The Valve Location Problem in Simple Network Topologies

Abstract

To control possible spills in liquid or gas transporting pipe systems, the systems are usually equipped with shutoff valves. In case of an accidental leak these valves separate the system into a number of pieces limiting the spill effect. In this paper, we consider the problem, for a given edge-weighted network representing a pipe system and for a given number of valves, to place the valves in the network in such a way that the maximum possible spill, i.e. the maximum total weight of a piece, is minimized. We show that the problem is NP-hard even if restricted to any of the following settings: (i) for series-parallel graphs and hence for graphs of treewidth two; (ii) if all edge weights equal one. If the network is a simple path, a cycle, or a tree, the problem can be solved in polynomial time. We also give a pseudo-polynomial time algorithm and a fully polynomial approximation scheme for networks of bounded treewidth.

Detecting critical node structures on graphs: A mathematical programming approach

“…First, if the set of node structures  is closed under inclusion (i.e., if T 1 ⊆ T 2 and T 2 ∈  , then T 1 ∈  ), constraints (13) will suffice because any solution  with overlapping critical node structures can be transformed into a solution  ′ with disjoint critical node structures (see the proofs of Theorems 1 and 2 in the Appendix A for the clique and star cases). On the contrary, if  is not closed under inclusion, constraints [13] can be replaced by the following constraints.…”

“…The idea behind this variant coincides with the one of the ordinary path constraints (6) in which if no node structure T ∈  having nodes along a path P ∈  is removed from G, such a path also exists in G[V ⧵ V()], implying that nodes i and j remain connected after the set of critical node structures  is removed from G. The relationship between variables x and z given by constraints (13) implies that any solution (x, y, z) for the CNP connectivity measure (or (x, y, z, u) for the LC connectivity measure) satisfying constraint (37) satisfies constraint (6) as well. To see this, simply replace (13) in (6) for any given path…”

“…That is, if the resulting optimal value is larger than , then the corresponding clique can be included in  ′ . Notice that the subproblem given by formulation (40)-(42) is that of finding a clique of maximum weight in G, where the weights are the dual variables of constraints (13). Finding maximum cliques in general graphs is known to be NP-hard [29], which implies that pricing new cliques could be potentially challenging for large instances.…”

“…In many of these studies, a wide variety of structural properties and their corresponding measures have been proposed to quantify the disruption inflicted on the graph G. In general, these measures are either associated with (1) optimal solutions to flow problems over G, such as shortest paths, maximum flow, or minimum cost flow problems [17,19,33,38,41,44,49,78,79]; (2) the sizes or relative weight of some topological node or edge structures in G, like spanning trees [28], dominating sets [47], central nodes-for example, the one-median or one-center nodes [10], matchings [80,81], independent sets [9], cliques [46], and node covers [9]; and (3) connectivity and cohesiveness properties of G, such as the total number of connected node pairs [1, 6, 21-23, 51, 52, 65, 69, 70, 72, 73], the weight of the connections between the node pairs [6,21,73], the size of the largest component [13,30,55,63,64,73], the total number of components [5,[63][64][65]68], distance-based connectivity metrics [74], and the graph information entropy [14,35,56,73].…”

Measures for graph integrity: a comparative survey

“…Все последние десятилетия и до настоящего времени проблеме исследования мер целостности графов уделяется особое внимание, поскольку она связана с вопросами анализа и синтеза отказоустойчивых сложных технических систем, включая вычислительные системы, коммуникационные и транспортные сети, трубопроводные системы [4,26,41]. Для таких систем отказоустойчивость, как правило, является важнейшим показателем качества функционирования [2,4,5,10].…”