic limitations. Together, these results advance the understanding of incremental network design under growing budgets, offering both tight tree-like guarantees and scalable, though complexity-influenced, general-graph approaches with clear directions for further polynomial-time improvements.

Abstract

We consider an incremental variant of the rooted prize-collecting Steiner-tree problem with a growing budget constraint. While no incremental solution exists that simultaneously approximates the optimum for all budgets, we show that a bicriterial $(α,μ)$-approximation is possible, i.e., a solution that with budget $B+α$ for all $B \in \mathbb{R}_{\geq 0}$ is a multiplicative $μ$-approximation compared to the optimum solution with budget $B$. For the case that the underlying graph is a tree, we present a polynomial-time density-greedy algorithm that computes a $(χ,1)$-approximation, where $χ$ denotes the eccentricity of the root vertex in the underlying graph, and show that this is best possible. An adaptation of the density-greedy algorithm for general graphs is $(γ,2)$-competitive where $γ$ is the maximal length of a vertex-disjoint path starting in the root. While this algorithm does not run in polynomial time, it can be adapted to a $(γ,3)$-competitive algorithm that runs in polynomial time. We further devise a capacity-scaling algorithm that guarantees a $(3χ,8)$-approximation and, more generally, a $\smash{\bigl((4\ell - 1)χ, \frac{2^{\ell + 2}}{2^{\ell}-1}\bigr)}$-approximation for every fixed $\ell \in \mathbb{N}$.

Figures (9)

• Figure 1: Instance of the prize-collecting Steiner-tree problem with $\delta \ll 1$ and no $(0,\mu)$-competitive incremental solution: Staying competitive for budget $B = \chi / 2$ requires to build the left edge first, but then we are not competitive for the budget $B = \chi$.
• Figure 2: The left figure shows graph $G$ with tree $T\in \mathcal{T}$ and the right figure shows the contracted graph $G/T$. The contracted subgraph $S/T$ contains parallel edges and loops. The extension $C^+$ of the connected subgraph $C\subset G/T$ is not connected.
• Figure 3: For the tree $T$ with root $r_T$, the branch $T_e$ rooted in edge $e$ is shown in red and the branch $T_v$ rooted in vertex $v$ is shown in blue.
• Figure 4: Lower bound used in the proof of \ref{['thm:result_greedy_trees']}
• Figure 5: Lower bounds used in the proof of \ref{['thm:result_LB_trees']}

Theorems & Definitions (40)

• Theorem 1
• Theorem 2
• Theorem 3
• Corollary 3
• Theorem 4
• Theorem 5
• Lemma 6: cf. AlpernL13, Lemma 2
• Lemma 6
• proof
• Lemma 6