D. Bienstock and P. Seymour. Monotonicity in graph searching. J. Algorithms, 12:239--245, 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....searchers leading from v to a contaminated vertex. In some applications it is required that recontamination should never occur, but for most of the search game variants considered it can be shown, by some very clever techniques, that the resource usage does not increase in spite of this constraint [15, 16, 4, 7]. The classical goal of the search problem is to nd the search program such that the maximum number of searchers in use at any move is minimized. The minimum number of searchers needed to clear the graph is related to the parameter called pathwidth. Dendris et al. 7] studied a variation of the ....

D. Bienstock and P. Seymour, Monotonicity in graph searching, J. Algorithms, 12 (1991), pp. 239 - 245.

....of a network is equal to its intervalwidth [18] and to its vertex separator plus one [19] Seymour and Thomas [28] showed that the t search number is equal to the tree width plus one. Takahashi, Ueno and Kajitani [35] showed that the mixed search number is equal to the proper path width. In [3], Bienstock and Seymour simpli ed the proof of Lapaugh s result [21] stating that there is a minimal search strategy that does not recontaminate any link (see also [2] Thilikos [36] used graph minors to derive a linear time algorithm that checks whether a network has a search number at most 2. ....

....a monotone contiguous search strategy using cs(T ) searchers. Moreover, this strategy satis es that all searchers are initially placed at the same node x0 , and the rst step consists to clear a link incident to x0 . The proof is inspired from the short and elegant proof of Bienstock and Seymour [3] for the monotonicity theorems of Lapaugh [21] and Kirousis and Papadimitriou [18, 19] The structure of the proof as well as the notions of crusades and progressive crusades are directly borrowed from [3] The proof is nevertheless reported below for its non obvious adaptation to the weighted ....

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D. Bienstock and P. Seymour. Monotonicity in graph searching. Journal of Algorithms 12, 239-245, 1991.

....not known, have been considered in the past. We have the deterministic search games, where a fugitive that possesses some properties (it is agile or inert) hides in the nodes or edges of a graph and the aim is to locate it using as few searchers as possible (for definitions and relevant theory see [5, 14, 18]) Also, the problem of exploring an unknown graph has been considered (see [2, 19, 20] for example) Closer to the spirit of our work are the search problems that are defined and studied in [4] where the authors consider problems of locating points in the plane using incomplete knowledge about ....

D. Bienstock and P. Seymour, Monotonicity in graph searching, Journal of Algorithms 12 (1991), 239--245.

.... characteristics that define the search variant (e.g. it may be moving constantly or it may move only when some searcher is about to visit the node at which it resides) hides in the nodes of the graph and the aim is to capture it using the smallest possible number of searchers (see, for example, [6, 10, 15]) Deterministic algorithms that explore graphs with unknown topology have been studied in [3, 17] There also exists a class of games called stochastic games in which the opponents employ move strategies involving a probability transition matrix (see [20] A description of games that involve ....

D. Bienstock and P. Seymour (1991) Monotonicity in graph searching. Journal of Algorithms, 12, 239--245.

....the best possible route. Search problems in graphs, where the identity of the node that contains the information sought is not known, have been considered before. These include deterministic search games, where a fugitive that possesses some properties hides in the nodes or edges of a graph [5, 13, 14]) and the problem of exploring an unknown graph [2, 12, 16] Our model is similar in spirit to the model in [4] where the authors propose algorithms to search for a point on a line or on a lattice drawn on the plane. However, in that model, the nodes have limited if any knowledge of where the ....

D. Bienstock and P. Seymour, Monotonicity in graph searching, Journal of Algorithms 12 (1991), 239-245.

.... This research was partially supported by ESPRIT Long Term Research Project 20244 (project ALCOM IT: Algorithms and Complexity in Information Technology) The second author was supported by the Training and Mobility of Researchers (TMR) Program, EU contract no ERBFMBICT950198) and Seymour in [2]. Linear width can be seen as a linear variant of branch width , in the same way as pathwidth can be seen as a linear variant of treewidth . In [25] it is proved that several variants of problems appearing on graph searching can be reduced to the problem of computing linear width. In a graph ....

....ns(G) and ms(G) The first graph searching game was introduced by Breisch [6] and Parsons [18] and is the one of edge searching. Node searching appeared as a variant of edge searching and was introduced by Kirousis and Papadimitriou in [14] Finally, mixed searching was introduced in [24] and [2] and is a natural generalisation of the two previous variants (for the formal definitions see Subsection 4.1 for other results concerning search games on graphs see [1, 7, 9, 15, 16, 23] The problems of computing es(G) ns(G) ms(G) or linear width(G) is NP complete (see [16, 14, 24, 25] ....

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D. Bienstock and P. Seymour. Monotonicity in graph searching. J. Algorithms, 12:239 -- 245, 1991.

....every i = 1; r Gamma 1, there are at most k vertices incident to edges that belong both to fe 1 ; e i g and to fe i 1 ; e r g. Linear width was first mentioned by Thomas in [36] and is strongly connected with the notion of crusades introduced by Bienstock and Seymour in [3]. In this paper we prove that several variants of problems appearing on graph searching can be reduced to the problem of computing linear width. In a graph searching game a graph represents a system of tunnels where an agile, fast, and invisible fugitive is resorting. We desire to capture this ....

....ns(G) and ms(G) The first graph searching game was introduced by Breisch [9] and Parsons [27] and is the one of edge searching. Node searching appeared as a variant of edge searching and was introduced by Kirousis and Papadimitriou in [22] Finally, mixed searching was introduced in [35] and [3] and is a natural generalisation of the two previous variants (for the formal definitions see Subsection 5.1) The problem of computing es(G) ns(G) ms(G) or linear width(G) is NP complete (see [24, 22, 35] and Theorem 5.i of this paper) On the other hand, since all of these parameters is ....

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D. Bienstock and P. Seymour. Monotonicity in graph searching. J. Algorithms, 12:239 -- 245, 1991.

....All, i] A[i 4 2, JAIl. ii) If the sequence contains two elements ai and aj such that j i 2 and Vi k j ai ak aj or Vi j ai ak aj, then set A A(1,i) A(j, A[ We define the compression r(A) of a sequence A 5, as the unique minimum length ele ment of B [ B A . For example, [5,5,6,7,7, 7,4,4, 3,5,4,6, 8, 2, 9,3,4,6,7, 2, 7,5,4,4,6, 4]) 5,7,3,8,2,9,2,7,4] We call a sequence A typical if A ,5 and q (A) A. The following results have been proved in [12] Lemmata 3.3 and 3.5 respectively) LEMMA 2.4. If A 5 and max A k, then r(A) contains at most 2k 1 elements. LEMMA 2.5. The number of different typical sequences ....

....ai and aj such that j i 2 and Vi k j ai ak aj or Vi j ai ak aj, then set A A(1,i) A(j, A[ We define the compression r(A) of a sequence A 5, as the unique minimum length ele ment of B [ B A . For example, 5,5,6,7,7, 7,4,4, 3,5,4,6, 8, 2, 9,3,4,6,7, 2, 7,5,4,4,6, 4] [5,7,3,8,2,9,2,7,4]. We call a sequence A typical if A ,5 and q (A) A. The following results have been proved in [12] Lemmata 3.3 and 3.5 respectively) LEMMA 2.4. If A 5 and max A k, then r(A) contains at most 2k 1 elements. LEMMA 2.5. The number of different typical sequences consisting of integers ....

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D. Bienstock and P. Seymour. Monotonicity in graph searching. J. Algorithms, 12:239 245, 1991.

....cost of clearing an edge by far exceeds the cost of traversing an edge. Hence each edge should be cleared only once. Lapaugh [21] has proved that for every G there is always a monotone search strategy that uses s(G) searchers; a similar positive result exists also for node search and mixed search [2, 4]. A search strategy is contiguous if the set of clear edges is always connected. The necessity for contiguity arises e.g. in applications where communication between the searchers can occur only within completely clear areas of the network; hence connectivity is required for their coordination. ....

....a singular point in the zoology of search problems. The natural open problem is thus determining the properties of search strategies which are both monotone and internal. Surprisingly, unlike the case of monotone strategies for which there exist detailed studies and characterizations (e.g. [2, 4, 15, 21, 36]) very little is known about contiguous search strategies and monotone internal strategies. Unfortunately, the existing techniques and results for (the many variants of) the problem not only cannot be employed but do not even provide any direct insight on these two important properties. In this ....

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D. Bienstock and P. Seymour. Monotonicity in graph searching. Journal of Algorithms 12, 239-245, 1991.

....problem considered here. In terms of the restaurant finding problem, routing under faults corresponds to the problem where the exact address of the restaurant is known and one wishes avoid roadblocks or traffic jams. Other related lines of research including searching for a fugitive in a graph [4, 12, 15] or exploring an unknown graph [2, 7, 16] In neither case do the nodes contain information of the type considered here that might direct the search. In the restaurant analogy, there are no informed policeman. We feel our work is closer in spirit to that of computing with uncertainty or with noisy ....

D. Bienstock and P. Seymour, "Monotonicity in graph searching," J. of Alg., 12 (1991), 239-245.

.... some characteristics that define the search variant (e.g. it may be moving constantly, or it may move only when some searcher is about to visit the node at which it resides) hides in the nodes of the graph; the aim is to capture it using the smallest possible number of searchers (see, for example, [12, 13, 14]) Deterministic algorithms that explore graphs with unknown topology have been studied in [15, 16] There also exists a class of games called stochastic games in which the opponents employ move strategies involving a probability transition matrix (see [17] A description of games called ....

Bienstock, D. and Seymour, P. (1991) Monotonicity in graph searching. J. Algorithms, 12, 239--245.

.... pathwidth, developed in [20] and [3] were the cornerstones for proving the monotonicity of the corresponding graph searching variants (see also [7] Our paper is motivated and constitutes an extension of the ideas in the proofs of the monotonicity of the agile fugitive search games examined in [12, 13, 17, 22, 2, 4, 3, 15] as well as the proofs of the min max theorems in [21] and [19] Our main observation is that the kernel argument of all these proofs is based on the fact that, in any game variant, the cost of the search can be expressed by a connectivity function that is a nonnegativevalued function on the set ....

....it provides obstruction characterizations, game counterparts, and monotonicity proofs for the parameters of linear width, cutwidth and their extensions. Finally, our general min max theorem implies in a uniform way the monotonicity proofs of all the agile fugitive search games developed so far in [12, 17, 22, 4, 15]. To illustrate the main motivation of our research let us give a simple example of an expansion game. Suppose that we have a set of countries subject to join some organization. At every moment of time we can either add a bounded number of countries to the union or expel an arbitrary number of ....

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D. Bienstock and P. Seymour, Monotonicity in graph searching, J. Algorithms, 12 (1991), pp. 239 - 245.

.... ] Q h 1 ( Q h 1 (1) Q h 1 ( Q h 1 (2) Q h 1 = # 6 6 6 6 6 6 8 7 6 6 9 7 # 9 9 9 9 9 9 9 9 9 9 9 9 6 7 4 7 3 8 1 ] w h = z m h 1 m h 2 z z m h 3 z m h 4 m h 5 z ] Q h 2 = [ 1 5 2 3 6 8 8 8 8 8 8 8 8 5 6 6 5 6 2 5 2 2 3 7 7 7 7 7 7 7 7 ] B h 2 = 1 8 8 2 7 7 ] l = z m h 1 z m h 3 z m h 4 ] Q h 2 ) 1 8 2 7 ] Q h 2 = 1 5 2 3 6 8 5 6 6 5 6 2 5 2 2 3 7 ] FIG. 4. An example of the proof of Lemma 3.1. QG i ;l (r) i = 1; 2, and Q = Q G;l (0) QG i ;l (r) ....

....to the minimum k for which the result of this check is positive. To do this, we use the construction of Lemma 5.1, and get a tree decomposition (Y; U) of G D with treewidth dw. The result now follows from Theorem 5.1, taking in mind that (G D ) 2. For a proof of the following, see [5]. Lemma 5.3. If G n is the graph obtained from G by replacing every edge in G with two edges in parallel, then pathwidth(G) linear width(G n ) 33 It is easy to see that there exists a procedure that given an edge ordering of a graph G with width k, transforms it to a path decomposition ....

D. Bienstock and P. Seymour. Monotonicity in graph searching. J. Algorithms, 12:239 - 245, 1991.

.... The search number of a graph refers to the minimum number of pursuers needed to solve a pursuit evasion problem, and has been closely related to other graph properties such as cutwidth [16, 18] It has also been shown that a graph can be searched monotonically (i.e. without recontamination) in [2, 12]. Pursuit evasion scenarios in continuous spaces have arisen in a variety of applications 1 such as air traffic control [1] military strategy [11] and trajectory tracking [10] This has resulted in the formal study of general decision problems in which two decision makers have diametrically ....

....is also obtained in [6] It states that there exist examples that require recontaminating some portion of the free space a linear number of times. This result is surprising because for Parsons problem it was shown in [12] that no recontamination is necessary (a shorter proof of this appears in [2]) Theorem 5 establishes that a linear number of recontaminations can be needed, and it still remains open to determine whether the number of recontaminations can be bounded from above by a polynomial, which would imply that the problem of deciding whether H(F ) 1 lies in NP . Theorem 5 There ....

D. Bienstock and P. Seymour. Monotonicity in graph searching. J. Algorithms, 12:239--245, 1991.

....Introduction Communication networks are vulnerable to privacy violations. Surveillance of the network is one way to deter eavesdroppers. This gives rise to various models of pursuit and evasion on graphs and corresponding complexity considerations. One problem that has been examined in depth (see [5, 16] and references) is the search of a graph by a team of searchers traversing the edges of the graph in pursuit of a mobile fugitive. The minimum number of searchers necessary to detect the fugitive with certainty is called the search number of the graph. Computing it is easy for trees but NP hard ....

D. Bienstock and P. Seymour (1991), Monotonicity in graph searching. Journal of Algorithms 12, 230--245.

....scanned and it has reached a final state. Although a graph automaton has a finite number of states and finitely many instructions, the number of heads is unbounded. It depends on the size and the structure of the graph given to the input and is bounded from below by the edge search number [BS91]. The heads are placed on nodes and edges, where they guard nodes and clear edges. At any time, the set of currently visited edges is an edge separator between the visited and the yet undiscovered part of the input graph. In a move, the border of separating edges continuously moves beyond the read ....

....between the visited and the yet undiscovered part of the input graph. In a move, the border of separating edges continuously moves beyond the read subgraph. Thus already cleared nodes and edges cannot be recontaminated. This means a monotone search strategy, graph searching without recontamination [BS91, LaP93, MHGJP88]. Hence, graph automata are plans for monotone search strategies on graphs. The search strategies are special. They are given by a finite set of instructions and can be executed by nondeterministic finite state machines. Our main result states that graph automata are equivalent to linear graph ....

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D. Bienstock, P. Seymour. Monotonicity in graph searching. J. Algorithms 12 (1991), 239-245.

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D. Bienstock and P. Seymour. Monotonicity in graph searching. J. Algorithms, 12:239--245, 1991.

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D. Bienstock and P. Seymour. Monotonicity in graph searching. Journal of Algorithms 12:239-- 245, 1991.

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D. Bienstock and P. Seymour. Monotonicity in graph searching. J. Algorithms, 12:239 -- 245, 1991.

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D. Bienstock and P. Seymour. Monotonicity in graph searching. Journal of Algorithms, 12(2):239--245, 1991.

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D. Bienstock and P. Seymour. Monotonicity in graph searching. J. Algorithms, 12:239 -- 245, 1991.

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D. Bienstock and P. Seymour. Monotonicity in graph searching. Journal of Algorithms 12, 239--245, 1991.

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D. Bienstock, P. Seymour, Monotonicity in graph searching, J. Algorithms 12 (1991) 239-245.

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D. Bienstock and P. Seymour, Monotonicity in graph searching, J. Algorithms, 12 (1991), pp. 239 -- 245.

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D. Bienstock and P. Seymour, Monotonicity in graph searching, J. Algorithms, 12 (1991), pp. 239 -- 245.

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