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Asymmetric Number Partitioning with Splitting and Interval Targets

Samuel Bismuth, Erel Segal-Halevi, Dana Shapira

TL;DR

This work studies three asymmetric, bounded-splitting variants of the NP-hard $n$-way number partitioning problem: SplitItem (limit $s$ split items), Splitting (limit $t$ splittings), and Inter (bin sums constrained to a $u$-slack interval). It establishes a comprehensive complexity landscape: with unbounded $n$ all three are strongly NP-hard, while for fixed $n$ each admits tractable cases when $s\ge n-2$, $t\ge n-1$, or $u\ge (n-2)/n$, and a polynomial-time, two-way reduction exists between SplitItem and Inter. The authors leverage FPTAS techniques (notably Woeginger’s framework) to design approximation-based algorithms, including a complete $O(poly(m,\log(AS)))$-time scheme for Inter when $n\ge 3$ and $u\ge n-2$, and binary-search-based strategies for SplitItem. They also prove strong NP-hardness of the Dec-Inter, Dec-SplitItem, and Dec-Splitting problems via 3-partition reductions, clarifying the boundary between tractable and intractable instances. The results have practical implications for fair division among asymmetric agents and for scheduling on uniform machines, offering both exact, approximate, and heuristic avenues depending on the parameter regime.

Abstract

The n-way number partitioning problem, a fundamental challenge in combinatorial optimization, has significant implications for applications such as fair division and machine scheduling. Despite these problems being NP-hard, many approximation techniques exist. We consider three closely related kinds of approximations, and various objectives such as decision, min-max, max-min, and even a generalized objective, in which the bins are not considered identical anymore, but rather asymmetric (used to solve fair division to asymmetric agents or uniform machine scheduling problems). The first two variants optimize the partition such that: in the first variant some fixed number s of items can be split between two or more bins and in the second variant we allow at most a fixed number t of splittings. The third variant is a decision problem: the largest bin sum must be within a pre-specified interval, parameterized by a fixed rational number u times the largest item size. When the number of bins n is unbounded, we show that every variant is strongly NP-complete. When the number of bins n is fixed, the running time depends on the fixed parameters s,t,u. For each variant, we give a complete picture of its running time. For n=2, the running time is easy to identify. Our main results consider any fixed n>=3. Using a two-way polynomial-time reduction between the first and the third variant, we show that n-way number-partitioning with s split items can be solved in polynomial time if s>=n-2, and it is NP-complete otherwise. Also, n-way number-partitioning with t splittings can be solved in polynomial time if t>=n-1, and it is NP-complete otherwise. Finally, we show that the third variant can be solved in polynomial time if u>=(n-2)/n, and it is NP-complete otherwise. Our positive results for the optimization problems consider both asymmetric min-max and asymmetric max-min versions.

Asymmetric Number Partitioning with Splitting and Interval Targets

TL;DR

This work studies three asymmetric, bounded-splitting variants of the NP-hard -way number partitioning problem: SplitItem (limit split items), Splitting (limit splittings), and Inter (bin sums constrained to a -slack interval). It establishes a comprehensive complexity landscape: with unbounded all three are strongly NP-hard, while for fixed each admits tractable cases when , , or , and a polynomial-time, two-way reduction exists between SplitItem and Inter. The authors leverage FPTAS techniques (notably Woeginger’s framework) to design approximation-based algorithms, including a complete -time scheme for Inter when and , and binary-search-based strategies for SplitItem. They also prove strong NP-hardness of the Dec-Inter, Dec-SplitItem, and Dec-Splitting problems via 3-partition reductions, clarifying the boundary between tractable and intractable instances. The results have practical implications for fair division among asymmetric agents and for scheduling on uniform machines, offering both exact, approximate, and heuristic avenues depending on the parameter regime.

Abstract

The n-way number partitioning problem, a fundamental challenge in combinatorial optimization, has significant implications for applications such as fair division and machine scheduling. Despite these problems being NP-hard, many approximation techniques exist. We consider three closely related kinds of approximations, and various objectives such as decision, min-max, max-min, and even a generalized objective, in which the bins are not considered identical anymore, but rather asymmetric (used to solve fair division to asymmetric agents or uniform machine scheduling problems). The first two variants optimize the partition such that: in the first variant some fixed number s of items can be split between two or more bins and in the second variant we allow at most a fixed number t of splittings. The third variant is a decision problem: the largest bin sum must be within a pre-specified interval, parameterized by a fixed rational number u times the largest item size. When the number of bins n is unbounded, we show that every variant is strongly NP-complete. When the number of bins n is fixed, the running time depends on the fixed parameters s,t,u. For each variant, we give a complete picture of its running time. For n=2, the running time is easy to identify. Our main results consider any fixed n>=3. Using a two-way polynomial-time reduction between the first and the third variant, we show that n-way number-partitioning with s split items can be solved in polynomial time if s>=n-2, and it is NP-complete otherwise. Also, n-way number-partitioning with t splittings can be solved in polynomial time if t>=n-1, and it is NP-complete otherwise. Finally, we show that the third variant can be solved in polynomial time if u>=(n-2)/n, and it is NP-complete otherwise. Our positive results for the optimization problems consider both asymmetric min-max and asymmetric max-min versions.
Paper Structure (34 sections, 22 theorems, 43 equations, 1 figure, 1 table, 6 algorithms)

This paper contains 34 sections, 22 theorems, 43 equations, 1 figure, 1 table, 6 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Partition with Interval Target
  4. AsymDec$\textsc{-Inter}[$$n,u$$]($${\@fontswitch\mathcal{X}}, {\@fontswitch\mathcal{R}}$$)$: an algorithm for $n = 2$ and $u > 0$
  5. AsymDec$\textsc{-Inter}[$$n,u$$]($${\@fontswitch\mathcal{X}}, {\@fontswitch\mathcal{R}}$$)$: an algorithm for $n \geq 3$ and $u \ge n-2$
  6. Structure of partitions with two or more almost-full bins
  7. Algorithm sketch
  8. Algorithm for bins 1 and 2
  9. Complete algorithm
  10. Hardness for $n \geq 3$ bins and $u<n-2$
  11. Partition with Split Items
  12. Binary Search to solve AsymMinMax$\textsc{-SplitItem}[$$n,s$$]($${\@fontswitch\mathcal{X}}, {\@fontswitch\mathcal{R}}$$)$
  13. AsymMaxMin$\textsc{-SplitItem}[$$n, s$$]($${\@fontswitch\mathcal{X}}, {\@fontswitch\mathcal{R}}$$)$
  14. Partition with Splittings
  15. Strong hardness proofs
  16. ...and 19 more sections

Key Result

Lemma 1

For any $n\geq 2$, $v>0$, $\epsilon>0$, if, for all $i\in[n]$, FPTAS(AsymMinMax$\textsc{-Part}[$$n,v,i$$]($${\@fontswitch\mathcal{X}}, {\@fontswitch\mathcal{R}}$$)$, $\epsilon) > (1+v)\cdot S$, then in any $v$-feasible $n$-way partition of ${\@fontswitch\mathcal{X}}$, at least two bins are almost-f

Figures (1)

  • Figure 1: Difference between the optimal and the perfect partition, in percent. The plots show results for uniform distribution with $r=0$ (left plot) and exponential distribution (right plot). In both plots, $m=13$ and the item size has 16 bits. The results for other values of the parameters are qualitatively similar; we present them in the supplementary information.

Theorems & Definitions (50)

  • Definition 1
  • Definition 2
  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Corollary 1
  • proof
  • Remark 1
  • Lemma 2
  • ...and 40 more