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Minimum Spanning Tree Verification Under Uncertainty
Abstract
In the verification under uncertainty setting, an algorithm is given, for each input item, an uncertainty area that is guaranteed to contain the exact input value, as well as an assumed input value. An update of an input item reveals its exact value. If the exact value is equal to the assumed value, we say that the update verifies the assumed value. We consider verification under uncertainty for the minimum spanning tree (MST) problem for undirected weighted graphs, where each edge is associated with an uncertainty area and an assumed edge weight. The objective of an algorithm is to compute the smallest set of updates with the property that, if the updates of all edges in the set verify their assumed weights, the edge set of an MST can be computed. We give a polynomial-time optimal algorithm for the MST verification problem by relating the choices of updates to vertex covers in a bipartite auxiliary graph. Furthermore, we consider an alternative uncertainty setting where the vertices are embedded in the plane, the weight of an edge is the Euclidean distance between the endpoints of the edge, and the uncertainty is about the location of the vertices. An update of a vertex yields the exact location of that vertex. We prove that the MST verification problem in this vertex uncertainty setting is NP-hard. This shows a surprising difference in complexity between the edge and vertex uncertainty settings of the MST verification problem.
... Test results for MST-U are provided by Focke et al. in [7]. In [3], Erlebach and Hoffmann deal with the verification problem for MST-U, i.e. the problem of computing an optimal query set if the uncertainty sets as well as the exact edge weights are given. They show that the verification problem for MST-U is solvable in polynomial time while the verification problem for the vertex uncertainty problem V-MST-U is NP-hard. ...
... input : An instance I of V-MST-U with uniform query costs, a graph G = (V, E) and uncertainty sets A v , v ∈ V such that no two cycles share a non-trivial vertex output: A feasible query set Q 1 Compute the associated edge instance; 2 Initialize Q = ∅; 3 Preprocess the instance such that T L = T U ; 4 Index the edges f 1 , ..., f m−n+1 in R := E \ T L arbitrarily ; 5 for i ← 1 to m − n + 1 do 6 Let C i be the unique cycle in T L + f i ; 7 Let X(f i ) be the set of edges g ∈ T L ∩ C i with U g > L fi ; 8 Compute the size a of a smallest vertex cover of X(f 1 ) which does not intersect f 1 and consists of non-trivial vertices only; 9 if no edge in C i is always maximal then 10 if a = 1 then ...
This article studies the Minimum Spanning Tree Problem under Explorable Uncertainty as well as a related vertex uncertainty version of the problem. We particularly consider special instance types, including cactus graphs, for which we provide randomized algorithms. We introduce the problem of finding a minimum weight spanning star under uncertainty for which we show that no algorithm can achieve constant competitive ratio.
... Note that the Minimum problem is equivalent to the problem of determining the maximum element of each of the sets in S, e.g., by simply negating all the numbers involved. A motivation for studying the Minimum problem thus arises from the minimum spanning tree problem with uncertain edge weights [11,14,18,28]: Determining the maximum-weight edge of each cycle of a given graph allows one to determine a minimum spanning tree. Therefore, there is a connection between the problem of determining the maximum of each set in a family of possibly overlapping sets (which could be the edge sets of the cycles of a given graph) and the minimum spanning tree problem. ...
... After this initial foundation, many classic discrete problems were studied in this framework, including geometric problems [7,9], shortest paths [16], network verification [4], minimum spanning tree [11,14,18,28], cheapest set and minimum matroid base [13,30], linear programming [27,32], traveling salesman [34], knapsack [20], and scheduling [2,3,10]. The concept of witness sets was proposed by Bruce et al. [7], and identified as a pattern in many algorithms by Erlebach and Hoffmann [12]. ...
In computing with explorable uncertainty, one considers problems where the values of some input elements are uncertain, typically represented as intervals, but can be obtained using queries. Previous work has considered query minimization in the settings where queries are asked sequentially (adaptive model) or all at once (non-adaptive model). We introduce a new model where k queries can be made in parallel in each round, and the goal is to minimize the number of query rounds. Using competitive analysis, we present upper and lower bounds on the number of query rounds required by any algorithm in comparison with the optimal number of query rounds for the given instance. Given a set of uncertain elements and a family of m subsets of that set, we study the problems of sorting all m subsets and of determining the minimum value (or the minimum element(s)) of each subset. We also study the selection problem, i.e., the problem of determining the i -th smallest value and identifying all elements with that value in a given set of uncertain elements. Our results include 2-round-competitive algorithms for sorting and selection and an algorithm for the minimum value problem that uses at most ( 2 + ε ) · opt k + O 1 ε · lg m query rounds for every 0 < ε < 1 , where opt k is the optimal number of query rounds.
... Note that the Minimum problem is equivalent to the problem of determining the maximum element of each of the sets in S, e.g., by simply negating all the numbers involved. A motivation for studying the Minimum problem thus arises from the minimum spanning tree problem with uncertain edge weights [11,14,17,26]: Determining the maximum-weight edge of each cycle of a given graph allows one to determine a minimum spanning tree. Therefore, there is a connection between the problem of determining the maximum of each set in a family of possibly overlapping sets (which could be the edge sets of the cycles of a given graph) and the minimum spanning tree problem. ...
... After this initial foundation, many classic discrete problems were studied in this framework, including geometric problems [7,9], shortest paths [15], network verification [4], minimum spanning tree [11,14,17,26], cheapest set and minimum matroid base [13,28], linear programming [25,30], traveling salesman [32], knapsack [19], and scheduling [2,3,10]. The concept of witness sets was proposed by Bruce et al. [7], and identified as a pattern in many algorithms by Erlebach and Hoffmann [12]. ...
The area of computing with uncertainty considers problems where some information about the input elements is uncertain, but can be obtained using queries. For example, instead of the weight of an element, we may be given an interval that is guaranteed to contain the weight, and a query can be performed to reveal the weight. While previous work has considered models where queries are asked either sequentially (adaptive model) or all at once (non-adaptive model), and the goal is to minimize the number of queries that are needed to solve the given problem, we propose and study a new model where k queries can be made in parallel in each round, and the goal is to minimize the number of query rounds. We use competitive analysis and present upper and lower bounds on the number of query rounds required by any algorithm in comparison with the optimal number of query rounds. Given a set of uncertain elements and a family of m subsets of that set, we present an algorithm for determining the value of the minimum of each of the subsets that requires at most rounds for every , where is the optimal number of rounds, as well as nearly matching lower bounds. For the problem of determining the i-th smallest value and identifying all elements with that value in a set of uncertain elements, we give a 2-round-competitive algorithm. We also show that the problem of sorting a family of sets of uncertain elements admits a 2-round-competitive algorithm and this is the best possible.
... The main contribution in [15] is a randomized algorithm with expected competitive ratio of 1 + 1/ √ 2 ≈ 1.707 whereas the best-known lower bound is 1.5. The offline problem of finding the optimal query set for a given realization of edge weights can be solved in polynomial time [5]. ...
... A similar instance evokes the reverse performance behavior. Consider a cycle C with k edges e i with interval (0, 3), one edge g with interval (0, 4) and one edge f with interval (1,5). We choose the weights as w ei = 2 and w g = w f = 3. ...
We consider the minimum spanning tree (MST) problem in an uncertainty model where uncertain edge weights can be explored at extra cost. The task is to find an MST by querying a minimum number of edges for their exact weight. This problem has received quite some attention from the algorithms theory community. In this paper, we conduct the first practical experiments for MST under uncertainty, theoretically compare three known algorithms, and compare theoretical with practical behavior of the algorithms. Among others, we observe that the average performance and the absolute number of queries are both far from the theoretical worst-case bounds. Furthermore, we investigate a known general preprocessing procedure and develop an implementation thereof that maximally reduces the data uncertainty. We also characterize a class of instances that is solved completely by our preprocessing. Our experiments are based on practical data from an application in telecommunications and uncertainty instances generated from the standard TSPLib graph library.
... They also show that this ratio is optimal and can be generalized to the problem of finding a minimum weight basis of a matroid with uncertain weights [6]. According to Erlebach [4] it remained a major open problem whether randomized algorithms can beat competitive ratio 2. The offline problem of finding the optimal query set for given exact edge weights can be solved optimally in polynomial time [5]. Further problems studied in this uncertainty model include finding the k-th smallest value in a set of uncertainty intervals [9, 11, 12] (also with non-uniform query cost [9]), caching problems in distributed databases [15], computing a function value [13], and classical combinatorial optimization problems, such as shortest path [8], finding the median [9], and the knapsack problem [10]. ...
... One key observation is that the minimum spanning tree problem under uncertainty can be interpreted as a generalized online bipartite vertex cover problem. A similar connection for a given realization of edge weights was established in [5] for the related MST verification problem. In our case, new structural insights allow for a preprocessing which suggests a unique bipartition of the edges for all realizations simultaneously. ...
We consider the problem of finding a minimum spanning tree (MST) in a graph with uncertain edge weights given by open intervals on the edges. The exact weight of an edge in the corresponding uncertainty interval can be queried at a given cost. The task is to determine a possibly adaptive query sequence of minimum total cost for finding an MST. For uniform query cost, a deterministic algorithm with best possible competitive ratio 2 is known [7].
We solve a long-standing open problem by showing that randomized query strategies can beat the best possible competitive ratio 2 of deterministic algorithms. Our randomized algorithm achieves expected competitive ratio . This result is based on novel structural insights to the problem enabling an interpretation as a generalized online bipartite vertex cover problem. We also consider arbitrary, edge-individual query costs and show how to obtain algorithms matching the best known competitive ratios for uniform query cost. Moreover, we give an optimal algorithm for the related problem of computing the exact weight of an MST at minimum query cost. This algorithm is based on an interesting relation between different algorithmic approaches using the cycle-property and the cut-property characterizing MSTs. Finally, we argue that all our results also hold for the more general setting of matroids. All our algorithms run in polynomial time.
Given subsets of uncertain values, we study the problem of identifying the subset of minimum total value (sum of the uncertain values) by querying as few values as possible. This set selection problem falls into the field of explorable uncertainty and is of intrinsic importance therein as it implies strong adversarial lower bounds for a wide range of interesting combinatorial problems such as knapsack and matchings. We consider a stochastic problem variant and give algorithms that, in expectation, improve upon these adversarial lower bounds. The key to our results is to prove a strong structural connection to a seemingly unrelated covering problem with uncertainty in the constraints via a linear programming formulation. We exploit this connection to derive an algorithmic framework that can be used to solve both problems under uncertainty, obtaining nearly tight bounds on the competitive ratio. This is the first non-trivial stochastic result concerning the sum of unknown values without further structure known for the set. With our novel methods, we lay the foundations for solving more general problems in the area of explorable uncertainty.Keywordsexplorable uncertaintyqueriesset selectionset cover
Decision-making under uncertainty is a major challenge in logistics. Mathematical optimization has a long tradition in providing powerful methods for solving logistics problems. While classical optimization models for uncertainty in the input data do not consider the option to actively query the precise value of uncertain input elements, this option is in practice often available at a certain cost. The recent line of research on optimization under explorable uncertainty develops methods with provable performance guarantees for such scenarios. In this chapter, we highlight some recent results from the mathematical optimization perspective and outline the potential power of such model and techniques for solving logistics problems.
We study problems with stochastic uncertainty information on intervals for which the precise value can be queried by paying a cost. The goal is to devise an adaptive decision tree to find a correct solution to the problem in consideration while minimizing the expected total query cost. We show that, for the sorting problem, such a decision tree can be found in polynomial time. For the problem of finding the data item with minimum value, we have some evidence for hardness. This contradicts intuition, since the minimum problem is easier both in the online setting with adversarial inputs and in the offline verification setting. However, the stochastic assumption can be leveraged to beat both deterministic and randomized approximation lower bounds for the online setting.
We study the problem of sorting under incomplete information, when queries are used to resolve uncertainties. Each of n data items has an unknown value, which is known to lie in a given interval. We can pay a query cost to learn the actual value, and we may allow an error threshold in the sorting. The goal is to find a nearly-sorted permutation by performing a minimum-cost set of queries.
We show that an offline optimum query set can be found in polynomial time, and that both oblivious and adaptive problems have simple query-competitive algorithms. The query-competitiveness for the oblivious problem is n for uniform query costs, and unbounded for arbitrary costs; for the adaptive problem, the ratio is 2.
We then present a unified adaptive strategy for uniform query costs that yields the following improved results: (i) a 3/2-query-competitive randomized algorithm; (ii) a 5/3-query-competitive deterministic algorithm if the dependency graph has no 2-components after some preprocessing, which has query-competitive ratio 3/2+O(1/k) if the components obtained have size at least k; and (iii) an exact algorithm if the intervals constitute a laminar family. The first two results have matching lower bounds, and we have a lower bound of 7/5 for large components.
We also give a randomized adaptive algorithm with query-competitive factor 1+433≈1.7698 for arbitrary query costs, and we show that the 2-query competitive deterministic adaptive algorithm can be generalized for queries returning intervals and for a more general graph problem (which is also a generalization of the vertex cover problem), by using the local ratio technique. Furthermore, we prove that the advice complexity of the adaptive problem is ⌊n/2⌋ if no error threshold is allowed, and ⌈n/3⋅lg3⌉ for the general case.
Finally, we present some graph-theoretical results regarding co-threshold tolerance graphs, and we discuss uncertainty variants of some classical interval problems.
We consider the minimum spanning tree (MST) problem in an uncertainty model where interval edge weights can be explored to obtain the exact weight. The task is to find an MST by querying the minimum number of edges. This problem has received quite some attention from the algorithms theory community. In this article, we conduct the first practical experiments for MST under uncertainty, theoretically compare three known algorithms, and compare theoretical with practical behavior of the algorithms. Among others, we observe that the average performance and the absolute number of queries are both far from the theoretical worst-case bounds. Furthermore, we investigate a known general preprocessing procedure and develop an implementation thereof that maximally reduces the data uncertainty. We also characterize a class of instances that is solved to optimality by our preprocessing. Our experiments are based on practical data from an application in telecommunications and uncertainty instances generated from the standard TSPLib graph library.
Explorable uncertainty refers to settings where parts of the input data are initially unknown, but can be obtained at a certain cost using queries. In a typical setting, initially only intervals that contain the exact input values are known, and queries can be made to obtain exact values. An algorithm must make queries one by one until it has obtained sufficient information to solve the given problem. We discuss two lines of work in this area: In the area of query-competitive algorithms, one compares the number of queries made by the algorithm with the best possible number of queries for the given input. In the area of scheduling with explorable uncertainty, queries may correspond to tests that can reduce the running-time of a job by an a priori unknown amount and are executed on the machine that also schedules the jobs, thus contributing directly to the objective value of the resulting schedule.
We consider the minimum spanning tree problem in a setting where information about the edge weights of the given graph is uncertain. Initially, for each edge e of the graph only a set Ae, called an uncertainty area, that contains the actual edge weight we is known. The algorithm can ‘update ’ e to obtain the edge weight we ∈ Ae. The task is to output the edge set of a minimum spanning tree after a minimum number of updates. An algorithm is k-update competitive if it makes at most k times as many updates as the optimum. We present a 2-update competitive algorithm if all areas Ae are open or trivial, which is the best possible among deterministic algorithms. The condition on the areas Ae is to exclude degenerate inputs for which no constant update competitive algorithm can exist. Next, we consider a setting where the vertices of the graph correspond to points in Euclidean space and the weight of an edge is equal to the distance of its endpoints. The location of each point is initially given as an uncertainty area, and an update reveals the exact location of the point. We give a general relation between the edge uncertainty and the vertex uncertainty versions of a problem and use it to derive a 4-update competitive
We present a framework for computing with input data specified by intervals,
representing uncertainty in the values of the input parameters. To compute a
solution, the algorithm can query the input parameters that yield more refined
estimates in form of sub-intervals and the objective is to minimize the number
of queries. The previous approaches address the scenario where every query
returns an exact value. Our framework is more general as it can deal with a
wider variety of inputs and query responses and we establish interesting
relationships between them that have not been investigated previously. Although
some of the approaches of the previous restricted models can be adapted to the
more general model, we require more sophisticated techniques for the analysis
and we also obtain improved algorithms for the previous model.
We address selection problems in the generalized model and show that there
exist 2-update competitive algorithms that do not depend on the lengths or
distribution of the sub-intervals and hold against the worst case adversary. We
also obtain similar bounds on the competitive ratio for the MST problem in
graphs.
We consider the problems of computing maximal points and the convex hull of a set of points in two dimensions, when the points are “in motion.” We assume that the point locations (or trajectories) are not known precisely and determining these values exactly is feasible, but expensive. In our model the algorithm only knows areas within which each of the input points lie, and is required to identify the maximal points or points on the convex hull correctly by updating some points (i.e., determining their location exactly). We compare the number of points updated by the algorithm on a given instance to the minimum number of points that must be updated by a nondeterministic strategy in order to compute the answer provably correctly. We give algorithms for both of the above problems that always update at most three times as many points as the nondeterministic strategy, and show that this is the best possible. Our model is similar to that in [3] and [5].
We consider the problem of estimating the length of the shortest path from a vertex s to a vertex t in a DAG whose edge lengths are known only approximately but can be determined exactly at a cost. Initially, for each edge e, the length of e is known only to lie within an interval [le,he]; the estimation algorithm can pay we to find the exact length of e. We study the problem of finding the cheapest set of edges such that, if exactly these edges are queried, the length of the shortest s–t path will be known, within an additive κ⩾0, an input parameter. An actual s–t path, whose true length exceeds that of the shortest s–t path by at most κ, will be obtained as well. The problem of finding a cheap set of edge queries is in neither NP nor co-NP unless NP = co-NP. We give positive and negative results for two special cases and for the general case, which we show is in Σ2.
We study the complexity of geometric minimum spanning trees under a stochastic model of input: Suppose we are given a master set of points s1,s_2,...,sn in d-dimensional Euclidean space, where each point si is active with some independent and arbitrary but known probability pi. We want to compute the expected length of the minimum spanning tree (MST) of the active points. This particular form of stochastic problems is motivated by the uncertainty inherent in many sources of geometric data but has not been investigated before in computational geometry to the best of our knowledge. Our main results include the following. We show that the stochastic MST problem is SPHARD for any dimension d ≥ 2. We present a simple fully polynomial randomized approximation scheme (FPRAS) for a metric space, and thus also for any Euclidean space. For d=2, we present two deterministic approximation algorithms: an O(n4)-time constant-factor algorithm, and a PTAS based on a combination of shifted quadtrees and dynamic programming. We show that in a general metric space the tail bounds of the distribution of the MST length cannot be approximated to any multiplicative factor in polynomial time under the assumption that P ≠ NP. In addition to this existential model of stochastic input, we also briefly consider a locational model where each point is present with certainty but its location is probabilistic.
An optimum rectilinear Steiner tree for a set A of points in the plane is a tree which interconnects A using horizontal and vertical lines of shortest possible total length. Such trees correspond to single net wiring patterns on printed backplanes which minimize total wire length. We show that the problem of determining this minimum length, given A, is NP-complete. Thus the problem of finding optimum rectilinear Steiner trees is probably computationally hopeless, and the emphasis of the literature for this problem on heuristics and special case algorithms is well justified. A number of intermediary lemmas concerning the NP-completeness of certain graph-theoretic problems are proved and may be of independent interest.
Computing the median with uncertainty
- T Feder
- R Motwani
- R Panigrahy
- C Olston
- J Widom