Competitive Search in a Network
CCompetitive Search in a Network
Spyros Angelopoulos ∗ and Thomas Lidbetter CNRS and Sorbonne Université, Laboratoire d’Informatique de Paris 6, 4 Place Jussieu, Paris,France 75252. [email protected] Department of Management Science and Information Systems, Rutgers Business School,Newark, NJ 07102, USA. [email protected]
August 7, 2019
Abstract
We study the classic problem in which a
Searcher must locate a hidden point, also called the
Hider in a network, starting from a root point. The network may be either bounded or unbounded,thus generalizing well-known settings such as linear and star search. We distinguish between pathwisesearch, in which the Searcher follows a continuous unit-speed path until the Hider is reached, andexpanding search, in which, at any point in time, the Searcher may restart from any previously reachedpoint. The former has been the usual paradigm for studying search games, whereas the latter is amore recent paradigm that can model real-life settings such as hunting for a fugitive, demining a field,or search-and-rescue operations. We seek both deterministic and randomized search strategies thatminimize the competitive ratio , namely the worst-case ratio of the Hider’s discovery time, divided bythe shortest path to it from the root. Concerning expanding search, we show that a simple searchstrategy that applies a “waterfilling” principle has optimal deterministic competitive ratio; in contrast,we show that the optimal randomized competitive ratio is attained by fairly complex strategies even in avery simple network of three arcs. Motivated by this observation, we present and analyze an expandingsearch strategy that is a 5/4 approximation of the randomized competitive ratio. Our approach is alsoapplicable to pathwise search, for which we give a strategy that is a 5 approximation of the randomizedcompetitive ratio, and which improves upon strategies derived from previous work.
Keywords:
Game Theory; Search games; Competitive analysis; Networks.
We consider the classic setting in which a mobile
Searcher must locate a stationary hidden object, calledthe
Hider , in a network Q with given arc lengths. This general problem goes back to early work in (Isaacs,1965) and (Gal, 1979), who introduced it in the context of the standard, pathwise search; namely, in thisusual setting, the Searcher moves at unit speed starting from a given point O of the network that we ∗ Corresponding author. a r X i v : . [ m a t h . O C ] A ug all the root , and the search time is defined as the first time at which the Searcher reaches the Hider.A different approach was recently introduced in (Alpern & Lidbetter, 2013), and allows the Searcherto move at infinite speed within any region of the network that it has already visited; see Section 2.1for a formal definition. This paradigm captures several situations in which the cost of re-exploration isnegligible, compared to the cost of first-time exploration, and thus can model settings such as mining forcoal, hunting a fugitive, or searching for a missing person.The above works take the approach of seeking mixed, i.e., randomized search strategies, with theobjective of minimizing the expected search time, in the worst case; that is, the maximum expected searchtime over all hiding points in the network. This is accomplished by studying a zero-sum game with payoffthe search time, between a minimizing Searcher and a maximizing Hider. In this paper, instead, we studya normalized variant of the search time, in which the search time for reaching a point p in Q is dividedby the shortest path from O to p in Q ; we call this the normalized search time of p . The objective thusbecomes to find strategies that minimize the worst-case (normalized) search time, by considering all pointsin the network Q .This normalized formulation was first applied in search games over unbounded domains, such as the linear search (Beck & Newman, 1970) and star search (Gal, 1972) problems. Normalization is essential inunbounded domains, since otherwise the Hider can induce unbounded search times, by hiding arbitrarilyfar from O . Further motivation behind the study of normalized objectives is provided by competitiveanalysis of online algorithms in which the algorithm operates in a status of total uncertainty about theinput, and the normalized objective describes how much close the algorithm’s output is, in comparisonto an ideal solution with complete information on the input. For this reason, (Jaillet & Stafford, 1993)refer to searching under the competitive ratio as online searching . Competitive analysis has been appliedeven in search games over a bounded domain, as in (Koutsoupias, Papadimitriou, & Yannakakis, 1996;Fleischer, Kamphans, Klein, Langetepe, & Trippen, 2008; Angelopoulos, Dürr, & Lidbetter, 2019). Wewill refer to the competitive ratio of a strategy as the worst-case normalized search time among all points ofthe network . Lastly, we define the competitive ratio of a network Q (with a given root O ) as the minimumcompetitive ratio of any search strategy for Q . We will further distinguish between the deterministic andthe randomized competitive ratios, depending on whether we consider deterministic or randomized searchstrategies, respectively. In this work we study the competitive ratio of general networks, both in the expanding and the pathwisesearch paradigms, which are defined precisely in Section 2. For expanding search, we first show in Section 3that the deterministic competitive ratio is achieved by a simple strategy. This strategy can be visualized asthe frontier that is obtained by “flooding” the network starting at O , assuming that the arcs represent pipesof corresponding lengths. We then move to randomized strategies for expanding search in Section 4. Here, In (Koutsoupias et al., 1996; Angelopoulos et al., 2019) the term search ratio is used in order to refer to the competitiveratio. In this work we choose the latter, since it is more prevalent, and since it has been adopted both by the OperationsResearch and the Computer Science communities; see e.g., the discussion in (Alpern & Gal, 2003) and (Jaillet & Stafford,1993).
2e show that, unlike the deterministic case, optimal search strategies have a complex statement even ona very simple network that consists of three arcs. Motivated by this observation, we give approximationsto the value of the game. First, we show that the randomized competitive ratio of a network is within afactor of 2 of its deterministic competitive ratio, and this bound is tight. More importantly, we give a classof randomized strategies that approximate the randomized competitive ratio of a network within a factorof / . This class of strategies is based on iterative applications of Randomized Depth-First-Search, inrandomly chosen and increasingly large subsets of the network. This strategy is inspired by a randomizedstrategy used for tree graphs in the discrete setting, namely when the Hider can only hide over vertices of agiven, finite tree (Angelopoulos et al., 2019). We emphasize that, unlike (Angelopoulos et al., 2019), in thiswork, the search domain may be substantially more complex than a tree, and it may also be unbounded.Moreover, we give further approximations of the value of the game by relating the payoff of the searchstrategy to the function f Q , which informally gives the measure of the set of points within a certaingiven radius from the root. As a corollary, we show that if the function f Q is concave, the randomizedcompetitive ratio is identical to the deterministic one. This finding may have practical implications in thecontext of searching in a big city, since the road network is naturally much more dense in its center thanit its outskirts, and one expects this density to decrease the further we move from the city center.Our approach in studying expanding search, and more specifically, our lower bounds on the randomizedcompetitive ratio, have implications for pathwise search as well. More precisely, in Section 5 we give arandomized pathwise search strategy, inspired by the one for expanding search, which is a 5-approximationof the randomized competitive ratio. This is an improvement over the √ ≈ . -approximationthat can be derived from techniques in (Koutsoupias et al., 1996).In Section 6 we discuss some technicalities relating to the implementation of our search strategies, andin Section 7 we conclude with directions for future work.To illustrate the significance of the results and the approaches, consider the star-search problem inwhich the search domain consists of m infinite, concurrent rays (Figure 1a). Star search has a long historyof research, and several of variants of this problem have been studied under the competitive ratio (seeChapters 7 and 9 in (Alpern & Gal, 2003)). It is known that the deterministic competitive ratio is equalto mm − ) mm − (Gal, 1972). In contrast, the randomized competitive ratio is not known (in (Kao, Ma,Sipser, & Yin, 1998) optimality is shown under the fairly restrictive assumption of periodic strategies). Thestrategy we obtain in this work has randomized competitive ratio which is at most a factor of 5 from theoptimal one. Furthermore, the result applies to much more complicated unbounded domains, for instancesuch as the one depicted in Figure 1b, under the mild (and necessary) assumption that for any r > , thenumber of points at distance r from the root of the network is bounded. Expanding search on a network was introduced in (Alpern & Lidbetter, 2013), with the focus on theBayesian problem of minimizing the expected search time against a known Hider distribution. In a followuppaper (Alpern & Lidbetter, 2019), the same authors studied expanding search on general networks and gavetwo strategy classes that have expected search times that are within a factor close to 1 of the value of thegame. Both these works apply to the unnormalized search time. For normalized objectives (Angelopoulos3 (a)
A star domain in which m = 4 . O (b) An example of a search domain studied in this work.
Figure 1:
An illustration of different search domains. et al., 2019) recently studied expanding search in a fixed (finite) graph in which the Hider can only hideon vertices. In terms of finding a strategy of optimal deterministic competitive ratio (Angelopoulos et al.,2019) showed that the problem is NP-hard, and gave a approximation. Concerning the randomizedcompetitive ratio, the same work presented a strategy that is a / -approximation in the special case oftree graphs.The competitive ratio of pathwise search was first studied by Beck and Newman in the context of thelinear search problem (Beck & Newman, 1970) and later by Gal (Gal, 1972, 1974) for star search. For fixedgraphs, assuming that the Hider can only hide on vertices, it is NP-hard to approximate the deterministiccompetitive ratio (Koutsoupias et al., 1996). The same paper also gave constant-factor approximations forboth the deterministic and the randomized competitive ratio, assuming the graph is undirected. Extensionsto edge-weighted graphs were studied in (Ausiello, Leonardi, & Marchetti-Spaccamela, 2000), which alsoshowed connections between graph searching and classic optimization problems such as the TravelingSalesman problem and the Minimum Latency problem. The setting in which the search graph is notknown to the Searcher, but is rather revealed as the search progresses was studied in (Fleischer et al.,2008).The exact and approximate competitive ratio of pathwise search has been studied in many set-tings, mostly assuming a star-like search domain. Examples include multi-Searcher strategies (López-Ortiz & Schuierer, 2004; Angelopoulos, , Arsénio, Dürr, & López-Ortiz, 2016), searching with turncost (Demaine, Fekete, & Gal, 2006; Angelopoulos, Arsénio, & Dürr, 2017), searching with probabilis-tic information (Jaillet & Stafford, 1993), searching with upper/lower bounds on the distance of the Hiderfrom the root (Hipke, Icking, Klein, & Langetepe, 1999; López-Ortiz & Schuierer, 2001; Bose, Carufel,& Durocher, 2015), and searching for multiple hiders (Angelopoulos, López-Ortiz, & Panagiotou, 2014;McGregor, Onak, & Panigrahy, 2009; Kirkpatrick, 2009). All these works assume that the search domainis either the unbounded line or the unbounded star.4 Preliminaries
We consider a search domain that is represented by a connected network Q which consists of vertices andarcs, and which has a certain vertex O designated as its root. Moreover, Q is endowed with Lebesguemeasure corresponding to length. The measure of a subset A of Q is denoted by λ ( A ) , and in the casethat Q has finite measure, we will denote by µ = λ ( Q ) the total measure of Q . This defines a metric on Q , where d ( x, y ) is the length of the shortest path from x to y . We write d ( x ) for the distance d ( O, x ) from O to x . We denote by deg Q ( v ) the degree of v in Q , namely the number of arcs incident to v .We do not limit ourselves to bounded networks, but make the standing assumption that the network Q satisfies the condition that there exists some integer M such that for any r > , |{ x ∈ Q : d ( x ) = r }| ≤ M. (1)That is, there are at most M points at distance r from O . Any network with a finite number of arcsautomatically satisfies this condition. We will see that this condition ensures that the competitive ratioexists. As an example of an unbounded network, if Q is an m -ray star, we have |{ x ∈ Q : d ( x ) = r }| = m ,for all r , hence this quantity is bounded. In contrast, if Q is an unbounded full binary tree in which there are i vertices at distance i from the root, for all i ∈ N + , then this quantity is unbounded, and this implies thecompetitive ratio is also unbounded. Indeed, any deterministic search strategy (pathwise or expanding) onthis network must take at least time i to reach all points at distance i from the root, so the deterministiccompetitive ratio would be at least i /i , which is unbounded. We will see later (Proposition 7) that thisimplies that the randomized competitive ratio of this tree network is also unbounded.Given a network Q , and any r ≥ , we denote the closed disc of radius r around O by Q [ r ] = { x ∈ Q : d ( x ) ≤ r } . Let r max = max x ∈ Q d ( x ) be the distance of the furthest point in Q from O , where r max = ∞ if Q is unbounded. We define the real function f Q : [0 , r max ] → R given by f Q ( r ) = λ ( Q [ r ]) , so f Q ( r ) is themeasure of the set of points at distance no more than r from the root.We begin with preliminary definitions and results concerning expanding search, since it is a more recentparadigm, and somewhat more subtle to define. We then explain how these definitions change in whatconcerns pathwise search. In expanding search, we allow the search to move at no cost over any part of the network that it haspreviously explored. This is formalized in the following definition.
Definition 1 ((Alpern & Lidbetter, 2013)) . An expanding search on a network Q with root O is a familyof connected subsets S ( t ) ⊂ Q (for ≤ t ≤ µ ) satisfying: (i) S (0) = O ; (ii) S ( t ) ⊂ S ( t (cid:48) ) for all t ≤ t (cid:48) ; and(iii) λ ( S ( t )) = t for all t . If the context is clear, we will refer to an expanding search as a search search strategy. For a givenexpanding search S of Q and a point H ∈ Q , let T ( S, H ) = min { t : H ∈ S ( t ) } be the (expanding) searchtime of H under S . This was shown to be well defined in (Alpern & Lidbetter, 2013). For H (cid:54) = O , let ˆ T ( S, H ) be the ratio T ( S, H ) /d ( H ) of the search time of H to the distance of H from the root. We referto ˆ T ( S, H ) as the normalized search time . It is convenient to define ˆ T ( S, O ) to be equal to .5 efinition 2. The deterministic competitive ratio σ S = σ S ( Q ) of a deterministic expanding search S ofa network Q is given by σ S ( Q ) = sup H ∈ Q ˆ T ( S, H ) . The (deterministic, expanding) competitive ratio, σ = σ ( Q ) of Q is given by σ ( Q ) = inf S σ S ( Q ) , where the infinum is taken over all search strategies S . If σ S = σ we say that S is optimal. Note that the competitive ratio of a strategy S may be infinite. For example, suppose that Q consistsof two unit-length arcs a and b meeting at the root and suppose S searches a first and then b . If H lieson the arc b at distance x from the root then ˆ T ( S, H ) = (1 + x ) /x = 1 + 1 /x → ∞ as x → . It is notimmediately obvious whether or not the competitive ratio of a network is finite in general, but we willshow in Section 3 that this is indeed the case, by explicitly giving the optimal search strategy for anynetwork.In addition, we consider randomized search strategies: that is, search strategies that are chosen ac-cording to some probability distribution. We denote randomized strategies by lower case letters, and forrandomized strategies s and h for the Searcher and the Hider, respectively, we denote the expected searchtime by T ( s, h ) and the expected normalized search time by ˆ T ( s, h ) . Definition 3.
The randomized competitive ratio ρ s = ρ s ( Q ) of a randomized expanding search s of anetwork Q is given by ρ s ( Q ) = sup H ∈ Q ˆ T ( s, H ) . The randomized competitive ratio, ρ = ρ ( Q ) of Q is given by ρ ( Q ) = inf s ρ s ( Q ) , where the infimum is taken over all possible randomized search strategies s . If ρ s = ρ we say that s isoptimal. When clear from context, we omit Q for simplicity, e.g., we will use σ S instead of σ S ( Q ) .We can view the randomized competitive ratio of a network as the value of the following zero-sumgame Γ( Q, O ) . A strategy S for the Searcher is a search strategy as described above and a strategy H forthe Hider is a point on Q . The payoff of the game is the normalized search time ˆ T ( S, H ) . For randomizedstrategies s and h of the Searcher and Hider, respectively, the expected payoff is denoted by ˆ T ( s, h ) .In (Alpern & Lidbetter, 2013) the authors considered a similar zero-sum game on finite networks inwhich the players’ strategy sets are the same but the payoff is the unnormalized search time T ( S, H ) . Theyshowed that the strategy sets are compact with respect to the uniform Hausdorff metric and that T ( S, H ) is lower semicontinuous in S for fixed H . Since d ( H ) is a constant for fixed H , it follows that ˆ T ( S, H ) = T ( S, H ) /d ( H ) is also lower semicontinuous in S for fixed H , and by the Minimax Theorem (Alpern &Gal, 1988), we have the following theorem. 6 heorem 4. Let Q be a finite network with root O . The game Γ( Q, O ) has a value, which is equal to therandomized competitive ratio ρ ( Q ) . The Searcher has an optimal mixed strategy (with competitive ratio ρ ( Q ) ) and the Hider has ε -optimal mixed strategies. It is not so straightforward to show that the game has a value if Q is unbounded. Nonetheless, this isnot important for our analysis, and we will rely on the following general result for zero-sum games thatfor any mixed Hider strategy h , ρ ( Q ) ≥ sup S ˆ T ( S, h ) , (2)where the supremum is taken over all search strategies S . For pathwise search, which is the usual search paradigm, the Searcher follows a continuous, unit-speedpath: that is a trajectory S : [0 , ∞ ) → Q with S (0) = O and d ( S ( t ) , S ( t )) ≤ t − t for all t < t . Forsuch a pathwise search S and a point H on Q , the (pathwise) search time T ( S, H ) of H under S is thefirst time that H is reached by the Searcher, i.e., min { t ≥ S ( t ) = H } . The concepts of deterministicand randomized search times, as well as the deterministic and randomized competitive ratios are definedanalogously to Definitions 2 and 3.As in the case of expanding search, we may view the randomized competitive ratio of a network as thevalue of a game played between a minimizing Searcher and a maximizing Hider where the payoff is thesearch time. In the case of finite networks, it is easy to show that the value exists, whereas for unboundednetworks, it is again the inequality (2) which will be most essential in our analysis. In this section we show how to obtain an expanding search of optimal deterministic ratio, using a “waterfilling” principle. Informally, the network is searched in such a way that the set of points that have beensearched at any given time form an expanding disc around O . Recall the definition of f Q from Section 2. f Q is piecewise linear and strictly increasing so has an inverse g Q . The interpretation is that g Q ( t ) is theunique radius r for which Q [ r ] has measure t . Definition 5.
For a network Q with root O , consider the expanding search S ∗ defined by S ∗ ( t ) = Q [ g Q ( t )] for ≤ t ≤ r max . Thus, S ∗ ( t ) is an expanding disc of radius g Q ( t ) . It is easy to verify that S ∗ is indeed an expandingsearch. First, we note that S ∗ ( t ) is connected, since Q [ r ] is always connected. It also trivially satisfies (i)and (ii) from Definition 1, and (iii) is also satisfied since λ ( S ∗ ( t )) = λ ( Q [ g Q ( t )]) = f Q ( g Q ( t )) = t. We will show that S ∗ attains the optimal competitive ratio. First, note that the search time of a point H ∈ Q under S ∗ is the unique time t such that S ∗ ( t ) = Q [ d ( H )] , so T ( S ∗ , H ) = λ ( Q [ d ( H )]) = f Q ( d ( H )) .7ence, the competitive ratio of S ∗ is σ S ∗ = sup H ∈ Q −{ O } f Q ( d ( H )) d ( H )= sup r> f Q ( r ) r . (3)This has an intuitive interpretation as follows: if we draw the graph of f Q ( r ) then the competitive ratio isthe slope of the steepest straight line through the root that intersects with the graph of f Q ( r ) . Condition (1)ensures that σ S ∗ is finite for unbounded networks, since f Q ( r ) ≤ M r for all r . Theorem 6. The expanding search S ∗ is optimal and the competitive ratio σ of a network Q with root O is given by σ = sup r> f Q ( r ) r . (4) Proof. let S be an optimal search, and let t ( r ) = min { t > Q [ r ] ⊂ S ( t ) } be the first time that S contains Q [ r ] . Then the maximum search time of any point H at some fixed distance r from O is t ( r ) ,and it follows that σ = σ S is given by σ S = sup r> t ( r ) r . Clearly, t ( r ) ≥ f Q ( r ) , so σ S ∗ ≤ σ S , by (3). The optimality of S ∗ and the expression for σ follows. In this section we study the randomized competitive ratio of expanding search, which is significantly morechallenging to analyze than the deterministic one. We begin by showing that the randomized competitiveratio is at most half the deterministic competitive ratio and that there exist networks for which this boundis tight (Section 4.1). In Section 4.2 we give a Hider strategy that allows us to get useful lower bounds onthe randomized competitive ratio. We also obtain bounds on ρ that are parameterized by the function f Q ,from which we can deduce the randomized competitive ratio for networks with concave f Q . In Section 4.3we show that the randomized strategy may have a quite complex statement, even for very simple networksthat consist only of three arcs. We address this difficulty in Section 4.4, in which we give a strategy thatis within a factor at most / of the optimal randomized competitive ratio, for all networks. Recall that S ∗ denotes the optimal deterministic search strategy of Section 3. Proposition 7. For a network Q with root O , the randomized competitive ratio ρ satisfies σ/ ≤ ρ ≤ σ. Furthermore, the bounds are tight, in the sense that they are the best possible. This theorem appeared without proof as Theorem 6 of (Angelopoulos, Dürr, & Lidbetter, 2016). This proposition appeared without proof as Proposition 7 of (Angelopoulos, Dürr, & Lidbetter, 2016). roof. The right-hand inequality is clear, since every deterministic search strategy is also a randomizedsearch strategy. To prove the left-hand inequality, we first observe that since S ∗ is an optimal deterministicsearch, for any ε > , we can find some point H on Q such that ˆ T ( S ∗ , H ) ≥ σ − ε . Let r = d ( H ) so that σ ≤ f Q ( r ) /r + ε . Let h be the Hider strategy that hides on Q [ r ] uniformly: that is, it chooses a subset of Q [ r ] with probability proportional to the measure of that subset. For any search strategy S , the expectedsearch time T ( S, h ) is at least λ ( Q [ r ]) / , so ρ ≥ sup S ˆ T ( S, h ) ≥ λ ( Q [ r ]) / r (since every point in Q [ r ] is at distance no more than r from the root) = f Q ( r )2 r ≥ σ − ε . Since ε can be arbitrarily small, it follows that ρ ≥ σ/ .We will now argue that both bounds are tight. This is trivially true for the right-hand inequality sincethe network consisting of one arc with the root at its end has the same deterministic and randomizedcompetitive ratio.For the left-hand inequality, consider the network depicted in Figure 2. The normalized search time ˆ T ( S ∗ , H ) is maximized at leaf nodes X , so that σ = ˆ T ( S ∗ , X ) = ( n + n ) / ( n + 1) = n .Consider now the randomized strategy s that searches the arc of length n first before searching theremainder of the arcs in a uniformly random order. Then all points H at distance no greater than n haveexpected normalized search time ; a point H at distance d > n has ˆ T ( s, H ) = d + ( n − / d ≤ n − / n ≤ n/ , so ρ ≤ n/ σ/ . Since ρ ≥ σ/ , we must have that σ/ρ → , as n → ∞ . Figure 2:
A network for which ρ ≈ σ/ . A corollary of Proposition 7 is that the “water-filling” search S ∗ approximates the optimal randomizedsearch by a factor of . 9 .2 A Hider strategy, and lower bounds on the randomized competitive ratio For a general network Q , let A be a connected subset. Let d ( A ) be the distance from O to A and let u A be the Hider strategy (probability measure) that hides uniformly on A , so that u A ( X ) = λ ( X ) /λ ( A ) for a measurable subset X ⊂ A . Then denote the average distance from O to points in A by d ( A ) = (cid:82) x ∈ A d ( x ) du A ( x ) . Theorem 8.
Consider the Hider strategy h A given by dh A ( x ) = d ( x ) d ( A ) du A ( x ) . By adopting the strategy h A , the Hider ensures that the randomized competitive ratio ρ satisfies ρ ≥ d ( A ) + λ ( A ) / d ( A ) . Proof.
Let S be any search strategy, and note that T ( S, u A ) ≥ d ( A ) + λ ( A ) / . We have ρ ≥ ˆ T ( S, h A ) = (cid:90) x ∈ A T ( S, x ) d ( x ) dh A ( x )= 1 d ( A ) (cid:90) x ∈ A T ( S, x ) du A ( x )= 1 d ( A ) T ( S, u A ) ≥ d ( A ) + λ ( A ) / d ( A ) . To illustrate the applicability of Theorem 8, we show how to obtain, in a different way, the corollaryof Proposition 7 that the optimal deterministic search strategy approximates the optimal randomizedstrategy by a factor of . Corollary 9.
The optimal deterministic search S ∗ approximates the randomized competitive ratio by afactor of .Proof. Let ε > and let x be a point of Q such that σ ≤ ˆ T ( S ∗ , x ) + ε/ λ ( A ) /d ( x ) + ε/ , where A ≡ Q [ d ( x )] is the set of all points at distance at most d ( x ) . By Theorem 8, ρ ≥ λ ( A ) / (2 d ( A )) , so σρ ≤ λ ( A ) /d ( x ) + ε/ λ ( A ) / (2 d ( A )) = 2 d ( A ) d ( x ) + εd ( A ) λ ( A ) ≤ ε. Since ε can be arbitrarily small, the corollary follows.More importantly, Theorem 8 allows us to obtain the following lower bound on the randomized com-petitive ratio of Q . Lemma 10.
For any network Q with root O , it holds that ρ ≥ deg Q ( O ) . roof. Let E O denote the set of arcs in Q that are incident with O . Fix r > such that r ≤ min e ∈ E O λ ( e ) ;clearly, such an r must exist. Let A = Q [ r ] be the ball of points in Q that are at distance at most r from O , and let h A be the Hider strategy associated with A , and defined as in the statement of Theorem 8.We calculate the average distance d ( A ) from O to points in A by writing d ( A ) = (cid:90) r − u A ( Q [ r ]) dr = (cid:90) r − rr dr = r . Moreover, from the definition of A , we have that λ ( A ) = deg Q ( O ) · r . By Theorem 8, we have ρ ≥ λ ( A )2 d ( A ) = deg Q ( O ) . The above lemma implies a tight bound on the randomized competitive ratio for all networks Q forwhich the function f Q is concave, as shown in the following corollary. Corollary 11.
For any network Q for which f Q is concave, we have that σ = ρ = deg Q ( O ) , and strategy S ∗ is an optimal randomized strategy.Proof. The lower bound on ρ follows from Lemma 10. For the upper bound, by (4), we have ρ ≤ σ =sup r> f Q ( r ) /r , and for any network for which f Q is concave, it holds that sup r> f Q ( r ) /r = deg Q ( O ) .An example of a network for which f Q is concave is depicted in Figure 3, along with a plot of itsfunction f Q . O Figure 3:
An example of a network Q (left) and the function r (cid:55)→ f ( r ) ≡ f Q ( r ) (right). All arcs in Q areunit-length. More generally, we have established the following approximation.
Corollary 12.
Suppose that for the network Q it holds that sup r> f Q ( r ) /r ≤ α deg Q ( O ) , for some α > .Then S ∗ approximates the optimal randomized ratio of Q within a factor of at most α . .3 Optimal randomized strategies are complex: Y -networks We now consider a class of the simplest networks for which the function f Q is not concave, and thusCorollary 11 does not apply. In particular, we consider the Y -network depicted in Figure 4 consisting of anode v which is incident to three arcs of lengths , L and M ≥ L . The root node is the other endpoint ofthe arc of length . We refer to the arc of length L as the “left arc” and the arc of length M as the “rightarc”. Figure 4:
The Y -network. Clearly the optimal Hider strategy on the Y -network will hide on the arc incident to the root withprobability . Let A be the subset of Q consisting of all the points on the left arc at distance at most x from v and all the points on the right arc at distance at most y from v . From Theorem 8, by using thestrategy h A , the Hider ensures that the competitive ratio is at least ρ ≥ d ( A ) + λ ( A ) / d ( A )1 + ( x + y ) /
21 + ( x/ ( x + y ))( x/
2) + ( y/ ( x + y )) y/
2= 1 + 2 xyx ( x + 2) + y ( y + 2) . By elementary calculus, this bound is maximized for x = L and y = max { M, (cid:112) L ( L + 2) } , giving ρ ≥ V := 1 + 2 LM (cid:48) L ( L + 2) + M (cid:48) ( M (cid:48) + 2) , (5)where M (cid:48) = max { M, (cid:112) L ( L + 2) } .We show that the expression V given in (5) is indeed the randomized competitive ratio by giving anoptimal Searcher strategy. The optimal Searcher strategy mixes between four different strategies whichwe list below. (Each strategy begins by searching the arc incident to the root, so we do not mention thispart of the search.) A . Search the left arc first then search the right arc. B . Search the right arc up to length M (cid:48) first then the left arc then search the remainder of the rightarc. C . Search the left arc and the right arc at the same time, at speeds proportional to L and M respectively,until the whole of the left arc has been searched, then search the remainder of the right arc. In other12ords, in the time interval [1 , t ] , search tL/ ( L + M (cid:48) ) of the left arc and tM (cid:48) / ( L + M (cid:48) ) of theright arc, for t ≤ L + M (cid:48) , then search the remainder of the right arc. D . Begin by searching the right arc, but at some time chosen uniformly at random between and M (cid:48) ,search the whole of the left arc before completing the search of the right arc.In Table 1 we list probabilities that the Searcher should choose each of these four strategies along withthe expected search time of a point at distance a ≤ L from v on the left arc and a point at distance b ≤ M (cid:48) from v on the right arc. Search strategy Probability Expected search time on left Expected search time on right A ML ( L +2)+ M (cid:48) ( M (cid:48) +2) a L + bB L +2 L − M L ( L +2)+ M (cid:48) ( M (cid:48) +2) M (cid:48) + a bC L L ( L +2)+ M (cid:48) ( M (cid:48) +2) aL ( L + M (cid:48) ) 1 + bM (cid:48) ( L + M (cid:48) ) D M − L L ( L +2)+ M (cid:48) ( M (cid:48) +2) M (cid:48) + a bM (cid:48) ( L + M (cid:48) ) Table 1:
An optimal search strategy for the Y -network. A simple calculation shows that for points on the left arc at distance a ≤ L from v , the expected searchtime is V (1 + a ) and for points on the right arc at distance b ≤ M (cid:48) from v , the expected search time is V (1 + b ) . For points on the right arc at distance b > M (cid:48) from v (if such points exist), the expected searchtime is L + b , and it is easy to show that this is strictly less than V (1 + b ) . Hence the randomizedcompetitive ratio is V . / -approximation of the randomized competitive ratio In this section we give a search strategy that is a / -approximation of the optimal randomized search.This is inspired by the strategy of (Angelopoulos et al., 2019) for the discrete case, namely for searchingin a given graph when the Hider can only hide at a vertex.We first define the concept of a Randomized Depth-First Search ( RDFS ) of a tree T . Let S be anydepth-first search of T and let S − be the depth-first search that visits the leaf nodes of T in the reverseorder from S . Then the randomized search s that chooses between S and S − equiprobably is a RDFS of T . Lemma 13.
Let s be a RDFS of a tree T . Then the expected time T ( s, H ) at which a point H ∈ T isfound by s satisfies T ( s, H ) ≤ λ ( T ) + d ( H )2 . Proof.
Suppose s is an equiprobable mixture of the depth-first search S and its reverse S − . Let t = T ( S, H ) and t = T ( S − , H ) . Then T ( s, H ) = t + t λ ( S ( t )) + λ ( S ( t ))2 .
13t is easy to see that S ( t ) ∩ S ( t ) is the path from O to H , so λ ( S ( t )) + λ ( S ( t )) = λ ( S ( t ) ∪ S ( t )) + λ ( S ( t ) ∩ S ( t s )) ≤ λ ( T ) + d ( H ) . The lemma follows.Now we can define the randomized search that is a / -approximation. For an arbitrary network Q , let Q T be its shortest path tree. We will define a search of Q T which naturally translates to a search of Q . Firstwe partition Q T into infinitely many randomly chosen subsets R j , j ∈ Z . To define the sets R j , we choosenumbers d j uniformly at random from the interval [2 j − , j ] , and set R j := { x ∈ Q T : d j ≤ d ( x ) < d j +1 } .We call the R j the levels of the search.The randomized doubling strategy s is defined as follows. At the start of the j th iteration, ∪ i For any j ∈ Z , (cid:18) − d ( Q j )2 j (cid:19) λ ( Q j ) ≤ j − ρ. Proof. Applying Theorem 8 to Q j , (cid:18) − d ( Q j )2 j (cid:19) λ ( Q j ) ≤ (cid:18) − d ( Q j )2 j (cid:19) · d ( Q j ) ρ. Regarding the right-hand side of the expression above as a quadratic in d ( Q j ) , it is maximized when d ( Q j ) = 2 j − , and the lemma follows. Lemma 15. The expected measure of Q j ∩ R j − is (2 − d ( Q j ) / j − ) λ ( Q j ) .Proof. A point x ∈ Q j is contained in R j − if and only if d j > x . This occurs with probability (2 j − d ( x )) / j − . Therefore, the expected measure of Q j ∩ R j − is (cid:90) x ∈ Q j (cid:18) j − d ( x )2 j − (cid:19) λ ( Q j ) du ( x ) = (cid:18) − d ( Q j )2 j − (cid:19) λ ( Q j ) . Theorem 16. The randomized doubling strategy s is a / -approximation of the optimal randomizedsearch. In particular, ρ s ≤ (5 / ρ + 1 / .Proof. Suppose that the randomized competitive ratio of s is maximized at some point x which is containedin Q k , for some k . Let J be a random variable that takes the value k − or k depending on whether x iscontained in R k − or R k , respectively. Let L J = λ ( ∪ i Let s be a (pathwise) RDFS of a tree T . Then the expected time T ( s, H ) a point H ∈ T isfound by s satisfies T ( s, H ) ≤ λ ( T ) . The (pathwise) random doubling strategy then performs successive RDFS’s of unions of levels ∪ i ≤ j R j . Theorem 19. The (pathwise) random doubling strategy s is a -approximation for the optimal randomizedpathwise search. That is, ρ s ≤ ρ .Proof. As in the proof of Theorem 16, suppose that the randomized competitive ratio of s is maximized atsome point x which is contained in Q k , for some k . Again, we define J as the index of the level containing x . We use Lemma 18 to write down an expression for the expected search time of x , conditioned on J ,and which we will denote by T ( s, x | J ) . T ( s, x | J ) = λ ( ∪ i ≤ J R i ) + (cid:88) j ≤ J − λ ( ∪ i ≤ j R i ) . T ( s, x | J ) = λ ( R J ) + λ ( ∪ i ≤ J − R i ) + (cid:88) j ≤ J − λ ( R j ) + λ ( ∪ i ≤ j − R i ) + λ ( ∪ i ≤ j R i )= λ ( R J ) + (cid:88) j ≤ J − λ ( R j ) + λ ( ∪ i ≤ J − R i ) + (cid:88) j ≤ J − λ ( ∪ i ≤ j R i ) + (cid:88) j ≤ J − λ ( ∪ i ≤ j R i )= (cid:88) j ≤ J λ ( R i ) + 2 λ ( ∪ i ≤ j − R i ) . Now taking expectations, with respect to J , we obtain T ( s, x ) = (cid:88) j ≤ J E ( L j ) , (13)where L J = λ ( R J ) / λ ( ∪ i ≤ J − R i ) is defined as in the proof of Theorem 16, and similarly for L J − , L J − , etc. We showed in the proof of Theorem 16 that E ( L J ) d ( x ) ρ ≤ / , and it follows that E ( L J − j )2 − j d ( x ) ρ ≤ / . So by (13), ˆ T ( s, x ) ρ ≤ (cid:88) j ≥ · − j E ( L J − j )2 − j d ( x ) ρ ≤ (cid:88) j ≥ · − j · / 4= 5 . In this section we discuss issues related to the implementation of our search strategies. Infinitesimally small tours For the purposes of the analysis, we allow the doubling strategies ofSections 4.2 and 5 to start with an infinite number of infinitesimally small tours. This is a standard wayof getting around the technical complication that any strategy which starts by a search to a constantdistance c > from the root cannot be constant-competitive, and has been applied in the analysis ofsearching in the infinite line and the infinite star, e.g., (Gal, 1974). In practice, of course, the Searcherwill start its search at some small distance from the root, and we may assume, as often in the ComputerScience literature on search algorithms, that the Hider is at distance at least 1 from the origin. The overallanalysis remains the same, at the expense of some negligible additive contribution to the overall searchcost that does not affect the competitive ratios. 18 epresentation of the network If the network Q is bounded, then there is a straightforward way ofrepresenting it as an undirected, weighted graph, in which the edge weights correspond to arc lengths.However, if Q is unbounded, we need certain assumptions in regards to how the Searcher can access thenetwork. One such way is to assume an oracle that given a parameter r > returns the subnetwork Q [ r ] of Q that corresponds to all points in Q at a radius r around the root, and which in turn can be encodedas a weighted graph, since it is bounded. For a Hider H , and a given search strategy, we denote by O H the number of accesses to the oracle that the strategy requires (and which we aim to bound).With these observations in mind, we can now discuss the implementation of our strategies. Concerningthe “waterfilling” deterministic strategy of Section 3, it suffices for the Searcher to access the oracle a loga-rithmic number of times, namely O H = O (log( d ( H ))) . Specifically, the oracle will reveal the subnetworks Q [2 i ] , with i ∈ [1 , (cid:100) log( d ( H )) (cid:101) ] . Within any given subnetwork, the strategy can be implemented in timepolynomial in its graph representation, by simply keeping track of the “active” edges, namely arcs of thenetwork which have been only partially searched.Similarly, for the doubling strategies of Sections 4.2 and 5, a logarithimic number of oracle accesses, inthe distance of the Hider, will suffice. For a given level j in the execution of these algorithms, all associatedactions of the strategies, namely finding a shortest path tree, performing an RDFS traversal of the tree,or finding a Chinese Postman tour can be done in time polynomial in the size of the graph representationof the corresponding level. In this work we studied expanding and pathwise search in a general, possibly unbounded network. Wefocused on the competitive ratio of the network as a measure for the efficiency of a search strategy, andgave the first constant-approximation mixed strategies in these settings. In particular, we addressed twoopen questions from (Angelopoulos et al., 2019), namely how to derive efficient strategies that are i)randomized; and ii) apply to a general network and not only to discrete trees.The obvious open problem from our work is to further improve the approximation of the randomizedcompetitive ratios, or identify more classes of networks for which optimal strategies can be found (althoughour result of Section 4.3 shows that any such identification will unavoidably exclude some very simplenetworks). Another direction is to consider searching for multiple hiders, as an extension to multi-hidersearch in a star under the competitive ratio (Angelopoulos et al., 2014) or relaxations of the competitiveratio (McGregor et al., 2009; Kirkpatrick, 2009).Last, concerning bounded networks, and beyond competitive analysis of search strategies, an inter-esting, and perhaps surprisingly open problem is to find a pathwise search strategy that minimizes theexpected time to locate the Hider, assuming that the Hider’s distribution is known. Acknowledgements This research benefited from the support of the FMJH Program Gaspard Mongefor optimization and operations research and their interactions with data science, and from the supportfrom EDF, Thales and Orange. 19 eferences Alpern, S., & Gal, S. (1988). A mixed strategy minimax theorem without compactness. SIAM Journalon Control and Optimization , (6), 1357-1361.Alpern, S., & Gal, S. (2003). The theory of search games and rendezvous . Kluwer Academic Publishers.Alpern, S., & Lidbetter, T. (2013). Mining coal or finding terrorists: The expanding search paradigm. Operations Research , (2), 265–279.Alpern, S., & Lidbetter, T. (2019). Approximate solutions for expanding search games on general networks. Annals of Operations Research , , 259–279.Angelopoulos, S., , Arsénio, D., Dürr, C., & López-Ortiz, A. (2016). Multi-processor search and schedulingproblems with setup cost. Theory of Computing Systems , 1–34.Angelopoulos, S., Arsénio, D., & Dürr, C. (2017). Infinite linear programming and online searching withturn cost. Theoretical Computer Science , , 11–22.Angelopoulos, S., Dürr, C., & Lidbetter, T. (2016). The expanding search ratio of a graph. In Proceedingsof the 33rd International Symposium on Theoretical Aspects of Computer Science (stacs) (pp. 9:1–9:14).Angelopoulos, S., Dürr, C., & Lidbetter, T. (2019). The expanding search ratio of a graph. DiscreteApplied Mathematics , , 51–65.Angelopoulos, S., López-Ortiz, A., & Panagiotou, K. (2014). Multi-target ray searching problems. Theo-retical Computer Science , , 2–12.Ausiello, G., Leonardi, S., & Marchetti-Spaccamela, A. (2000). On salesmen, repairmen, spiders, and othertraveling agents. In Proceedings of 4th Italian Conference on Algorithms and Complexity, CIAC (pp.1–16).Beck, A., & Newman, D. (1970). Yet more on the linear search problem. Israel Journal of Mathematics , , 419–429.Bose, P., Carufel, J. D., & Durocher, S. (2015). Searching on a line: A complete characterization of theoptimal solution. Theoretical Computer Science , , 24–42.Demaine, E., Fekete, S., & Gal, S. (2006). Online searching with turn cost. Theoretical Computer Science , , 342-355.Fleischer, R., Kamphans, T., Klein, R., Langetepe, E., & Trippen, G. (2008). Competitive online approx-imation of the optimal search ratio. SIAM Journal on Computing , (3), 881-898.Gal, S. (1972). A general search game. Israel Journal of Mathematics , , 32–45.Gal, S. (1974). Minimax solutions for linear search problems. SIAM Journal on Applied Mathematics , , 17–30.Gal, S. (1979). Search games with mobile and immobile hider. SIAM Journal on Control and Optimization , (1), 99–122.Hipke, C. A., Icking, C., Klein, R., & Langetepe, E. (1999). How to find a point on a line within a fixeddistance. Discrete Applied Mathematics , (1), 67–73.Isaacs, R. (1965). Differential games . John Wiley and Sons, New York.Jaillet, P., & Stafford, M. (1993). Online searching. Operations Research , , 234–244.20ao, M.-Y., Ma, Y., Sipser, M., & Yin, Y. (1998). Optimal constructions of hybrid algorithms. Journalof Algorithms , (1), 142–164.Kirkpatrick, D. G. (2009). Hyperbolic dovetailing. In Proceedings of the 17th European Symposium onAlgorithms (ESA) (pp. 616–627).Koutsoupias, E., Papadimitriou, C., & Yannakakis, M. (1996). Searching a fixed graph. In