Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Searching by Heterogeneous Agents

Dariusz Dereniowski, Łukasz Kuszner, Robert Ostrowski

TL;DR

This work studies pursuit-evasion with heterogeneous searchers on edge-labeled graphs, where each searcher may traverse only edges of a designated color. It develops a formal framework for heterogeneous edge search, reveals that monotonicity can fail even on trees, and establishes NP-hardness for both monotone and non-monotone variants in trees, along with a polynomial-time tractable family when each color class forms a connected subtree. The results highlight fundamental differences from classical edge search and delineate clear boundaries between intractable and tractable instances, offering insights for heterogeneous robotic search and related applications. The paper also outlines open questions about membership in NP and extensions to richer color- and agent-models relevant for real-world robotic search systems.

Abstract

In this work we introduce and study a pursuit-evasion game in which the search is performed by heterogeneous entities. We incorporate heterogeneity into the classical edge search problem by considering edge-labeled graphs: once a search strategy initially assigns labels to the searchers, each searcher can be only present on an edge of its own label. We prove that this problem is not monotone even for trees and we give instances in which the number of recontamination events is asymptotically quadratic in the tree size. Other negative results regard the NP-completeness of the monotone, and NP-hardness of an arbitrary (i.e., non-monotone) heterogeneous search in trees. These properties show that this problem behaves very differently from the classical edge search. On the other hand, if all edges of a particular label form a (connected) subtree of the input tree, then we show that optimal heterogeneous search strategy can be computed efficiently.

Searching by Heterogeneous Agents

TL;DR

This work studies pursuit-evasion with heterogeneous searchers on edge-labeled graphs, where each searcher may traverse only edges of a designated color. It develops a formal framework for heterogeneous edge search, reveals that monotonicity can fail even on trees, and establishes NP-hardness for both monotone and non-monotone variants in trees, along with a polynomial-time tractable family when each color class forms a connected subtree. The results highlight fundamental differences from classical edge search and delineate clear boundaries between intractable and tractable instances, offering insights for heterogeneous robotic search and related applications. The paper also outlines open questions about membership in NP and extensions to richer color- and agent-models relevant for real-world robotic search systems.

Abstract

In this work we introduce and study a pursuit-evasion game in which the search is performed by heterogeneous entities. We incorporate heterogeneity into the classical edge search problem by considering edge-labeled graphs: once a search strategy initially assigns labels to the searchers, each searcher can be only present on an edge of its own label. We prove that this problem is not monotone even for trees and we give instances in which the number of recontamination events is asymptotically quadratic in the tree size. Other negative results regard the NP-completeness of the monotone, and NP-hardness of an arbitrary (i.e., non-monotone) heterogeneous search in trees. These properties show that this problem behaves very differently from the classical edge search. On the other hand, if all edges of a particular label form a (connected) subtree of the input tree, then we show that optimal heterogeneous search strategy can be computed efficiently.

Paper Structure

This paper contains 15 sections, 37 theorems, 14 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Related work
  3. Our work --- a short outline
  4. Preliminaries
  5. Problem formulation
  6. Additional notation and remarks
  7. Lack of monotonicity
  8. NP-hardness for trees
  9. NP-hardness of non-monotone searching of trees
  10. Preliminaries on non-monotone strategies for $\tilde{T}_{\textup{SAT}}$
  11. Some technical lemmas
  12. Adaptation to non-monotonicity --- there is no going back
  13. Conclusion
  14. Polynomially tractable instances
  15. Conclusions and open problems

Key Result

Lemma 2.1

Given a tree $T = (V(T), E(T), c)$ and any area $H$ in $T$, any search $\tilde{c}$-strategy for $T$ uses at least $\textup{s}(H)$ searchers of color $c(H)$.

Figures (4)

  • Figure 1: The construction of $T_{3}$ ($l=3$) from the trees $T_1'$, $T_2'$ and $T_{3}"$. Regular, heavy and dashed edges have labels $1,2$ and $3$, respectively.
  • Figure 2: Construction of $T$: (a) the subgraph $H_z$; (b) the subgraph $L_{x}$; (c) the subgraph $L'_{x'}$
  • Figure 3: Construction of $T$: attachment of subgraphs $H_z$ and the subgraphs $L_x$ and $L'_{x'}$ to the path $P$
  • Figure 4: Construction of $T$: (a) the variable component constructed from $S_{-p}$ and $S_p$; (b) the clause component that corresponds to $C_d$.

Theorems & Definitions (64)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Theorem 1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • ...and 54 more