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A tight 1/3-approximation algorithm and fully polynomial-time approximation schemes for the Colored Knapsack Problem

Fabio Ciccarelli, Fabio Furini

arXiv:2609.17713v1cs.DSmath.OC

TL;DR

ColKP adds a color-ordering constraint to knapsack, and negative-profit items can be necessary, complicating standard approximation techniques. The paper develops an LP-rounding approximation algorithm and profit-scaled schemes based on both dynamic programs. It establishes a 1/3 guarantee and FPTASs for nonnegative and arbitrary integer profits.

  • Problem

    ColKP requires a capacity-feasible item set whose items can be ordered with different colors consecutively, and negative-profit items can be necessary, complicating standard knapsack approximation techniques.

    Source: §1 · p. 2

  • Method

    The paper rounds an optimal basic LP solution for a linear-time approximation algorithm and adapts both exact dynamic programs to profit indexing and scaling for FPTASs.

    Source: §7 · p. 22

  • Results

    The LP-based algorithm has a worst-case ratio of 1/3, while item-by-item and color-by-color FPTASs cover nonnegative and arbitrary integer profits.

    Source: §7 · p. 22

  • Takeaways & Limitations

    ColKP admits both a constant-factor approximation and fully polynomial-time approximation schemes even with arbitrary integer profits.

    Source: §7 · p. 22

    Reducing FPTAS running times, particularly for instances with negative-profit items, remains a direction for further research.

    Source: §7 · p. 22

Abstract

from arXiv · show

The $\textit{Colored Knapsack Problem}$ (ColKP) generalizes the classical Knapsack Problem by partitioning the items into color classes and requiring the selected items to admit an ordering in which consecutive items have different colors. The problem is weakly $\mathcal{NP}$-hard and admits two pseudo-polynomial dynamic programming (DP) algorithms proposed in the literature. These two DP algorithms have worst-case running times $O(b \, n^4)$ and $O(b^2 \, n^3)$, respectively, where $b$ is the knapsack capacity and $n$ is the number of items. We develop the first approximation algorithm for the ColKP. By rounding an optimal basic solution of the linear programming relaxation of its natural integer programming formulation and repairing color feasibility, we obtain a linear-time approximation-algorithm whose worst-case performance ratio is $1/3$. We then reformulate both DP algorithms so that profit, rather than knapsack capacity, indexes their pseudo-polynomial dimension, and combine them with profit scaling to obtain two fully polynomial-time approximation schemes (FPTASs). The first FPTAS runs in $O(n^5/\varepsilon)$ time for nonnegative profits and in $O(n^6/\varepsilon)$ time for arbitrary integer profits. The second FPTAS runs instead in $O(n^5/\varepsilon^2)$ and $O(n^7/\varepsilon^2)$ time, respectively. The approximation guarantee, along with new structural insights, provides the bounds needed to control the scaled profit range and establish these running times.

1. Introduction

ColKP combines a knapsack capacity constraint with a color-ordering requirement, and its feasible sets can require negative-profit items. The paper addresses open approximability questions with a linear-time approximation algorithm and two FPTASs.

1. Introduction

  • ColKP seeks a maximum-profit capacity-feasible set whose items can be ordered with different colors consecutively; negative-profit items may be needed as separators.Color feasibility requires each color’s selected-item count to be at most one greater than the total count of all other colors.Source: §1 · p. 2
  • Earlier work established weak NP-hardness and two exact pseudo-polynomial DPs, but proposed no approximation algorithm or scheme for ColKP [11].The two-color results did not extend to the general m-color problem.Source: §1 · pp. 2–3

1.1. Approximation algorithms and approximation schemes for knapsack problems

  • Profit scaling adapts classical knapsack’s profit-indexed dynamic program by trading an additive loss of less than λ per selected item for a polynomially bounded profit range.The section reviews approximation guarantees and schemes as background for the ColKP algorithms.Source: §1.1 · pp. 3–4

1.2. Contributions and outline of the paper

  • Structural results bound relevant signed profits: items with profit at most −pmax can be removed, and optimal solutions containing negative-profit items have a unique critical color and at most (n −1)/2 such items.These bounds support control of the scaled profit ranges used by the schemes.Source: §1.2 · p. 5
  • Rounding an optimal basic LP solution and repairing color feasibility gives a linear-time 1/3-approximation with a tight worst-case ratio.The algorithm also gives lower and upper bounds on OPT within a factor of three, helping control scaled profit ranges.Source: §1.2 · p. 5
  • The item-by-item DP reformulation yields an FPTAS in O(n^5/ε) time for nonnegative profits and O(n^6/ε) for arbitrary integer profits.The approach combines profit indexing and scaling with bounds on OPT.Source: §1.2 · p. 5
  • The color-by-color DP also yields an FPTAS, with running times O(n^5/ε^2) for nonnegative profits and O(n^7/ε^2) for arbitrary integer profits.Under the direct scaling approach, the item-by-item recursion has better worst-case running times.Source: §1.2 · p. 5

2. Structural properties of ColKP solutions

The section establishes structural properties that bound relevant profits and explain when negative-profit items can occur in optimal solutions.

2. Structural properties of ColKP solutions

  • Every singleton is feasible, so the optimum is nonnegative and at least pmax; if pmax ≤0, the empty set is optimal.The analysis assumes wi ≤ b for every item.Source: §2 · pp. 5–6
  • Removing an item of a dominant color preserves color feasibility; removing such an item together with a nondominant item does too.These modifications are used to derive bounds on relevant negative profits.Source: §2 · p. 6
  • An optimal solution exists with no item whose profit is at most −pmax, so all such items can be deleted in O(n) time without changing the optimum.The result permits restricting the profits considered by later algorithms.Source: §2 · p. 6
  • If an optimal solution contains a negative-profit item, its unique dominant color is critical and it has at most (n −1)/2 negative-profit items.The bound follows from the solution’s critical-color structure.Source: §2 · p. 7

3. A linear-time 1/3-approximation

LP-Round rounds an optimal basic LP solution and repairs any color-feasibility violation by removing one item. It guarantees a feasible solution with at least one-third of the optimum in linear time, and this worst-case ratio is tight.

3. A linear-time 1/3-approximation

  • An optimal basic LP solution has at most two fractional variables and can be computed in O(n) time.The paper uses these properties of the LP relaxation in the algorithm's linear-time analysis.Source: §3 · pp. 7, 10
  • LP-Round rounds down an optimal basic LP solution and repairs a color-feasibility violation, if present, by removing one item.The repair removes an item of the violated dominant color.Source: §3 · pp. 7–8
  • The rounded, possibly repaired set has profit at least zLP − 2pmax; comparing it with a best-profit singleton yields the 1/3 guarantee.When pmax is at least zLP/3, the singleton suffices; otherwise the rounded set provides the bound.Source: §3 · p. 10
  • The worst-case performance ratio is exactly 1/3; LP-Round returns a feasible solution in O(n) time.Its profit is at least zLP/3, and zLP is an upper bound on OPT.Source: §3 · pp. 7, 10
  • A four-item, two-color instance family shows tightness: LP-Round returns profit ℓ while the optimum is 3ℓ−1.The family is used to establish that the 1/3 worst-case bound cannot be improved for this algorithm.Source: §3 · pp. 10–11

4. A first profit-indexed DP algorithm

The first exact dynamic program exchanges capacity and profit as the pseudo-polynomial dimension. It tracks color feasibility while storing the minimum weight needed for each state.

4. A first profit-indexed DP algorithm

  • The algorithm reformulates DP1 from [11] by exchanging the roles of capacity and the objective, making profit the indexed dimension.Items are sorted by color before processing.Source: §4 · p. 12
  • The state counters determine color feasibility for a partial solution and its extensions, so among matching states a lighter solution dominates a heavier one.This dominance justifies exchanging capacity and profit dimensions.Source: §4 · p. 12
  • For each prefix, item count, dominant-color count, current-color count, and profit, the DP stores the minimum weight of a realizing subset.States with no realizing subset have value +∞.Source: §4 · p. 12
  • The pack and do-not-pack transitions update the counters, resetting the current-color count when the item color changes.The algorithm returns the largest profit whose state meets the color constraint and whose minimum weight is at most capacity.Source: §4 · pp. 12–13

5. A first approximation scheme

The first FPTAS scales profits and runs the profit-indexed dynamic program over a bounded profit range. It returns a feasible solution worth at least (1 − ε)OPT, with running time depending on whether profits are nonnegative or arbitrary integers.

5. A first approximation scheme

  • The scheme sets λ = εLB/n, rounds item profits, and runs the profit-indexed DP over a restricted profit range.LB is a lower bound on OPT; the scaled profit range is polynomially bounded using the bounds developed in the paper.Source: §5 · pp. 13–14
  • The FPTAS guarantees value at least (1 − ε)OPT and runs in O(n^5/ε) time for nonnegative profits or O(n^6/ε) for arbitrary integer profits.The additional factor n for arbitrary profits arises from retaining prefixes whose profit can be negative or exceed the final profit.Source: §5 · pp. 14–16
  • For arbitrary integer profits, the DP must retain prefixes whose scaled profits can extend beyond the final solution's profit range.The two ends of the scaled range require an additional contribution of order npmax/λ.Source: §5 · pp. 14–16
  • With ε = 1/2, the example's FPTAS can return profit 16 from {2, 5}, satisfying the guarantee against OPT = 17.Item 4 has scaled profit zero in this run, though it still consumes capacity.Source: §5 · p. 16
  • With ε = 1/100, the example's FPTAS returns the optimum of 17, while the profit axis is fifty times longer.The optimal solution uniquely attains scaled profit 707, exceeding the scaled profit 666 of {2, 5}.Source: §5 · p. 16

6. Can the color-by-color recursion do better?

The color-by-color DP can be reindexed by profit and scaled to yield an FPTAS, but its convolution cost makes it slower than the item-by-item scheme under direct scaling. Its exact version combines per-color packings while tracking total cardinality, dominant-color cardinality, and profit.

6. Can the color-by-color recursion do better?

  • DPp2 indexes its inner and outer programs by profit and stores minimum weight; the outer program combines one packing per color while tracking total and dominant-color cardinalities.For each color, the inner program summarizes packings by cardinality and profit, retaining minimum weight for each pair.Source: §6 · p. 17
  • The exact DP returns an optimal ColKP solution by maximizing profit among states whose weight is at most b and whose cardinalities satisfy 2d≤t+1.A profit level is feasible exactly when such a terminal state exists.Source: §6 · pp. 18–19
  • For every ε∈(0,1), the scaled color-by-color scheme returns a feasible solution worth at least (1−ε)OPT.The scheme restricts both programs to a bounded scaled-profit range and discards states heavier than b.Source: §6 · p. 19
  • O(n^5/ε) versus O(n^5/ε²) for nonnegative profits, and O(n^6/ε) versus O(n^7/ε²) for arbitrary integer profits: item-by-item scaling is faster than color-by-color scaling.The color-by-color recursion offers no improvement once scaled profit, rather than capacity, is the pseudo-polynomial dimension.Source: §6 · p. 22
  • The color-by-color recursion trades fewer outer states for a convolution over attainable profits, making its running time O(n^3 q^2) rather than O(n^4 q).For the FPTAS, q=Π=Θ(n/ε) with nonnegative profits, so the extra factor in the transition cost outweighs the saved factor in the state space.Source: §6 · pp. 21–22

7. Conclusions

The paper establishes approximation guarantees for ColKP despite the color-ordering constraint and the possibility that optimal solutions need negative-profit items. It gives a linear-time 1/3-approximation and two profit-scaled FPTASs, with item-by-item recursion yielding the better stated running times.

7. Conclusions

  • The paper establishes a constant-factor approximation and FPTASs even when item profits are arbitrary integers.Negative-profit items can be necessary in an optimal solution, and the paper bounds how many can occur to control signed profits in the schemes.Source: §7 · p. 22
  • The linear-time LP-rounding algorithm has a worst-case performance ratio of 1/3.It rounds an optimal basic solution of the natural ILP's LP relaxation and repairs color feasibility.Source: §7 · p. 22
  • The item-by-item FPTAS runs in O(n^5/ε) for nonnegative profits and O(n^6/ε) for arbitrary integer profits; the color-by-color FPTAS runs in O(n^5/ε²) and O(n^7/ε²), respectively.Under the direct profit-scaling approach considered, item-by-item recursion has the better worst-case running times.Source: §7 · p. 22
  • The authors identify reducing FPTAS running times, particularly for negative-profit instances, and improving the 1/3 guarantee as directions for further research.A stronger guarantee would require exploiting color-feasible solution structure beyond the LP-rounding approach considered.Source: §7 · p. 22
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