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Search and Rescue on a Poset

Jan-Tino Brethouwer, Robbert Fokkink

TL;DR

The paper generalizes Search and Rescue (SR) games to posets, introducing OSR (ordered) and CSR (chained) variants and connecting them to Bayesian networks. It delivers polynomial-time solutions for uncorrelated Bernoulli rescue events, and shows how SR on trees corresponds to completely reducible correlated distributions with Lidbetter-style depth-first strategies when Bayesian factors are bounded by 1; it also provides network-flow LP formulations and bounds under correlation, plus broad generalizations to non-binary rewards and random rescue sets. Key contributions include explicit value formulas for multi-stage OSR/CSR, width- and antichain-based bounds via Dilworth’s theorem, and a framework linking SR games to Bayesian networks and graph-theoretic decompositions. The results offer practical algorithms and theoretical bounds for scheduling, object detection, and adaptive search, and open avenues for further study of general Bayesian-factor regimes and richer reward structures.

Abstract

A Search and Rescue game (SR game) is a new type of game on a graph that has quickly found applications in scheduling, object detection, and adaptive search. In this paper, we broaden the definition of SR games by putting them into the context of ordered sets and Bayesian networks, extending known solutions of these games and opening up the way to further applications.

Search and Rescue on a Poset

TL;DR

The paper generalizes Search and Rescue (SR) games to posets, introducing OSR (ordered) and CSR (chained) variants and connecting them to Bayesian networks. It delivers polynomial-time solutions for uncorrelated Bernoulli rescue events, and shows how SR on trees corresponds to completely reducible correlated distributions with Lidbetter-style depth-first strategies when Bayesian factors are bounded by 1; it also provides network-flow LP formulations and bounds under correlation, plus broad generalizations to non-binary rewards and random rescue sets. Key contributions include explicit value formulas for multi-stage OSR/CSR, width- and antichain-based bounds via Dilworth’s theorem, and a framework linking SR games to Bayesian networks and graph-theoretic decompositions. The results offer practical algorithms and theoretical bounds for scheduling, object detection, and adaptive search, and open avenues for further study of general Bayesian-factor regimes and richer reward structures.

Abstract

A Search and Rescue game (SR game) is a new type of game on a graph that has quickly found applications in scheduling, object detection, and adaptive search. In this paper, we broaden the definition of SR games by putting them into the context of ordered sets and Bayesian networks, extending known solutions of these games and opening up the way to further applications.
Paper Structure (6 sections, 12 theorems, 41 equations, 2 figures)

This paper contains 6 sections, 12 theorems, 41 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Definitions and notation
  3. Uncorrelated Search and Rescue
  4. Correlated Search and Rescue
  5. Further generalizations of SR games
  6. Conclusion

Key Result

Lemma 1

Suppose $(X,\prec)$ is an extension of $(X,<)$, i.e., $x<y$ implies $x\prec y$. Then $V(X,\prec)\leq V(X,<)$ for the OSR game and $V(X,\prec)\geq V(X,<)$ for the CSR game.

Figures (2)

  • Figure 1: A tree with three leaves $a,b,c$ root $r$ and internal node $i$. Success probabilities displayed in the nodes.
  • Figure 2: The tree of Fig. \ref{['fig:1']} revisited as a network with events $a,b,c$ and internal nodes $q,r$. The events $a,b,c$ represent successful rescue in these three locations. The weights of the directed edges are ratios of conditional probabilities.

Theorems & Definitions (28)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • ...and 18 more