Research Collection. Grid exploration. Master Thesis. ETH Library. Author(s): Wernli, Dino. Publication Date: 2012

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1 Research Collection Master Thesis Grid exploration Author(s): Wernli, Dino Publication Date: 2012 Permanent Link: Rights / License: In Copyright - Non-Commercial Use Permitted This page was generated automatically upon download from the ETH Zurich Research Collection. For more information please consult the Terms of use. ETH Library

2 Grid Exploration Master Thesis Dino Wernli August 1, 2012 Advisors: Prof. Dr. P. Widmayer, Dr. Y. Disser Department of Computer Science, ETH Zürich

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4 Abstract The grid exploration problem consists in finding a shortest possible tour which visits every cell of a grid at least once. Solutions to this problem can be applied in scenarios where an autonomous robot must perform tasks such as mowing the lawn or cleaning a room. We distinguish two variants of grid exploration. In the offline variant, the entire grid is given as input, whereas in the online variant the robot has limited visibility and must acquire information about the initially unknown grid as it explores it. We provide an overview of currently known results for offline and online grid exploration. We also develop new techniques to prove bounds for the length of shortest exploration tours and show how to combine them to get stronger bounds. Previous results for the online variant assume that the robot is initially located at the boundary of the grid. We give a new algorithm which does not make this assumption and achieves the same competitive ratio. i

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6 Contents Contents iii 1 Introduction The Exploration Problem Related Work Thesis Outline Preliminaries Traversing Graphs Grids Graph Exploration Approximation and Competitive Ratio Distance Graphs Grid Exploration and TSP Computing the Distance Graph Reducing Grid Exploration to TSP Online Grid Exploration Bounds for Excess Upper Bound Basic Lower Bounds Area Bounds Area Bounds using the Number of Ports Area Bounds using Port Multiplicities Combining Area Bounds Explaining Excess Offline Grid Exploration Hamiltonian Cycles in General Grids Related Hamiltonian Cycle Problems iii

7 Contents Embedding Graphs Proof of Hardness Hamiltonian Cycles in Solid Grids Two-Factors Alternating Sequences Hamiltonian Cycle Algorithm Approximating Planar Graph Exploration Algorithm Outline Spanner Step Thinning Step Dynamic Programming Step Lifting Step Combining the Steps Approximating Grid Exploration Online Grid Exploration: Known Results General Grid Exploration Depth-First Search Lower Bound for Competitive Ratio Solid Grid Exploration Improving Depth-First Search Further Improvements Lower Bound for Competitive Ratio Online Grid Exploration: General Starting Positions Lower Bound for Approximation Upper Bound for Competitive Ratio Finding the Boundary Composing Tours Proof of Upper Bound Conclusion Summary of Contributions Further Work A Details of Hardness Proof 57 A.1 Proof of Lemma A.2 Definition of Tentacles Bibliography 59 iv

8 Chapter 1 Introduction 1.1 The Exploration Problem Exploring an unknown environment is a fundamental path planning problem frequently faced by robots when performing tasks autonomously. Many different exploration problems have been studied in the past, differing in objective, environment and robot model [2, 4, 7, 8, 10, 12, 13]. In our setting, the objective of the robot is to find a shortest possible tour which covers the entire environment. Applications range from laser scanners trying to detect objects at specific locations to service robots which mow the lawn, prepare ice rinks or vacuum rooms. Many environments can be approximated by grids of square-shaped cells and in this thesis, we focus on the problem of exploring such grids. The robot is assumed to occupy one cell of the grid and, in each time step, to be able to move one cell up, down, left or right. The objective of the robot is to visit each cell at least once and then return to its starting position. In many realistic settings, the environment can have obstacles. For instance, if a robot is attempting to vacuum a room, there could be a solid piece of furniture somewhere in the room. The robot is then expected to move around the obstacles and explore the remaining parts of the room. We model the presence of obstacles as holes in the grid. It turns out that whether or not we allow holes in the grid influences the complexity of the problem significantly. There are multiple variants of the grid exploration problem. In the offline version, the robot receives the entire grid as input and can compute a tour beforehand. In the presence of holes, it has been shown that already the Hamiltonian cycle problem is NP-hard [15], which implies that grid exploration is NP-hard as well. However, if we do not allow holes in the grids, 1

9 1. Introduction finding Hamiltonian cycles can be done in polynomial time [21]. Whether or not the exploration problem for grids without holes is NP-hard is an important open question. In the online variant of grid exploration, the robot has limited visibility and must explore the environment from some starting location without any prior knowledge. We do not impose any memory restrictions on the robot, allowing it to store all the knowledge it accumulates while exploring the grid. In the presence of holes, it has been shown that best competitive ratio can be achieved by executing an online variant of depth-first search [14, 19]. In grids without holes, it is known that the best possible competitive ratio lies between 20/17 and 5/4 = 20/16 if the robot is allowed to start its tour at a boundary cell of the grid. It was unknown until now whether or not this assumption affects the competitive ratio of the problem. In this thesis, we show that the competitive ratio of general grid exploration is the same as the competitive ratio of grid exploration starting at the boundary of the grid. Therefore, it can be assumed without loss of generality that the robot starts its tour at the boundary and results carry over to the general setting. We also develop new offline techniques to prove bounds for the length of a shortest exploration tour in a given grid. Additionally, we present proofs of most known results for the different variants of grid exploration in a unified context, in order to provide an overview of the research area Related Work As we will see, the exploration problem is closely related to two classical problems from graph theory. One of them is the Hamiltonian cycle problem which consists in determining whether or not a given graph contains a Hamiltonian cycle, i.e., a cycle which visits every vertex exactly once. The other one is the traveling salesman problem, which consists in finding a minimum-weight Hamiltonian cycle in a weighted graph. Both problems are well-known to be NP-complete [18]. Umans and Lenhart presented an algorithm for finding Hamiltonian cycles in grids without holes in time O(n 4 ) [21]. In contrast, Itai et al. proved that the Hamiltonian cycle problem is NP-hard in general grids [15], which implies that the grid exploration problem is hard as well. This result is particularly interesting because it demonstrates that allowing holes in the input grids can make the problem much harder. Arkin et al. presented an offline algorithm which achieves an approximation factor of 6/5 = 48/40 in grids without holes and a factor of 53/40 if the input grid contains holes [4]. Later, Arora et al. developed an algorithm

10 1.3. Thesis Outline which produces a (1 + ε)-approximation for arbitrarily small ε in general planar graphs and runs in time O(n 1/ε ) [5]. Finally, the runtime for the approximation algorithm was improved to O(c 1/ε2 n) by Klein [16]. Klein also suggested that the runtime of the approximation algorithm could be reduced to O(c 1/ε n) for unweighted graphs such as grids [16]. For the online variant of grid exploration, Icking et al. showed that the best possible competitive ratio is 2 for general grids [14] and Miyazaki et al. proved that an online version of depth-first search achieves this ratio [19]. Therefore, the focus moved to exploration of grids without holes for which Icking et al. described an online 4/3-competitive algorithm which assumes that the robot starts at the boundary of the grid [14]. This ratio was then improved to 5/4 = 20/16 by Kolenderska et al., who also showed that the best possible ratio is at least 20/17 [17]. 1.3 Thesis Outline We start in Chapter 2 by introducing and defining the main objects and concepts which we require throughout the thesis. Then, in Chapter 3, we develop new ways to derive lower bounds for the length of an exploration tour in a given grid. We also demonstrate how multiple such bounds can be combined to obtain stronger ones. In Chapter 4, we describe the currently best known results for offline grid exploration and apply the idea mentioned by Klein, which improves the running time of his (1 + ε)-approximation algorithm from O(c 1/ε2 n) to O(c 1/ε n) for grids. In Chapter 5, we give proofs for the known upper and lower bounds of 2 on the competitive ratio of online grid exploration. We then describe the 5/4-competitive algorithm due to Kolenderska et al. for grids without holes under the assumption that the robot is initially located at the boundary of the grid. In Chapter 6, we develop an algorithm that shows that the competitive ratio remains the same even if the robot is not guaranteed to start at the boundary of the grid. This shows that, for competitive analysis, it can be assumed without loss of generality that the robot starts at the boundary and that results obtained under this assumption carry over to the general problem. 3

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12 Chapter 2 Preliminaries In this chapter, we formally introduce the objects we will be dealing with throughout this thesis. Some basic knowledge of graph theory is assumed. Many of the following definitions are only stated for weighted graphs, i.e., graphs of the form G = (V, E) where some weight function w : E R assigns a real-valued weight to every edge. In order to carry the definitions over to unweighted graphs, we consider an unweighted graph as weighted with unit edge weights. 2.1 Traversing Graphs We start by defining some common notions from graph theory concerning the traversal of graphs. Definition 2.1 (walk) A walk in a graph G is a sequence of vertices such that each vertex in the sequence is adjacent to the vertices preceding and following it, if they exist. Definition 2.2 (path) A path in a graph G is a walk in G which visits each vertex of G at most once. Definition 2.3 (tour) A tour in a graph G is a closed walk in G, i.e., a walk such that the first vertex is incident to the last vertex. Note that, since tours are closed, the number of vertices in a tour T is equal to the number of edges in T. Definition 2.4 (cycle) A cycle in a graph G is a closed path in G, i.e., a path which is a tour. 5

13 2. Preliminaries Figure 2.1: A grid in which the black tour is restricted to the blue subset S. The restriction consists of two walks. Definition 2.5 (Hamiltonian cycle) A cycle which contains every vertex of a graph G exactly once is called a Hamiltonian cycle. Consider Figure 2.1 along with the following definition. Definition 2.6 (restriction) Let S be an induced subgraph of a graph G and let T be a walk in G. The restriction T S of T to S consists of T where all parts not in S are omitted. Note that the restriction of a tour is generally no longer a tour, but a set 1 of walks which all start and end at the boundary of S. 2.2 Grids This thesis mainly deals with a special type of graph called grid graph. In this section we give a formal definition of grids and some other related concepts. Definition 2.7 (infinite grid) The infinite grid G is a graph on the vertex set Z Z. Two vertices are connected by an edge iff their Euclidean distance is 1. Definition 2.8 (grid) A grid is a finite, connected, induced subgraph of G. Since a grid is fully defined by its vertex set, we will sometimes use a grid G and its set of vertices interchangeably to simplify notation. For instance, we will use S G to denote that S is either an induced subgraph of G or a subset of the vertices of G and we will use G v to denote the subgraph of G induced by all vertices except v. 1 Technically, the restriction of a tour is a multiset because identical walks can appear multiple times in a tour. However, since we will be dealing with shortest tours, the restrictions will generally be sets. 6

14 2.2. Grids Figure 2.2: A grid G with its halo H(G) left and its 8-halo H 8 (G) right. Note that G is bipartite, planar and that every vertex in G has degree exactly 4. It follows that every grid is bipartite and planar as well. Since grids can be drawn as checker boards, we will use the terms vertex and cell interchangeably for grids. Also, we will refer to the partite sets as B and W for the colors black and white. In general, grids can have holes. The following definition formalizes this notion. Definition 2.9 (hole, solid grid) Let G be a grid. Every finite connected component of G G is called a hole of G. A grid G is called solid if it has no holes, i.e., if G G is connected. Definition 2.10 (halo) The halo H(v) of v G is defined as the set containing the 4 neighbors of v in G. Similarly, the 8-halo H 8 (v) of v contains the eight vertices nearest to v in Euclidean metric. The definition of halo can be generalized for sets of vertices as shown in Figure 2.2. More formally, we generalize halos as follows. Definition 2.11 Let G = (V, E) be a finite, induced subgraph of G. We define the halo and 8-halo of G, respectively, as H(G) := H(v) \ V, and v V H 8 (G) := v V H 8 (v) \ V. It follows from the definitions that H(G) H 8 (G) for every subgraph G of G. Note that in Definition 2.11 we do not require G to be connected. However, in most cases, we will only use halos of connected subgraphs. Sometimes it is useful to only consider certain parts of a halo. The following definitions allow us to conveniently intersect halos with sets of vertices. 7

15 2. Preliminaries Figure 2.3: A grid with its boundary left and its 8-boundary right. Definition 2.12 Let G be a finite, induced subgraph of G and let S Z Z be a set of vertices. We define H S (G) := H(G) S, and H S 8(G) := H 8 (G) S. Consider Figure 2.3 along with the following definitions. Definition 2.13 (border cell, boundary) A vertex v of a grid G is called a border cell if it has less than 4 neighbors in G. The set H(G G) of all border cells of G is called the boundary of G. This definition of boundary requires every vertex to have a neighbor outside the grid. This leads to boundaries not always inducing a connected subgraph. To circumvent this, we define the 8-boundary of a grid as follows. Definition 2.14 (8-boundary) We refer to H 8 (G G) as the 8-boundary of a grid G Graph Exploration In order to define the exploration problem formally, we introduce the notion of weights of tours. Definition 2.15 (weight of edge sets) The weight w(s) of a sequence S of edges is defined as the sum of the weights of the individual edges. The weight of a graph G is defined as the sum of all weights of the edges in G. In order to avoid distinguishing cases for weighted and unweighted graphs, the following notions are defined in terms of weight instead of length.

16 2.4. Approximation and Competitive Ratio Definition 2.16 (weight of tours) The weight w(t) of a tour T is the sum of the weights of all used edges, including the final edge from the last vertex back to the first vertex. In unweighted graphs, the weight of a tour T is equal to the number of edges in T because unweighted graphs are considered to have unit edge weights. In general, we will use the term length of a tour as a synonym for its weight. In unweighted graphs, we will also write T for the length of a tour T. Definition 2.17 (exploration tour) A tour T of a graph G is called an exploration tour for G if every vertex of G is contained in T at least once. Definition 2.18 (exploration problem) Given a graph G, the exploration problem consists in finding a shortest (i.e., minimum-weight) exploration tour for G. The grid exploration problem is the graph exploration problem where the input graphs are constrained to being grids. Definition 2.19 (set of shortest exploration tours) We denote by T opt (G) the set of shortest exploration tours of a graph G and by τ opt (G) the length of these tours. The following observation directly follows from the fact that an exploration tour must visit every vertex at least once and that unweighted graphs are considered to have unit weights. Proposition 2.20 For an unweighted graph G on n vertices, we have τ opt (G) n. Proposition 2.20 suggests that for unweighted graphs, it makes sense to measure the quality of an exploration tour by the number of times it revisits a cell. The following definition makes this precise. Definition 2.21 (excess) Let G be an unweighted graph on n vertices. The excess e(t) of an exploration tour T of G is defined as e(t) := T n. The excess e(g) of a grid G is the excess of a shortest exploration tour for G, i.e., e(g) := τ opt (G) n. Note that by Proposition 2.20, excess is a non-negative quantity, as every exploration tour for G has length at least n. 2.4 Approximation and Competitive Ratio Sometimes, when we are unable to solve a given problem exactly, it is useful to consider algorithms which produce almost-optimal solutions. In order to reason about such algorithms, we provide precise definitions of the terms approximation and competitive ratio of algorithms. 9

17 2. Preliminaries Definition 2.22 (approximation) An algorithm A for grid exploration is called an α-approximation, α 1, for a set S of grids if for every grid G in S, the solution produced by A on G has length at most α τ opt (G). In the following definition, the asymptotics are with respect to the number n of vertices of the grid. Definition 2.23 (competitive ratio) An algorithm is said to have competitive ratio r 1 if it is an (r + o(1))-approximation for grids of size n. An algorithm with competitive ratio r is called r-competitive. Proposition 2.24 If an algorithm is an α-approximation, it is also α-competitive. In later chapters we will also speak of the competitive ratio of a problem, which is defined as follows. Definition 2.25 (competitive ratio of a problem) The competitive ratio of an exploration problem P is the smallest r such that there exists an algorithm which solves P with competitive ratio r. 2.5 Distance Graphs In order to compare the graph exploration problem to other well-known problems, we are interested in the distance between all pairs of vertices in a graph. The distance function defined below captures this notion. Definition 2.26 (distance function) Let G be a graph. The distance function d G (u, v) for two vertices u and v in G is defined to be the length of a shortest u-v-path in G. Note that the distance function is symmetric if the graph G is undirected, i.e., d G (u, v) = d G (v, u) for any two vertices u and v in G. Definition 2.27 (distance graph) The distance graph K G of G = (V, E) is the complete graph on V where for all u, v V the edge {u, v} in K G is assigned weight d G (u, v) Grid Exploration and TSP The traveling salesman problem is the problem of finding a minimum-weight Hamiltonian cycle and is a well-known NP-hard problem [18]. In this section, we show that grid exploration can be reduced to the traveling salesman problem in polynomial time. Even though the traveling salesman problem is NP-hard, this fact is not trivial because it is not obvious that the exploration problem itself is in NP.

18 2.6. Grid Exploration and TSP In later chapters, we apply algorithms for grid exploration to solve the traveling salesman problem and vice versa. In order to analyze those algorithms precisely, we need to know the exact runtime of the reduction. Therefore, we will be very explicit about the runtime of the reductions derived in this section. In order to reduce grid exploration to the traveling salesman problem, we first demonstrate how to compute the distance graph K G of a grid G in quadratic time. Then, we prove that solving the traveling salesman problem in K G solves grid exploration in G by transforming a shortest traveling salesman tour in K G into a shortest exploration tour for G Computing the Distance Graph To determine the distance graph K G for a grid G, we need to compute the distance function d G, i.e., determine the distance between all pairs of vertices. In the following we show how to do so efficiently. Lemma 2.28 Let G be a grid on n vertices and let u be a vertex in G. It is possible in time O(n) to compute the distances d G (u, v) for all vertices v in G. Proof. Since G is an unweighted graph, computing the length of a shortest path from u to every other vertex v is as simple as running a breadth-first search starting at u. By the definition of grids, G is connected and thus the search reaches every vertex v. Since grids are planar, the algorithm runs in time O(n) as claimed. Corollary 2.29 The distance graph K G of an unweighted graph G on n vertices can be computed in time O(n 2 ) Reducing Grid Exploration to TSP Now that we can compute K G from G, we reduce grid exploration to the traveling salesman problem by showing that we can translate a traveling salesman tour in K G into an exploration tour for G. We also show that, given a shortest exploration tour for G, we can compute a minimum-weight Hamiltonian cycle in K G. The results presented in this section are used throughout the thesis to analyze algorithms which solve the traveling salesman problem by first solving graph exploration or vice versa. In such a situation, it is necessary to know exactly how much effort is necessary to convert a solution for one problem into a solution for the other problem. Therefore, we analyze the runtime of the reduction precisely. 11

19 2. Preliminaries 12 Lemma 2.30 Let G be a grid on n vertices and let H be a Hamiltonian cycle of weight w in K G. There is a grid exploration tour T in G of length w. Moreover, T can be constructed from H in time O(n 2 ). Proof. For any u, v V(G) we define S G (u, v) to be any shortest path in G from u to v. Clearly, S G (u, v) has length d G (u, v) and can be computed in time O(n) using breadth-first search starting at u. We now describe how to construct T from H. We go through the cycle H = (h 1,..., h n ) in order and for each pair (h i, h i+1 ) in H, we append the entire path S G (h i, h i+1 ) to T. This way, for every edge with weight µ taken in H, we append a path of weight exactly µ to T. Thus, the resulting exploration tour T has length w. In order to construct T, we need to run n breadth-first searches to obtain the shortest paths we are interested in. Appending each of these n paths to T can be done in time O(n) and thus the construction of T takes time O(n 2 ) as claimed. Lemma 2.31 Let T be a grid exploration tour of length w of a grid G. There exists a Hamiltonian cycle H in K G of weight at most w. Moreover, H can be constructed from T in time linear in the number of vertices in T. Proof. Let T = (t 1,..., t k ) for some k n. We construct H = (h 1,..., h n ) as a subsequence of T by going through T in order and adding every vertex to H if we see it for the first time. Clearly, this procedure takes linear time. Since all vertices of G are visited by T, our new sequence H contains all vertices of K G. Moreover, by construction of H, every vertex occurs exactly once in H and since K G is a complete graph, H is a valid Hamilton cycle. It remains to show that H has weight at most w. To do so, consider any two successive vertices h i, h i+1 in H. The fact that they occur next to each other in H implies, by construction of H, that some path from h i to h i+1 was taken in T. By definition of d G, this path has length at least d G (h i, h i+1 ). In this manner, every edge of weight µ taken in H can be mapped injectively to a path of length at least µ in T. Hence, H has total weight at most w. Theorem 2.32 The grid exploration problem is reducible to the traveling salesman problem in polynomial time. Proof. We prove the claim by producing an instance of the traveling salesman problem for any given instance of grid exploration. Suppose we are given a grid G. By Corollary 2.29 we can compute K G in polynomial time. By Lemma 2.30 and Lemma 2.31, we know that the length of a shortest grid exploration tour for G is equal to the minimum weight of a Hamiltonian cycle in K G. We now solve the traveling salesman problem in K G and obtain a minimum-weight Hamiltonian cycle of weight w in K G. By Lemma 2.30,

20 2.7. Online Grid Exploration we can then compute a grid exploration tour of length w in polynomial time and we are done. In Section 3.1, we will see that in a grid G the length τ opt (G) of a shortest exploration tour T can be bounded by τ opt (G) 2n 2. Therefore, the runtime in Lemma 2.31 becomes linear in n and the total runtime of the reduction is dominated by the O(n 2 )-term from Lemma Online Grid Exploration In the online variant of the grid exploration problem, the robot must explore an initially unknown grid G from some starting cell. As in the offline version, the goal of the robot is to follow edges in the grid and produce a short exploration tour for G. In this setting however, the vision of the robot is limited to the neighbors of the vertex it is currently standing on. More precisely, if the robot is located at a vertex v of G and w H(v) is a neighbor of v in the infinite grid, the robot can determine whether or not w belongs to G. If w does belong to G, the robot may move to w, at which point it obtains information about the neighbors of w. We impose no memory restrictions on the robot. This means that it can store all the information it gathers in memory while exploring the grid. By the time the robot finishes a tour, it knows the entire topology of the grid. We call the exploration problem in the setting described above online grid exploration. In the following we provide precise definitions of different variants of the online grid exploration problem. Definition 2.33 (online solid grid exploration) The online solid grid exploration problem is the online grid exploration problem restricted to solid grids. Definition 2.34 (boundary online grid exploration) The boundary online grid exploration problem is the online grid exploration problem where the robot always starts its exploration tour at a border cell of the grid. We define the boundary online solid grid exploration problem analogously. 13

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22 Chapter 3 Bounds for Excess When dealing with the grid exploration problem, it is natural to ask how well the problem can be solved for a specific grid G. Concretely, we are interested in bounding the excess of the grid in question. To do so, we will start by proving some basic bounds which rely on simple observations about the structure of grids. Some of these simple bounds are implicit in previous work [17], but we will make them explicit in this chapter. We also develop more involved lower bounds for which we consider subsets S G of G in isolation and determine that exploring S will produce at least a certain amount of excess. Finally, we show how multiple lower bounds can be combined to obtain better lower bounds. 3.1 Upper Bound Given a grid G, we first prove an upper bound for the length τ opt (G) of a shortest exploration tour. Lemma 3.1 For every grid G on n vertices we have τ opt (G) 2n 2. Proof. We choose an arbitrary vertex u in G and run a depth-first search starting at u. Let S be the corresponding spanning tree of G with S = n 1. We can traverse G by taking each edge of S exactly twice. This gives us an exploration tour T for G with w(t) = 2n 2 and the claim follows because a shortest tour has length at most w(t). Applying the definition of excess yields the following bound. Corollary 3.2 For every grid G on n vertices we have e(g) n 2. 15

23 3. Bounds for Excess 3.2 Basic Lower Bounds In this section, we present simple lower bounds for the length τ opt (G) of a shortest grid exploration tour for a graph G. These bounds essentially follow from basic properties of grids. Definition 3.3 (imbalance) Let G be a grid with partite sets B and W. The imbalance (G) of G is defined as the difference in size between B and W, i.e., (G) := B W. The grid G is said to be balanced if (G) = 0. If the grid G is clear from context, we write instead of (G) for convenience. For the following lower bound we use the observation that imbalance causes excess because a tour in a bipartite graph must visit the vertices from the two partite sets in an alternating manner. Lemma 3.4 For every grid G we have e(g) (G). Proof. Because G is bipartite, every walk in G has to alternate between vertices from B and vertices from W. Since every exploration tour T has to visit every vertex in the larger partite set at least once, the tour T will have to produce B W = units of excess by revisiting vertices from the smaller partite set. Lemma 3.4 demonstrates that if the sizes of the partite sets of a grid differ greatly, then exploring the grid is more expensive. In the next sections we will return to this notion and derive more powerful bounds using imbalance. Another important observation is that if a vertex in the input grid is a cut vertex, any exploration tour will have to visit it at least twice because the tour has to return to its starting point eventually. The following result is a generalization of this fact. A similar result can be found implicitly as part of a proof in previous work by Kolenderska et al. [17]. Definition 3.5 (split number) Let G be a grid and let v G be a vertex of the grid. We define the split number s(v) as the number of connected components in G v. We generalize the split number to the entire grid by setting s(g) := s(v). v G 16

24 3.3. Area Bounds Figure 3.1: A grid with a subset S shaded in blue. The imbalance (S) is 1 and the ports of S are marked by blue dots. Lemma 3.6 Every exploration tour for a grid G has length at least s(g), i.e., τ opt (G) s(g). Proof. Let T be an exploration tour for G. It suffices to show that, for every vertex v in G, the tour T visits v at least s(v) times. If s(v) = 1, this is obvious because, by definition, T must visit each vertex of G. Suppose s(v) 2 for some vertex v and let s := s(v) for convenience. Let c 1,..., c s be the connected components of G v and we arbitrarily fix some vertex in c 1 as the starting vertex of T. Because T is an exploration tour, the components c 2,..., c s must all be visited by T. The only way to reach these components goes through v, so T has to visit v at least s 1 times. Finally, the tour has to be closed and thus T will have to go through v at least once more to reach c 1. Overall, T visits v at least s times. 3.3 Area Bounds For a special type of grid, Afrati develops a linear time algorithm which finds Hamiltonian cycles [1]. A key building block in the development of the algorithm is the notion of imbalance combined with horizontal and vertical edge cuts. This combination has already been used in the past to show lower bounds for the excess of grids [9]. In this section, we develop new methods to derive lower bounds for the excess of grids which are not only applicable in all grids, but they also generalize to cuts which are neither horizontal nor vertical. The bounds rely on the observation that local imbalance causes excess. If we are able to isolate a region of a grid with high imbalance, any tour will have to produce excess in that region in order to visit all vertices. Consider Figure 3.1 together with the following definition. 17

25 3. Bounds for Excess Definition 3.7 (port) Let G and S G be grids and let B be the larger of the two partite sets of S. A vertex p B is called a port of S in G if p is adjacent to a vertex in G S. We define P G (S) to be the set of such ports. Definition 3.8 Let T be an exploration tour for some graph G. We say that T produces k excess in some subset S G if the number of times T visits a vertex in S is exactly k + S. Definition 3.9 (area bound) Let G be a grid. A tuple (b, S) with b R, S G, is called an area bound for G if every grid exploration tour for G produces at least b units of excess in S G. We call b the strength and S the influence region of the bound. The following proposition directly follows from the definition of area bounds. Proposition 3.10 If (b, S) is an area bound for some grid G, then e(g) b. Proposition 3.10 gives us a tool for obtaining lower bounds for the excess of grids by proving the existence of area bounds. In the following, we derive two types of area bounds Area Bounds using the Number of Ports As we have seen in Lemma 3.4, imbalance causes excess in a grid. In this section, we show that ports play a central role in compensating for imbalance. More specifically, an exploration tour can only compensate for imbalance in a region of the grid by entering and leaving the region through ports. Once all ports are used, compensating for the remaining imbalance can only be done by producing excess. Theorem 3.11 Let G and S G be grids and let Then (b, S) is an area bound in G. b := (S) P G (S). Proof. Let T be a grid exploration tour for G. As illustrated in Figure 3.2a, the restriction T := T S of T to vertices of S consists of a series of walks in S starting and ending at the boundary of S. Note that since T visits every vertex in G, the sequences T visit all vertices in S. Let B and W be the partite sets of S such that B W. Since every walk has to visit vertices from B and W in an alternating manner, the imbalance (S) can only be compensated by either producing excess within S or by walks in T starting and ending at ports of S. We call these walks port walks, an example is shown in Figure 3.2b. Each port walk which does not produce excess allows to compensate for one unit of imbalance.

26 3.3. Area Bounds (a) A tour T and its restriction T S to the blue subset S. Note that the imbalance (S) of S is 1. (b) Since the red port walk starts and ends at a port, it allows T to compensate for one unit of imbalance without producing excess. Figure 3.2: Illustration of how port walks can compensate for imbalance. Figure 3.3: A grid with subset S shaded in blue and the ports marked by blue dots. Restricting the black tour to S causes the port in the middle of the grid to be a degenerate port walk. Note that, as illustrated in Figure 3.3, a degenerate port walk could consist of a single port. Therefore, in the worst case, each port walk uses only one port and thus P G (S) units of imbalance can be compensated without producing excess. Each remaining unit of imbalance requires T to produce one additional excess, and thus the excess produced in S is at least b Area Bounds using Port Multiplicities The following area bound is similar but slightly stronger than the one established in Theorem The additional strength of the bound comes at the price of accepting that the excess could also be produced in the halo of the region under consideration, effectively increasing the influence region of the 19

27 3. Bounds for Excess area bound. As we will see later, this can be a disadvantage when applying the bound to multiple regions of a grid in order to obtain a better bound. The following definition introduces a way to count how many times a region can be entered or left through a port without producing excess. Note that by definition of ports, a port p of some subset S in a grid G has at least one edge to G S. Definition 3.12 (port multiplicity) Let G and S G be grids. The multiplicity µ(p) of a port p P G (S) is defined as µ(p) := { 1 if p has exactly one edge to G S. 2 otherwise We define the port multiplicity of the subgraph S as µ G (S) := µ(p). p P G (S) Theorem 3.13 Let G and S G be grids and let b := (S) µg (S). 2 Then ( b, S H G (S) ) is an area bound in G. Proof. Let T be a grid exploration tour for G and let µ := µ G (S) for convenience. In order to prove the claim, we argue that the excess e produced by T in S H G (S) is at least e b = (S) µ 2. (3.14) 20 We consider the restriction T := T S of T to vertices of S which consists of a series of walks in S starting and ending at the boundary of S. Let B and W be the partite sets of S such that B W. Let p B be the number of port walks in T, i.e., the number of walks in T which start and end at a port. Note that a port walk visits a vertex from B one more time than it visits a vertex from W. All other walks in T visit a vertex from W at least as often as they visit a vertex from B. Therefore, each of the p B port walks allows to compensate for one unit of the imbalance (S) without producing excess. The rest of the imbalance has to be compensated by producing excess. We express the excess e produced by T in S H G (S) in terms of the imbalance (S), the number p B of port walks and an additional variable c which

28 3.3. Area Bounds we use to bound the excess produced by T to compensate for imbalance. This gives us e (S) p B + c. Therefore, we can obtain Equation 3.14 (and thus the claim of the theorem) by showing that which is equivalent to p B + c µ 2, p B µ 2 + c. (3.15) In the following we derive Equation 3.15 by analyzing the contribution of each walk in T to the inequality. First, note that all port walks in T must enter and leave S using an edge which connects a port to G S. 1. We first consider a maximal set P of port walks which produces no excess, i.e., in which no vertex occurs more than once. Let p B,1 := P be the number port walks in P. Each of these walks must enter and leave S by taking edges between ports and G S. Note that each such edge can be used at most once, otherwise excess is produced. For the same reason, at most two 1 edges can be used this way at each port in total. In other words, at most µ(p) edges can be used at each port p and since each port walk uses two edges, P has size at most µ/2 and thus p B,1 µ/2. 2. We consider all other port walks. Let p B,2 be the number of these remaining port walks. Each one of these port walks produces at least one excess, otherwise P is not maximal. Note that this excess is produced in S H G (S) which gives us p B,2 c. Using the two variables p B,1 and p B,2 introduced above, we can bound the number p B of port walks and establish Equation 3.15 via p B = p B,1 + p B,2 µ 2 + c. Corollary 3.16 Let G and S G be grids. Then e(g) (S) µg (S). 2 Note that Lemma 3.4 provides a lower bound for excess based on the imbalance of the entire grid. That statement can be viewed as a special case of Corollary 3.16 in which the subset S is chosen to be the entire grid. 1 In the degenerate case where a port walk consists of only one port (see Figure 3.3), two edges can be taken without excess. 21

29 3. Bounds for Excess Figure 3.4: A grid with a blue split and a purple split, each proving 1 excess by Theorem A dot in a cell indicates that it is part of the influence region of the area bound Combining Area Bounds The definition of area bounds (Definition 3.9) suggests a natural and intuitive way of combining multiple area bounds if their influence regions do not overlap. The following lemma expresses this notion more formally. Lemma 3.17 Let (b 1, r 1 ) and (b 2, r 2 ) be area bounds in a grid G such that r 1 r 2 =. Then (b 1 + b 2, r 1 r 2 ) is an area bound in G. Proof. Let T T opt (G) be a shortest grid exploration tour for G. Since T is a grid exploration tour, T must visit both regions r 1 and r 2. By definition of area bounds, the tour T produces at least b 1 units of excess in r 1 and at least b 2 units of excess in r 2. Thus, we can conclude that T produces at least b 1 + b 2 excess in total in the region r 1 r 2. An example application of multiple area bounds to the same grid is shown in Figure 3.4. Applying Theorem 3.11 to the two regions yields a lower bound of 0. However, if we apply Theorem 3.13, we obtain a lower bound of 1 for each of the two regions. By Lemma 3.17 we can combine the bounds and conclude that any tour produces an excess of at least Explaining Excess The results above show that local patches with imbalance in grids cause excess. It remains an interesting question whether or not there always exists a set of area bounds obtainable from Theorem 3.11 or Theorem 3.13 which explain the entire excess of a grid. 22

30 Chapter 4 Offline Grid Exploration In the offline variant of the grid exploration problem, the entire grid is provided as input and the goal is to determine a shortest exploration tour. Even though we will see that it is NP-hard to solve grid exploration in general grids, the difficulty of the problem seems to vary greatly depending on whether or not the grids are allowed to have holes. In this chapter we describe the best currently known results for the different variants of the problem. These range from NP-hardness results and approximation schemes for general grids to polynomial time algorithms for solid grids. We first present a proof by Itai et al. that the Hamiltonian cycle problem is NP-complete in general grids, i.e., in grids with holes. We then describe an algorithm due to Umans and Lenhart which finds Hamiltonian cycles in polynomial time, provided the input grid is solid. We then move on to approximation algorithms, where we summarize a (1 + ε)-approximation algorithm by Klein for the traveling salesman problem in planar graphs. Finally, we show how the algorithm can be used to approximate grid exploration and how the runtime can be improved in the special case of grids. 4.1 Hamiltonian Cycles in General Grids In the Hamiltonian cycle problem, the goal is to decide whether or not a given graph admits a Hamiltonian cycle. Clearly, solving the graph exploration problem also solves the Hamiltonian cycle problem. The latter is wellknown to be NP-complete in general graphs and Itai et al. prove that, even in the restricted setting of grids, the Hamiltonian cycle problem remains NP-complete [15]. This section intends to provide insight on the proof. 23

31 4. Offline Grid Exploration Figure 4.1: A strip with corners a, b, c, d on the left and a tentacle consisting of 4 strips on the right. the tentacle has corners a 1, b 1, c 4 and d Related Hamiltonian Cycle Problems The Hamiltonian cycle problem is NP-hard in general graphs, even with considerable restrictions on the graph structure. In particular, this remains true if the input graph is a directed, planar graph in which the sum of indegree and out-degree is exactly 3 for each vertex [11]. Using this fact, Itai et al. start by deriving the following lemma for another specific Hamiltonian cycle problem [15]. We will later use this problem to show that grid exploration is NP-hard. Lemma 4.1 [15] The Hamiltonian cycle problem for undirected, planar, bipartite graphs with maximum degree 3 is NP-complete. For a proof of Lemma 4.1 see Appendix A Embedding Graphs Definition 4.2 (internally vertex disjoint paths) A set P of paths in a graph is called internally vertex disjoint if, for every path p = (p 1, p 2,..., p k 1, p k ) P, the vertices {p 2,..., p k 1 } are not contained in any other path of P. Definition 4.3 (parity-preserving embedding) Let H be a bipartite graph with partite sets C and D and let G be a grid with partite sets B and W. Let f be a function from (C D) to vertices of G and from edges of H to paths in G. Then f is called a parity-preserving embedding if the following conditions hold: 1. If c C, then f (c) B and if d D, then f (d) W. 2. Edges of H are mapped to internally vertex-disjoint paths between the corresponding endpoints in G. Parity-preserving embeddings are a useful tool for reducing problems in general graphs to grids. The following lemma makes this explicit.

32 4.1. Hamiltonian Cycles in General Grids Figure 4.2: The grid gadget G 3,3. Lemma 4.4 [15] Let H be a bipartite graph on n vertices with maximum degree 3. Then it is possible in polynomial time to find a parity-preserving embedding of H in a 3n 3n rectangular grid. Proof sketch. As shown in Shiloach [20], any planar graph on n vertices in which all vertices have degree 3 can be embedded into the n n square grid G. However, if this is applied directly to the bipartite graph H, we are not guaranteed to get a parity-preserving embedding. This problem can be remedied by increasing the dimensions of G to 3n 3n, effectively replacing every vertex v by a 3 3 grid. Then, it becomes possible to select a vertex from the right partite set to represent v, thus making the embedding paritypreserving. Another component of the hardness proof is the notion of tentacles [15], which are concatenations of rectangular strips as illustrated in Figure 4.1a and in Figure 4.1b. For formal definitions see Appendix A. The following lemma is stated in Itai et al. and can be easily proven by induction over the number of strips [15], but the proof is more cumbersome than insightful. Therefore, we omit it here. Lemma 4.5 [15] Let T be a tentacle and let s and t be corners of T. Then there exists a Hamiltonian s-t-path in T iff s and t belong to different partite sets. The last ingredient necessary for the proof is the 3 3 rectangular grid gadget G 3,3 shown in Figure 4.2. We need the property that a Hamiltonian path can easily enter and leave such a gadget. More formally, we require the following statement for the gadget G 3,3 shown in Figure 4.2. Lemma 4.6 For all i, j there exists a Hamiltonian p i -p j -path in G 3,3 which uses all the edges e 1, e 2, e 3, e 4. Proof. By inspection. 25

33 4. Offline Grid Exploration Proof of Hardness In this section, we sketch how to put the pieces together and obtain a hardness proof for the Hamiltonian cycle problem in general grids. The proof idea consists in reducing the problem of finding Hamiltonian cycles in undirected, planar, bipartite graphs with maximum degree 3 to the problem of finding Hamiltonian cycles in grids. This is sufficient because, by Lemma 4.1, the former is NP-hard. Theorem 4.7 [15] The Hamilton cycle problem in general grids is NP-hard. Proof sketch. Suppose we are given an undirected, planar, bipartite graph H on n vertices with maximum degree 3. We now construct in polynomial time a grid G such that H has a Hamiltonian cycle if and only if G has one. Together with Lemma 4.1, this implies the claim. The construction of G is done as follows. First, we apply Lemma 4.4 and obtain a parity-preserving embedding of H into the 3n 3n grid G. From the definition of embeddings, we know that each pair of adjacent vertices in H is connected by a path in G. The next step is to replace these paths with tentacles. In order to do so, the dimensions of G are increased to 9n 9n, allowing us to replace each vertex in G by the gadget shown in Figure 4.2. We call this new grid G. The designated edges e 1, e 2, e 3, e 4 highlighted in Figure 4.2 are the locations where the tentacles are attached. The precise construction is described in [15]. Using the special Hamiltonian path properties of tentacles from Lemma 4.5 and the fact from Lemma 4.6 that every vertex gadget can only be crossed once with a Hamiltonian path, Itai et al. then show that H is Hamiltonian if and only if G is Hamiltonian. Finally, the proof is concluded by observing that G can be constructed from H in polynomial time. Corollary 4.8 The grid exploration problem is NP-hard Hamiltonian Cycles in Solid Grids As we have seen in Theorem 4.7, the Hamiltonian cycle problem is NP-hard in general grids. However, it turns out that if the input grids are known to be solid, the problem becomes solvable in polynomial time. In this chapter we present an algorithm due to Umans and Lenhart which runs in time O(n 4 ) and finds Hamiltonian cycles in solid grids [21]. Note that even though a hardness result for the Hamiltonian cycle problem directly implies the hardness of grid exploration in the same class of grids, a feasibility result for Hamiltonian cycles leaves the grid exploration problem open. It is currently not known whether or not the solid grid exploration problem is NP-hard.

34 4.2. Hamiltonian Cycles in Solid Grids The Hamiltonian cycle algorithm starts by partitioning the grid into a set of disjoint cycles which cover all vertices. It then applies a series of modifications to these cycles in order to obtain a single cycle which visits all vertices, i.e., a Hamiltonian cycle. One of the main results used in the development of the algorithm states that such sequences of operations exist and can be found efficiently Two-Factors Definition 4.9 Let G = (V, E) be a graph and let F E. We define G F := (V, F). Definition 4.10 (k-factor) Let G = (V, E) be a graph. We call F E a k-factor in G if every vertex v V has degree exactly k in G F. The starting point of the algorithm is a 2-factor for the input solid grid G. Intuitively, a 2-factor represents a decomposition of G into a set of cycles which, together, visit each vertex exactly once. It follows that every Hamiltonian cycle is a 2-factor, so if there is no 2-factor in G, then we can safely conclude that G is not Hamiltonian. Finding a 2-factor can be easily formulated as a linear programming problem. However, since most polynomial time algorithms for linear programming tend to take a long time in practice, Umans and Lenhart present a different approach based on perfect matchings [21]. This method builds a new graph H from G by adding a constant size gadget for every vertex in G. Whether or not two vertices are adjacent in G then determines how the corresponding gadgets are connected in H. This new graph H is bipartite and has the property that every perfect matching in H corresponds to a 2-factor in G and vice versa. Therefore, to find a 2-factor in G, it suffices to solve the maximum bipartite matching problem in H. This in turn is a well-studied problem which can be solved in time O (n E ) using an algorithm by Edmonds [18] Alternating Sequences Once a 2-factor T has been identified in G, finding a Hamiltonian cycle reduces to transforming T into another 2-factor with only one connected component. In this section we describe a basic operation on a 2-factor which yields a new 2-factor. The key result by Umans and Lenhart states that if a grid G is Hamiltonian, then any 2-factor can be transformed into a Hamiltonian cycle by a series of these operations [21]. In order to describe the operation, we introduce alternating cycles. 27