Capacity inverse minimum cost flow problem

Given a directed graph G=(N,A) with arc capacities u _ij and a minimum cost flow problem defined on G, the capacity inverse minimum cost flow problem is to find a new capacity vector [^(u)]\hat{u} for the arc set A such that a given feasible flow [^(x)]\hat{x} is optimal with respect to the modified capacities. Among all capacity vectors [^(u)]\hat{u} satisfying this condition, we would like to find one with minimum ||[^(u)]-u||\|\hat{u}-u\| value. We consider two distance measures for ||[^(u)]-u||\|\hat{u}-u\| , rectilinear (L ₁) and Chebyshev (L _∞) distances. By reduction from the feedback arc set problem we show that the capacity inverse minimum cost flow problem is NP\mathcal{NP} -hard in the rectilinear case. On the other hand, it is polynomially solvable by a greedy algorithm for the Chebyshev norm. In the latter case we propose a heuristic for the bicriteria problem, where we minimize among all optimal solutions the number of affected arcs. We also present computational results for this heuristic.     

Complexity analysis for maximum flow problems with arc reversals

We provide a comprehensive study on network flow problems with arc reversal capabilities. The problem is to identify the arcs to be reversed in order to achieve a maximum flow from source(s) to sink(s). The problem finds its applications in emergency transportation management, where the lanes of a road network could be reversed to enable flow in the opposite direction. We study several network flow problems with the arc reversal capability and discuss their complexity. More specifically, we discuss the polynomial time algorithms for the maximum dynamic flow problem with arc reversal capability having a single source and a single sink, and for the maximum (static) flow problem. The presented algorithms are based on graph transformations and reductions to polynomially solvable flow problems. In addition, we show that the quickest transshipment problem with arc reversal capability and the problem of minimizing the total cost resulting from arc switching costs are NP\mathcal{NP} -hard.     

Polynomial Time Approximation Scheme for the Rectilinear Steiner Arborescence Problem

Given a set N of n terminals in the first quadrant of the Euclidean plane E ², find a minimum length directed tree rooted at the origin o, connecting to all terminals in N, and consisting of only horizontal and vertical arcs oriented from left to right or from bottom to top. This problem is called rectilinear Steiner arborescence problem, which has been proved to be NP-complete recently (Shi and Su, 11th ACM-SIAM Symposium on Discrete Algorithms (SODA), January 2000, to appear). In this paper, we present a polynomial time approximation scheme for this problem.     

On Split-Coloring Problems

We study a new coloring concept which generalizes the classical vertex coloring problem in a graph by extending the notion of stable sets to split graphs. First of all, we propose the packing problem of finding the split graph of maximum size where a split graph is a graph G = (V,E) in which the vertex set V can be partitioned into a clique K and a stable set S. No condition is imposed on the edges linking vertices in S to the vertices in K. This maximum split graph problem gives rise to an associated partitioning problem that we call the split-coloring problem. Given a graph, the objective is to cover all his vertices by a least number of split graphs. Definitions related to this new problem are introduced. We mention some polynomially solvable cases and describe open questions on this area. An erratum to this article is available at .     

A novel approach to subgraph selection with multiple weights on arcs

In this paper, an extension of the minimum cost flow problem is considered in which multiple incommensurate weights are associated with each arc. In the minimum cost flow problem, flow is sent over the arcs of a graph from source nodes to sink nodes. The goal is to select a subgraph with minimum associated costs for routing the flow. The problem is tractable when a single weight is given on each arc. However, in many real-world applications, several weights are needed to describe the features of arcs, including transit cost, arrival time, delay, profit, security, reliability, deterioration, and safety. In this case, finding an optimal solution becomes difficult. We propose a heuristic algorithm for this purpose. First, we compute the relative efficiency of the arcs by using data envelopment analysis techniques. We then determine a subgraph with efficient arcs using a linear programming model, where the objective function is based on the relative efficiency of the arcs. The flow obtained satisfies the arc capacity constraints and the integrality property. Our proposed algorithm has polynomial runtime and is evaluated in rigorous experiments.

     

Arcs in $$\mathbb Z^2_{2p}$$

An arc in \(\mathbb Z^2_n\) is defined to be a set of points no three of which are collinear. We describe some properties of arcs and determine the maximum size of arcs for some small n.     

Minimum entropy coloring

We study an information-theoretic variant of the graph coloring problem in which the objective function to minimize is the entropy of the coloring. The minimum entropy of a coloring is called the chromatic entropy and was shown by Alon and Orlitsky (IEEE Trans. Inform. Theory 42(5):1329–1339, 1996) to play a fundamental role in the problem of coding with side information. In this paper, we consider the minimum entropy coloring problem from a computational point of view. We first prove that this problem is NP-hard on interval graphs. We then show that, for every constant ε>0, it is NP-hard to find a coloring whose entropy is within (1−ε)log n of the chromatic entropy, where n is the number of vertices of the graph. A simple polynomial case is also identified. It is known that graph entropy is a lower bound for the chromatic entropy. We prove that this bound can be arbitrarily bad, even for chordal graphs. Finally, we consider the minimum number of colors required to achieve minimum entropy and prove a Brooks-type theorem. S. Fiorini acknowledges the support from the Fonds National de la Recherche Scientifique and GERAD-HEC Montréal. G. Joret is a F.R.S.-FNRS Research Fellow.     

On Integrality, Stability and Composition of Dicycle Packings and Covers

Given a digraph D, the minimum integral dicycle cover problem (known also as the minimum feedback arc set problem) is to find a minimum set of arcs that intersects every dicycle; the maximum integral dicycle packing problem is to find a maximum set of pairwise arc disjoint dicycles. These two problems are NP-complete.Assume D has a 2-vertex cut. We show how to derive a minimum dicycle cover (a maximum dicycle packing) for D, by composing certain covers (packings) of the corresponding pieces. The composition of the covers is simple and was partially considered in the literature before. The main contribution of this paper is to the packing problem. Let be the value of a minimum integral dicycle cover, and * () the value of a maximum (integral) dicycle packing. We show that if = then a simple composition, similar to that of the covers, is valid for the packing problem. We use these compositions to extend an O(n³) (resp., O(n⁴)) algorithm for finding a minimum integral dicycle cover (resp., packing) from planar digraphs to K_3,3-free digraphs (i.e., digraphs not containing any subdivision of K_3,3).However, if , then such a simple composition for the packing problem is not valid. We show, that if the pieces satisfy, what we call, the stability property, then a simple composition does work. We prove that if = * holds for each piece, then the stability property holds as well. Further, we use the stability property to show that if = * holds for each piece, then = * holds for D as well.     

Approximating the Minimum k-way Cut in a Graph via Minimum 3-way Cuts

For an edge weighted undirected graph G and an integer k > 2, a k-way cut is a set of edges whose removal leaves G with at least k components. We propose a simple approximation algorithm to the minimum k-way cut problem. It computes a nearly optimal k-way cut by using a set of minimum 3-way cuts. We show that the performance ratio of our algorithm is 2 – 3/k for an odd k and 2 – (3k – 4)/(k ² – k) for an even k. The running time is O(kmn ³ log(n ²/m)) where n and m are the numbers of vertices and edges respectively.     

PLANNING FOR TRUCK FLEET SIZE IN THE PRESENCE OF A COMMON-CARRIER OPTION

This paper considers a class of network optimization problems in which certain directed arcs must be covered by a set of cycles. Our study was motivated by a distribution planning problem of a commercial firm that had to make deliveries over several origin-destination pairs (directed arcs) and that could service any demand arc by using a vehicle in its own fleet or by paying a common carrier. The problem is to determine an optimal fleet size and the resulting vehicle routes while satisfying maximum route-time restrictions. We formulate the problem, describe some approximate solution strategies, and discuss important implementation issues.     

Scheduling arc shut downs in a network to maximize flow over time with a bounded number of jobs per time period

We study the problem of scheduling maintenance on arcs of a capacitated network so as to maximize the total flow from a source node to a sink node over a set of time periods. Maintenance on an arc shuts down the arc for the duration of the period in which its maintenance is scheduled, making its capacity zero for that period. A set of arcs is designated to have maintenance during the planning period, which will require each to be shut down for exactly one time period. In general this problem is known to be NP-hard, and several special instance classes have been studied. Here we propose an additional constraint which limits the number of maintenance jobs per time period, and we study the impact of this on the complexity.     

A hybrid beam search looking-ahead algorithm for the circular packing problem

In this paper, we study the circular packing problem. Its objective is to pack a set of n circular pieces into a rectangular plate R of fixed dimensions L×W. Each piece’s type i, i=1,…,m, is characterized by its radius r _i and its demand b _i . The objective is to determine the packing pattern corresponding to the minimum unused area of R for the circular pieces placed. This problem is solved by using a hybrid algorithm that adopts beam search and a looking-ahead strategy. A node at a level ℓ of the beam-search tree contains a partial solution corresponding to the circles already placed inside R. Each node is then evaluated using a looking-ahead strategy, based on the minimum local-distance heuristic, by computing the corresponding complete solution. The nodes leading to the best solutions at level ℓ are then chosen for branching. A multi-start strategy is also considered in order to diversify the search space. The computational results show, on a set of benchmark instances, the effectiveness of the proposed algorithm.     

Online bin packing of fragile objects with application in cellular networks

We study a specific bin packing problem which arises from the channel assignment problems in cellular networks. In cellular communications, frequency channels are some limited resource which may need to share by various users. However, in order to avoid signal interference among users, a user needs to specify to share the channel with at most how many other users, depending on the user’s application. Under this setting, the problem of minimizing the total channels used to support all users can be modeled as a specific bin packing problem as follows: Given a set of items, each with two attributes, weight and fragility. We need to pack the items into bins such that, for each bin, the sum of weight in the bin must be at most the smallest fragility of all the items packed into the bin. The goal is to minimize the total number of bins (i.e., the channels in the cellular network) used. We consider the on-line version of this problem, where items arrive one by one. The next item arrives only after the current item has been packed, and the decision cannot be changed. We show that the asymptotic competitive ratio is at least 2. We also consider the case where the ratio of maximum fragility and minimum fragility is bounded by a constant. In this case, we present a class of online algorithms with asymptotic competitive ratio at most of 1/4+3r/2, for any r>1. A preliminary version of this paper appeared in Proc. of Workshop on Internet and Network Economics (WINE’05, pp. 564–573). The research of W.-T.C. was supported in part by Hong Kong RGC grant HKU5172/03E. The research of F.Y.-L.C. was supported in part by Hong Kong RGC Grant HKU7142/03E. The research of D.Y. was supported by NSFC (10601048). The research of G.Z. was supported in part by NSFC (60573020).     

The maximum flow problem with disjunctive constraints

We study the maximum flow problem subject to binary disjunctive constraints in a directed graph: A negative disjunctive constraint states that a certain pair of arcs in a digraph cannot be simultaneously used for sending flow in a feasible solution. In contrast to this, positive disjunctive constraints force that for certain pairs of arcs at least one arc has to carry flow in a feasible solution. It is convenient to represent the negative disjunctive constraints in terms of a so-called conflict graph whose vertices correspond to the arcs of the underlying graph, and whose edges encode the constraints. Analogously we represent the positive disjunctive constraints by a so-called forcing graph. For conflict graphs we prove that the maximum flow problem is strongly $\mathcal{NP}$ -hard, even if the conflict graph consists only of unconnected edges. This result still holds if the network consists only of disjoint paths of length three. For forcing graphs we also provide a sharp line between polynomially solvable and strongly $\mathcal{NP}$ -hard instances for the case where the flow values are required to be integral. Moreover, our hardness results imply that no polynomial time approximation algorithm can exist for both problems. In contrast to this we show that the maximum flow problem with a forcing graph can be solved efficiently if fractional flow values are allowed.     

Minimum Cost Edge Subset Covering Exactly k Vertices of a Graph

Given a graph G with nonnegative edge costs and an integer k, we consider the problem of finding an edge subset S of minimum total cost with respect to the constraint that S covers exactly k vertices of G. An O(n ³) algorithm is presented where n is the order of G. It is based on the author's previous paper dealing with a similar problem asking S to cover at least k vertices.     

Lower bounds and a tabu search algorithm for the minimum deficiency problem

An edge coloring of a graph G=(V,E) is a function c:E→ℕ that assigns a color c(e) to each edge e∈E such that c(e)≠c(e′) whenever e and e′ have a common endpoint. Denoting S _v (G,c) the set of colors assigned to the edges incident to a vertex v∈V, and D _v (G,c) the minimum number of integers which must be added to S _v (G,c) to form an interval, the deficiency D(G,c) of an edge coloring c is defined as the sum ∑ _v∈V D _v (G,c), and the span of c is the number of colors used in c. The problem of finding, for a given graph, an edge coloring with a minimum deficiency is NP-hard. We give new lower bounds on the minimum deficiency of an edge coloring and on the span of edge colorings with minimum deficiency. We also propose a tabu search algorithm to solve the minimum deficiency problem and report experiments on various graph instances, some of them having a known optimal deficiency.     

Incrementing Bipartite Digraph Edge-Connectivity

This paper solves the problem of increasing the edge-connectivity of a bipartite digraph by adding the smallest number of new edges that preserve bipartiteness. A natural application arises when we wish to reinforce a 2-dimensional square grid framework with cables. We actually solve the more general problem of covering a crossing family of sets with the smallest number of directed edges, where each new edge must join the blocks of a given bipartition of the elements. The smallest number of new edges is given by a min-max formula that has six infinite families of exceptional cases. We discuss a problem on network flows whose solution has a similar formula with three infinite families of exceptional cases. We also discuss a problem on arborescences whose solution has five infinite families of exceptions. We give an algorithm that increases the edge-connectivity of a bipartite digraph in the same time as the best-known algorithm for the problem without the bipartite constraint: O(km log n) for unweighted digraphs and O(nm log (n ²/m)) for weighted digraphs, where n, m and k are the number of vertices and edges of the given graph and the target connectivity, respectively.     

Optimal design and augmentation of strongly attack-tolerant two-hop clusters in directed networks

We consider the problems of minimum-cost design and augmentation of directed network clusters that have diameter 2 and maintain the same diameter after the deletion of up to R elements (nodes or arcs) anywhere in the cluster. The property of a network to maintain not only the overall connectivity, but also the same diameter after the deletion of multiple nodes/arcs is referred to as strong attack tolerance. This paper presents the proof of NP-completeness of the decision version of the problem, derives tight theoretical bounds, as well as develops a heuristic algorithm for the considered problems, which are extremely challenging to solve to optimality even for small networks. Computational experiments suggest that the proposed heuristic algorithm does identify high-quality near-optimal solutions; moreover, in the special case of undirected networks with identical arc construction costs, the algorithm provably produces an exact optimal solution to strongly attack-tolerant two-hop network design problem, regardless of the network size.     

Parameterized lower bound and inapproximability of polylogarithmic string barcoding

String barcoding is a method that can identify microorganisms by analyzing their genome sequences. In this paper, we study the polylogarithmic string barcoding problem, where the lengths of the substrings in the testing set are polylogarithmically bounded. In particular, we show that the polylogarithmic string barcoding problem remains NP-hard and moreover, for a problem instance with n sequences, it is NP-hard to achieve an approximate ratio within dln n in polynomial time, where d is some constant. We then consider the parameterized polylogarithmic string barcoding problem, where the number of substrings in the test set is considered to be a fixed parameter k. We show that, unless W[2]=FPT, there does not exist a 2 ^O(k) n ^c algorithm that can decide whether a test set of size k exists or not, where c is a constant independent of n and k.