Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Complexity of Graph Exploration Games

Janosch Fuchs, Christoph Grüne, Tom Janßen

TL;DR

The work addresses the complexity of Graph Exploration and Treasure Hunt when the searcher receives an unlabeled map, formulating online game variants to study worst-case performance under adversarial revelation. It introduces the Online Traveling Salesman Game and the Online $s$-$t$ Path Game, and proves $PSPACE$-completeness for most variants across directed/undirected graphs, with or without edge costs and under path/trail/walk constraints, using reductions from the $TQBF$ game. The core technique relies on carefully designed gadgets (variables, clauses, sinks, and reveal edges) to encode truth assignments and clause satisfaction, ensuring the online algorithm must follow specific traversals to succeed. The results connect to prior work on online graph problems with maps and provide a broad view of the intractability landscape for map-informed exploration, informing both theory and potential algorithm design. Overall, the paper clarifies fundamental limits on online exploration when a priori unlabeled map information is available and identifies open directions in constrained variants and related problems.

Abstract

Graph Exploration problems ask a searcher to explore an unknown environment. The environment is modeled as a graph, where the searcher needs to visit each vertex beginning at some vertex. Treasure Hunt problems are a variation of Graph Exploration, in which the searcher needs to find a hidden treasure, which is located at a designated vertex. Usually these problems are modeled as online problems, and any online algorithm performs poorly because it has too little knowledge about the instance to react adequately to the requests of the adversary. Thus, the impact of a priori knowledge is of interest. One form of a priori knowledge is an unlabeled map, which is an isomorphic copy of the graph. We analyze Graph Exploration and Treasure Hunt problems with an unlabeled map that is provided to the searcher. For this, we formulate decision variants of both problems by interpreting the online problems as a game between the online algorithm (the searcher) and the adversary. The map, however, is not controllable by the adversary. The question is whether the searcher is able to explore the graph completely or find the treasure for all possible decisions of the adversary. We analyze these games in multiple settings, with and without costs on the edges, on directed and undirected graphs and with different constraints (allowing multiple visits to vertices or edges) on the solution. We prove PSPACE-completeness for most of these games. Additionally, we analyze the complexity of related problems that have additional constraints on the solution.

The Complexity of Graph Exploration Games

TL;DR

The work addresses the complexity of Graph Exploration and Treasure Hunt when the searcher receives an unlabeled map, formulating online game variants to study worst-case performance under adversarial revelation. It introduces the Online Traveling Salesman Game and the Online - Path Game, and proves -completeness for most variants across directed/undirected graphs, with or without edge costs and under path/trail/walk constraints, using reductions from the game. The core technique relies on carefully designed gadgets (variables, clauses, sinks, and reveal edges) to encode truth assignments and clause satisfaction, ensuring the online algorithm must follow specific traversals to succeed. The results connect to prior work on online graph problems with maps and provide a broad view of the intractability landscape for map-informed exploration, informing both theory and potential algorithm design. Overall, the paper clarifies fundamental limits on online exploration when a priori unlabeled map information is available and identifies open directions in constrained variants and related problems.

Abstract

Graph Exploration problems ask a searcher to explore an unknown environment. The environment is modeled as a graph, where the searcher needs to visit each vertex beginning at some vertex. Treasure Hunt problems are a variation of Graph Exploration, in which the searcher needs to find a hidden treasure, which is located at a designated vertex. Usually these problems are modeled as online problems, and any online algorithm performs poorly because it has too little knowledge about the instance to react adequately to the requests of the adversary. Thus, the impact of a priori knowledge is of interest. One form of a priori knowledge is an unlabeled map, which is an isomorphic copy of the graph. We analyze Graph Exploration and Treasure Hunt problems with an unlabeled map that is provided to the searcher. For this, we formulate decision variants of both problems by interpreting the online problems as a game between the online algorithm (the searcher) and the adversary. The map, however, is not controllable by the adversary. The question is whether the searcher is able to explore the graph completely or find the treasure for all possible decisions of the adversary. We analyze these games in multiple settings, with and without costs on the edges, on directed and undirected graphs and with different constraints (allowing multiple visits to vertices or edges) on the solution. We prove PSPACE-completeness for most of these games. Additionally, we analyze the complexity of related problems that have additional constraints on the solution.
Paper Structure (3 sections, 3 theorems, 1 figure)

This paper contains 3 sections, 3 theorems, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Path Problems

Key Result

Theorem 4

If $P^{SSP} \in NP$, then $P_O^{SSP} \in \textsl{PSPACE}$.

Figures (1)

  • Figure 1: Variable gadget for the reduction from TQBF Game to Online s-t Path Game for a variable $x_i$. The thick solid black edges have to be used by the algorithm, while the thin solid black edges are optional. Any non-solid colored edge cannot be used, and only exists for later recognition of the vertex it connects to. These properties are ensured by the sinks (red) that allow the adversary to trap the algorithm whenever it uses a non-solid non-black edge.

Theorems & Definitions (7)

  • Definition 1: Search Sequence
  • Definition 2: TQBF Game
  • Definition 3: Online Search Sequence Game
  • Theorem 4
  • Definition 5: Online Undirected s-t Path Game
  • Theorem 6
  • Lemma 6