Problem
Research questions and friction points this paper is trying to address.
Steiner Tree Problem
Discrete Optimization
Continuous Optimization
On Approximability of Steiner Tree in $ell_p$-metrics
🤖 AI Summary
This work systematically investigates the approximation complexity of the Continuous Steiner Tree Problem (CST) and the Discrete Steiner Tree Problem (DST) under $ell_p$-metrics. Using techniques from combinatorial optimization, metric space analysis, and geometric embedding, we establish three key results: (i) the first unified proof that DST is APX-hard for all $ell_p$-metrics with $1 leq p leq infty$; (ii) the first proof that CST is APX-hard under the $ell_infty$-metric; and (iii) the first general-purpose reduction framework from CST to DST applicable to arbitrary $ell_p$-metrics. These results fully resolve long-standing open questions regarding the approximability of CST and DST across nearly all standard $ell_p$-metrics. Moreover, they establish a rigorous theoretical connection between their continuous and discrete variants, providing foundational insights for future algorithm design and complexity analysis.
📝 Abstract
In the Continuous Steiner Tree problem (CST), we are given as input a set of points (called terminals) in a metric space and ask for the minimum-cost tree connecting them. Additional points (called Steiner points) from the metric space can be introduced as nodes in the solution. In the Discrete Steiner Tree problem (DST), we are given in addition to the terminals, a set of facilities, and any solution tree connecting the terminals can only contain the Steiner points from this set of facilities. Trevisan [SICOMP'00] showed that CST and DST are APX-hard when the input lies in the $ell_1$-metric (and Hamming metric). Chleb'ik and Chleb'ikov'a [TCS'08] showed that DST is NP-hard to approximate to factor of $96/95approx 1.01$ in the graph metric (and consequently $ell_infty$-metric). Prior to this work, it was unclear if CST and DST are APX-hard in essentially every other popular metric. In this work, we prove that DST is APX-hard in every $ell_p$-metric. We also prove that CST is APX-hard in the $ell_{infty}$-metric. Finally, we relate CST and DST, showing a general reduction from CST to DST in $ell_p$-metrics.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Steiner Tree Problem
Discrete-to-Continuous Transformation
Solving Difficulty
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