Exact cardinality approximation for max min metric t join problem
Remote Matching: Exact-Cardinality Approximation and Tight UGC Hardness
Data Structures and Algorithms
Summary
This paper studies a complicated math problem where you want to connect specific points in a network so that the total connecting cost is as large as possible, under certain rules. Earlier work gave a way to approximate this solution within 1.5 times the best possible, and this paper shows that this is the best you can do unless a famous complexity conjecture is false. They also handle a more exact version where the number of points to connect is fixed and give methods that change based on that number. Their algorithms use advanced techniques involving cutting networks into pieces and smart ways to combine these pieces.
What this means in practice
- •For network designers: Use the approximation bounds to understand limits when designing cost-optimized network connections with specified terminal counts.
- •For optimization software engineers: Incorporate exact-cardinality approximation algorithms to improve heuristics for network design tools requiring fixed-size terminal sets.
A theory result. No direct application yet.
Authors
Arash Ahadi, Morteza Alimi, Sharareh Alipour, Shayan Tayefeh
Abstract
In the unrestricted max--min metric $T$-join problem, one seeks an even terminal set $T$ maximizing the cost of a minimum $T$-join. Iwata and Ravi gave a factor-$3/2$ approximation for this problem. We show that this guarantee is tight under the Unique Games Conjecture: no polynomial-time approximation with factor strictly smaller than $3/2$ exists under UGC. We then consider the exact-cardinality variant, which prescribes an even number \(k\) of terminals. Writing \(p:=k/n\), we give a deterministic polynomial-time \(ρ(p)\)-approximation for every feasible cardinality, where \[ ρ(p)= \begin{cases} 7/2, & \makebox[1.5em][r]{$0$}<p\le2/7,\\ 1/p, & 2/7\le p\le2/3,\\ 1/[2(1-p)], & 2/3\le p\le7/8,\\ 4, & 7/8\le p<1. \end{cases} \] In particular, a factor-\(4\) approximation holds throughout the entire feasible cardinality range, the factor is at most \(7/2\) whenever \(0<k\le 6n/7\), and equals \(3/2\) at \(k=2n/3\). The algorithmic framework is based on optimal laminar cut packings, their weighted tree representations, exact-cardinality rounding, and tree dynamic programming.