Augmentation Problem Research Articles

Structures of subpartitions related to a submodular function minimization

Letf be a submodular function on 2 E for a nonempty finite setE and λ be a parameter that takes on values between −∞ and ∞. In this paper, we show that the set of the subpartitions π ofE on which ΣX ∈Π(f - λ)(X) attains the minimum has a structure similar to that of the intersecting family. Moreover, we apply this result to the minimum augmentation problem with respect to the edge-connectivity. We reveal that when a given graphG=(V, A) is notK-edge-connected, the set of the subpartitions π of the vertex set except {V} on which ΣX ∈Π(d -K)(X) attains the minimum has a structure like the cointersecting family, whered(X) denotes the number of edges betweenX andV−X.

Deterministic Õ(nm) Time Edge-Splitting in Undirected Graphs

This paper presents a deterministic O (nm log n + n2log2n) = O (nm) time algorithm for splitting off all edges incident to a vertex s of even degree in a multigraph G, where n and m are the numbers of vertices and links (= vertex pairs between which G has an edge) in G, respectively. Based on this, many graph algorithms using edge-splitting can run faster. For example, the edge-connectivity augmentation problem in an undirected multigraph can be solved in O (nm) time, which is an improvement over the previously known randomized O (n3) bound and deterministic O (n2m) bound.

On balanced edge connectivity and applications to some bottleneck augmentation problems in networks

Letw i ∶V ×V →Q,i = 1, 2 be two weight functions on the possible edges of a directed or undirected graph with vertex setV such that for the cut function, the inequality $$\delta _{w_2 } (T): = \sum\limits_{\scriptstyle i \in T \hfill \atop \scriptstyle j \notin T \hfill} {w_2 (ij) \ge 0.} $$ holds for everyT e 2 V We consider the computation of the value λ(W 1,W 2,K) defined by $$\tilde \lambda (w_{\text{1}} ,w_2 ,k): = \min \{ l|\delta _{w_1 } (T) + l\delta _{w_2 } (T) \geqslant k \forall \emptyset \subset T \subset V\} .$$ We show that the associated decision problem is NP-complete, but for a class of instances we can give a polynomial time algorithm. This class is closely related to the following bottleneck augmentation problem. Consider a networkN = (V, E, c) with a rational valued capacity functionc∶ V ×V →Q +, and letk be a positive, rational number. Consider the problem of finding a capacity functionc′∶V ×V →Q + such that, in the resulting networkN′ = (V, E, c + c′) the edge connectivity number λc+c′, is at leastk, and the maximal increasec′(ij) is minimal. We give an algorithm which computes such an augmentation in strongly polynomial time.

A location modeling approach for groundwater monitoring network augmentation

A heuristic approach based on facility location theory effectively augments groundwater quality monitoring configurations through the strategic siting of additional wells. The approach is an alternative to variance‐based schemes previously established for the augmentation problem. The method developed herein is practical in that it (1) is relatively easily understood, (2) is not difficult to implement and solve, and (3) does not require a large number of preexisting observation points. The approach has been applied to a case study involving a landfill‐contaminated buried valley aquifer in southwest Ohio. Configurations derived by the heuristic approach exhibit two key characteristics: (1) location of wells near areas of high estimated contaminant concentration and (2) interwell separation distances that facilitate areal plume coverage.

Augmenting Graphs to Meet Edge-Connectivity Requirements

What is the minimum number $\gamma$ of edges to be added to a given graph G so that in the resulting graph the edge-connectivity between every pair $\{ u,v \}$ of its nodes is at least a prescribed value $r( u,v )$? Generalizing earlier results of S. Sridhar and R. Chandrasekaran [Integer Programming and Combinatorial Optimization, R. Kannan and W. Pulleyblank, eds., Proceedings of a conference held at the University of Waterloo, University of Waterloo Press, Waterloo, Ontario, Canada, 1990, pp. 467–484] (when G is the empty graph), of K. P. Eswaran and R. E. Tarjan [SIAM Journal on Computing, 5 (1976), pp. 653–665] (when $r ( u,v ) \equiv 2$), and of G.-R. Cai and Y.-G. Sun [Networks, 19 (1989 ), pp. 151–172 ] (when $r ( u,v ) \equiv k\geqq 2$, we derive a min-max formula for $\gamma$ and describe a polynomial time algorithm to compute $\gamma$. The directed counterpart of the problem is solved in the same sense for the case when $r ( u,v ) \equiv k\geqq 1$ and is shown to be NP-complete if $r ( u,v ) \equiv 1$ for $u,v \in T$, and $r( u,v ) \equiv 0$ otherwise where T is a specified subset of nodes. A fundamental tool in the proof is the splitting theorems of W. Mader [Annals of Discrete Mathematics, 3 (1978), pp. 145–164] and L. Lovász [lecture, Prague, 1974; North–Holland, Amsterdam, 1979]. We also rely extensively on techniques concerning submodular functions. The method makes it possible to solve a degree-constrained version of the problem. The minimum-cost augmentation problem can also be solved in polynomial time provided that the edge-costs arise from node-costs, while the problem for arbitrary edge-costs was known to be NP-complete even for $r( u,v ) \equiv 2$.

Minimum augmentation of a tree to a K‐edge‐connected graph

AbstractThis paper solves the minimum augmentation problem for a given tree and positive integer k, that is, to make a tree k‐edge‐connected by adding the minimum number of edges. It is shown that the minimum number of edges is the least integer not less than a half of the deficiency of the tree which is defined as the sum of k‐(degree) over all the vertices whose degrees are less than k. The proof is constructive and gives a polynomial‐time algorithm for constructing such an augmentation.

Optimal Mixed Graph Augmentation

In this paper we consider an augmentation problem on mixed graphs that generalizes and unifies two augmentation problems considered by Eswaran and Tarjan [this Journal, 5 (1976), pp. 653–665]. The mixed augmentation problem has applications in the design of communication networks, and forms of the mixed augmentation problem are central in statistical data security problems discussed in [Yale Univ. Tech. Report, August 1984]. We solve the augmentation problem on mixed graphs in time linear in the number of edges of the graph, the same time bound obtained in [ET] for each of the two special cases.

An optimal time algorithm for the k-vertex-connectivity unweighted augmentation problem for rooted directed trees

For a given digraph G=( V, A) and a positive integer k, the k-vertex-connectivity unweighted augmentation problem ( k-VCUAP) for G is to find a minimum set of arcs A′ ( A′⊆( V× V− A)) such that the digraph ( V,A∪ A′) is k-vertex-connected. It is known that the time-complexity of 1-VCUAP for every digraph is θ(| V|+| A|). However, it remains still open whether or not there exist polynomial time algorithms for k-VCUAP's ( k≥2) for digraphs. This paper shows that the time-complexity of k-VCUAP ( k≥2) is θ( k| V|) for every rooted directed tree.

An 0(log n) parallel algorithm for strong connectivity augmentation problem

A synchronised parallel algorithm for the strong connectivity augmentation problem is presented. Its depth is 0(log n) using 0(n 3) processors on a concurrent read, concurrent write parallel random access machine.

Optimal‐Time Algorithm for the k‐Node‐Connectivity Augmentation Problem for Ternary Trees

AbstractGiven a graph and a positive integer k, in the k‐node‐connectivity augmentation problem (k‐NCAP), a set of edges is determined to be added to convert the graph into a k‐node‐connected graph with the minimum sum of the added edge‐weights. It is known that 1‐NCAP, for the directed acyclic graph where the weight is restricted to 1 and 2, is NP‐complete. It is also known that when the weight of the edges are all equal, 1‐NCAP for any directed graph can be solved in O([V] + [E]) time, and when the graph is restricted to the directed binary tree, k‐NCAP (k ≥ 2) is solved in O(k |y|) time. In the preceding, |V and |E are the number of nodes and the number of edges of the given graph, respectively. This paper discusses k‐NCAP (k ≥ 2) for the directed graph, where all edge weights are equal. A solution for k‐NCAP is shown for the family of graphs properly containing the directed ternary trees. It is shown that the problem can be solved in (k ≥ 3) time O(k |V) for the directed ternary tree, which is the optimal‐time algorithm within a constant factor. In this paper, a new family of regular and k‐node‐connected graphs called a k‐bouquet is defined. The solution for k‐NCAP is obtained by constructing a k‐bouquet by adding edges to the given graph.

Approximation of the Consecutive Ones Matrix Augmentation Problem

In this publication we will prove a number of negative results concerning the approximation of the NP-complete CONSECUTIVE ONES MATRIX AUGMENTATION problem. We will characterize a large class of simple algorithms that do not find a near optimum augmentation of their input matrices. We will show that there are matrices for which these algorithms find augmentations that are even far from optimal. These results are important for the analysis of a sparse matrix storage scheme.

On the relationship between the biconnectivity augmentation and travelling salesman problems

A strategy for solving the traveling salesman problem is adapted to the problem of finding a biconnected subgraph of a weighted graph whose cost function satisfies the triangle inequality. An approximation algorithm similar to Christofides' algorithm [5] for the traveling salesman problem is shown to possess the same worst-case bound of 3 2 when applied to the biconnectivity augmentation problem. A tight inequality is derived relating the cost of an optimal traveling salesman tour to the cost of an optimal biconnection.