Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Set Selection with Uncertain Weights: Non-Adaptive Queries and Thresholds

Christoph Dürr, Arturo Merino, José A. Soto, José Verschae

TL;DR

This work addresses set selection with uncertain weights by introducing thresholds under uncertainty and showing their fundamental equivalence to minimum-cost non-adaptive queries (admissible queries). It develops efficient threshold computation methods for MSTs, matroids, and tree matchings, and translates these into fast algorithms for minimum cost admissible queries; it also proves NP-hardness for thresholds in s-t shortest paths and bipartite matching. The approach leverages the blue/red threshold taxonomy and non-trivial bottleneck concepts to connect optimization structure with uncertainty, enabling both exact and approximate solutions. The results clarify when non-adaptive querying suffices to guarantee universal optimality and when the problem becomes computationally intractable, with implications for exploring uncertainty in combinatorial optimization.

Abstract

We study set selection problems where the weights are uncertain. Instead of its exact weight, only an uncertainty interval containing its true weight is available for each element. In some cases, some solutions are universally optimal; i.e., they are optimal for every weight that lies within the uncertainty intervals. However, it may be that no universal optimal solution exists, unless we are revealed additional information on the precise values of some elements. In the minimum cost admissible query problem, we are tasked to (non-adaptively) find a minimum-cost subset of elements that, no matter how they are revealed, guarantee the existence of a universally optimal solution. We introduce thresholds under uncertainty to analyze problems of minimum cost admissible queries. Roughly speaking, for every element e, there is a threshold for its weight, below which e is included in all optimal solutions and a second threshold above which e is excluded from all optimal solutions. We show that computing thresholds and finding minimum cost admissible queries are essentially equivalent problems. Thus, the analysis of the minimum admissible query problem reduces to the problem of computing thresholds. We provide efficient algorithms for computing thresholds in the settings of minimum spanning trees, matroids, and matchings in trees; and NP-hardness results in the settings of s-t shortest paths and bipartite matching. By making use of the equivalence between the two problems these results translate into efficient algorithms for minimum cost admissible queries in the settings of minimum spanning trees, matroids, and matchings in trees; and NP-hardness results in the settings of s-t shortest paths and bipartite matching.

Set Selection with Uncertain Weights: Non-Adaptive Queries and Thresholds

TL;DR

This work addresses set selection with uncertain weights by introducing thresholds under uncertainty and showing their fundamental equivalence to minimum-cost non-adaptive queries (admissible queries). It develops efficient threshold computation methods for MSTs, matroids, and tree matchings, and translates these into fast algorithms for minimum cost admissible queries; it also proves NP-hardness for thresholds in s-t shortest paths and bipartite matching. The approach leverages the blue/red threshold taxonomy and non-trivial bottleneck concepts to connect optimization structure with uncertainty, enabling both exact and approximate solutions. The results clarify when non-adaptive querying suffices to guarantee universal optimality and when the problem becomes computationally intractable, with implications for exploring uncertainty in combinatorial optimization.

Abstract

We study set selection problems where the weights are uncertain. Instead of its exact weight, only an uncertainty interval containing its true weight is available for each element. In some cases, some solutions are universally optimal; i.e., they are optimal for every weight that lies within the uncertainty intervals. However, it may be that no universal optimal solution exists, unless we are revealed additional information on the precise values of some elements. In the minimum cost admissible query problem, we are tasked to (non-adaptively) find a minimum-cost subset of elements that, no matter how they are revealed, guarantee the existence of a universally optimal solution. We introduce thresholds under uncertainty to analyze problems of minimum cost admissible queries. Roughly speaking, for every element e, there is a threshold for its weight, below which e is included in all optimal solutions and a second threshold above which e is excluded from all optimal solutions. We show that computing thresholds and finding minimum cost admissible queries are essentially equivalent problems. Thus, the analysis of the minimum admissible query problem reduces to the problem of computing thresholds. We provide efficient algorithms for computing thresholds in the settings of minimum spanning trees, matroids, and matchings in trees; and NP-hardness results in the settings of s-t shortest paths and bipartite matching. By making use of the equivalence between the two problems these results translate into efficient algorithms for minimum cost admissible queries in the settings of minimum spanning trees, matroids, and matchings in trees; and NP-hardness results in the settings of s-t shortest paths and bipartite matching.
Paper Structure (19 sections, 20 theorems, 12 equations, 7 figures)

This paper contains 19 sections, 20 theorems, 12 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Minimum cost queries
  3. Thresholds of inclusion and exclusion
  4. Our contribution
  5. Further related work
  6. Threshold preliminaries
  7. Thresholds without uncertainty
  8. Thresholds with uncertainty
  9. From thresholds to minimizing queries, and back
  10. The query problem reduces to threshold computation
  11. Threshold computation reduces to the query problem.
  12. Minimum spanning trees
  13. Thresholds via greedy
  14. Thresholds via bottleneck paths
  15. Shortest paths
  16. ...and 4 more sections

Key Result

Lemma 1

Let $P:= \mathop{\mathrm{conv}}\nolimits(\{\chi_F \mid F \in \mathcal{F} \})$, $w \in \mathbb{R}^E$, and $e\in E$. There exists $x',y' \in \{0,1\}^{E}$ such that $w\cdot x' = \min \{ w\cdot x \mid x\in P, \; x_e = 0 \}$, $w \cdot y' = \min \{ w\cdot x \mid x\in P, \; x_e = 1 \}$, and $xy$ is an edge

Figures (7)

  • Figure 1: (a) An instance of $s$-$t$ paths with uncertain weights. (b)+(c) Two realizations of the instance in (a) with different optimal solutions marked in blue.
  • Figure 2: (a) A minimum-sized admissible query marked in green. (b)+(c) Different true weights of the queried edges give rise to different universally optimal solutions (marked in blue).
  • Figure 3: (a) This instance has no universally optimal solution. (b) If the edge marked in green has weight less (resp. more) than 2, then it is contained in every (resp. no) $s$-$t$ shortest path. (c) If the green edge has weight smaller than 1 (resp. larger than $2$), then it is included in all (resp. no) shortest $s$-$t$ paths, regardless of the true weights.
  • Figure 4: (a) Behaviour of $\textrm{OPT}^{-e}$ and $\textrm{OPT}^{+e}$ when we change the weight of $e$. The threshold $T_e$ is exactly the point at which these two values are the same. (b) Schematic view of this threshold as a function of the weight of some other element $f$.
  • Figure 5: Reduction from 3-SAT. Solid edges have uncertainty intervals $[0,1]$, while dashed edges trivial uncertainty intervals $0$.
  • ...and 2 more figures

Theorems & Definitions (34)

  • Lemma 1
  • proof
  • Corollary 2
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • Lemma 7
  • proof
  • ...and 24 more