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Efficient Many-To-Many Matching of Points with Demands in One Dimension

Fatemeh Rajabi-Alni, Behrouz Minaei-Bidgoli

TL;DR

This work addresses the minimum-cost many-to-many matching with demands (MMD) and its capacity-enabled variant in one dimension, where the goal is to pair points from two sets on a line so that each point meets its demand (and capacity) with minimal total distance. The authors develop a dynamic-programming framework that partitions the merged point set into alternating blocks along the line and uses states $C(q,j)$ to capture the minimum cost up to a point with exact demand accounting, augmented by structural lemmas about optimal pairings. They present the first $O(n^2)$-time algorithms for both the one-dimensional OMMD and the extended OMMDC problem, enabling efficient solutions for large instances. The methods have potential applications in wireless networks and line-based data association, where per-point demand/capacity constraints and geometric costs are natural modeling choices.

Abstract

Given two point sets $S$ and $T$, the minimum-cost many-to-many matching with demands (MMD) problem is the problem of finding a minimum-cost many-to-many matching between $S$ and $T$ such that each point of $S$ (respectively $T$) is matched to at least a given number of the points of $T$ (respectively $S$). We propose the first $O\left(n^2\right)$-time algorithm for computing a one dimensional MMD (OMMD) of minimum cost between $S$ and $T$, where $\left|S\right|+\left|T\right|=n$. In an OMMD problem, the input point sets $S$ and $T$ lie on the real line and the cost of matching a point to another point equals the Euclidean distance between the two points. We also study a generalized version of the MMD problem, the many-to-many matching with demands and capacities (MMDC) problem, that in which each point has a limited capacity in addition to a demand. We give the first $O(n^2)$-time algorithm for the minimum-cost one dimensional MMDC (OMMDC) problem.

Efficient Many-To-Many Matching of Points with Demands in One Dimension

TL;DR

This work addresses the minimum-cost many-to-many matching with demands (MMD) and its capacity-enabled variant in one dimension, where the goal is to pair points from two sets on a line so that each point meets its demand (and capacity) with minimal total distance. The authors develop a dynamic-programming framework that partitions the merged point set into alternating blocks along the line and uses states to capture the minimum cost up to a point with exact demand accounting, augmented by structural lemmas about optimal pairings. They present the first -time algorithms for both the one-dimensional OMMD and the extended OMMDC problem, enabling efficient solutions for large instances. The methods have potential applications in wireless networks and line-based data association, where per-point demand/capacity constraints and geometric costs are natural modeling choices.

Abstract

Given two point sets and , the minimum-cost many-to-many matching with demands (MMD) problem is the problem of finding a minimum-cost many-to-many matching between and such that each point of (respectively ) is matched to at least a given number of the points of (respectively ). We propose the first -time algorithm for computing a one dimensional MMD (OMMD) of minimum cost between and , where . In an OMMD problem, the input point sets and lie on the real line and the cost of matching a point to another point equals the Euclidean distance between the two points. We also study a generalized version of the MMD problem, the many-to-many matching with demands and capacities (MMDC) problem, that in which each point has a limited capacity in addition to a demand. We give the first -time algorithm for the minimum-cost one dimensional MMDC (OMMDC) problem.

Paper Structure

This paper contains 6 sections, 5 theorems, 7 figures, 5 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Our algorithms
  4. The OMMD algorithm
  5. The OMMDC algorithm
  6. Conclusions and Future Research

Key Result

Lemma 3.1

Let $b<c$ be two points in $S$, and $a<d$ be two points in $T$ such that $a\le b<c\le d$. If a minimum-cost OMMD, denoted by $M$, contains both of $(a,c)$ and $(b,d)$, then $(a,b) \in M$ or $(c,d) \in M$.

Figures (7)

  • Figure 1: $S \cup T$ is partitioned into maximal subsets $A_0,A_1,A_2,\dots$.
  • Figure 2: A minimum-cost OMMD between the points $p\leq a_s$ has been computed, and now we want to compute $C(b_i,k)$ for all $1 \leq i\leq t$ and $1 \leq k\leq k'$ for $Demand(b_i)\leq k'$.
  • Figure 3: $(a,c)$ and $(b,d)$ do not both belong to an optimal matching.
  • Figure 4: If $(a,d) \in M$, then either $(a,b) \in M$ or $(c,d) \in M$ or both.
  • Figure 5: Illustration of the situation $(a,b) \in M$ for all $b \in A_{i+1} \cup A_{i+3}\cup \dots \cup A_{j-2}$. The sets matched in a minimum-cost OMMD are connected with a line.
  • ...and 2 more figures

Theorems & Definitions (5)

  • Lemma 3.1
  • Lemma 3.2
  • Corollary 3.3
  • Theorem 3.4
  • Theorem 3.5