Postman Problem Research Articles

Postman tours and cycle covers

Let G be a bridgeless graph. We show that the length of a shortest postman tour is at most |E(G)| + |V(G)| 3 and that, if G is a minimally 2-edge connected graph, then the length is at most 2|V(G)|...

Postman tours and cycle covers

Let G be a bridgeless graph. We show that the length of a shortest postman tour is at most |E(G)| + |V(G)| − 3 and that, if G is a minimally 2-edge connected graph, then the length is at most 2|V(G)| − 2. We then deduce results concerning the length of a shortest cycle cover for graphs containing no subdivision of the Petersen graph.

An algorithm for min-cost edge-disjoint cycles and its applications

The problems of finding minimum-cost and maximum-cost sets of edge-disjoint cycles in a weighted undirected graph are studied. The importance of this problem is that it presents a ‘middle station’ in two reductions for planar graphs — one between the max-cut problem and that of max-weight matching, and the other between the Chinese Postman Problem and max-weight matching. The introduction of negative edge costs makes the reductions simple and efficient. We obtain new simpler algorithms for these two problems for planar graphs (where the max-weight matching problem can be solved very efficiently). We conclude that, in the case of planar weighted graphs (with arbitrary costs), all the three problems are mutually reducible and equivalent in terms of complexity.

Tight integral duality gap in the Chinese Postman problem

LetG = (V, E) be a graph and letw be a weight functionw:E →Z+. Let\(T \subseteq V\) be an even subset of the vertices ofG. AT-cut is an edge-cutset of the graph which dividesT into two odd sets. AT-join is a minimal subset of edges that meets everyT-cut (a generalization of solutions to the Chinese Postman problem). The main theorem of this paper gives a tight upper bound on the difference between the minimum weightT-join and the maximum weight integral packing ofT-cuts. This difference is called the (T-join) integral duality gap. Letτw be the minimum weight of aT-join, and letvw be the maximum weight of an integral packing ofT-cuts. IfF is a non-empty minimum weightT-join, andnF is the number of components ofF, then we prove thatτw—vw≤nF−1.

On the convex hull of feasible solutions to certain combinatorial problems

By Weyl's Theorem, the convex hull of the set of (characteristic vectors of) feasible solutions to combinatorial problems is a convex polyhedron when there is a finite number of feasible solutions. This need not be the case of infinite solution sets, even when they are finitely generated. We demonstrate this on examples of Steiner graphical travelling salesman problems, Steiner Chinese postman problems and single machine scheduling problems, either nonpreemptive with changeover times, or preemptive with deadlines or precedence constraints. We present a necessary and sufficient condition for the convex hull of a finite union of (unbounded) convex polyhedra to be closed, and hence a convex polyhedron itself. We apply this result to some of the above problems, and discuss conditions under which the convex hull of their feasible solutions is a convex polyhedron.

Evaluation and improvement of fault coverage of conformance testing by UIO sequences

The fault coverage of testing protocols using unique input/output (UIO) sequences is analyzed. UIO sequences can be efficiently employed in checking the conformance specifications of protocols by using transition testing. The test sequence is found using the rural Chinese postman tour algorithm. A comprehensive fault model is developed, and analytical expressions are given for the fault coverage. The conditions for undetectability are analyzed, and a new algorithm is proposed. Simulation results and illustrative examples are presented. Overhead issues are discussed, and significant improvements are shown for achieving 100% fault coverage. The major advantage of the proposed approach is that it provides the theoretical basis for fault coverage evaluation of protocol testing using UIO sequences.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

An optimization technique for protocol conformance test generation based on UIO sequences and rural Chinese postman tours

A method for generating test sequences for checking the conformance of a protocol implementation to its specification is described. A rural Chinese postman tour problem algorithm is used to determine a minimum-cost tour of the transition graph of a finite-state machine. It is shown that, when the unique input/output sequence (UIO) is used in place of the more cumbersome distinguishing sequence, both the controllability and observability problems of the protocol testing problem are addressed, providing an efficient method for computing a test sequence for protocol conformance testing.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Minimum cycle coverings and integer flows

AbstractIt was conjectured by Fan that if a graph G = (V,E) has a nowhere‐zero 3‐flow, then G can be covered by two even subgraphs of total size at most |V| + |E| ‐ 3. This conjecture is proved in this paper. It is also proved in this paper that the optimum solution of the Chinese postman problem and the solution of minimum cycle covering problem are equivalent for any graph admitting a nowhere‐zero 4‐flow.

Shortest Circuit Covers and Postman Tours in Graphs with a Nowhere Zero 4

Let G be a graph with a nowhere zero 4-flow. It is shown that the length of a shortest circuit cover of $E(G)$ is equal to the length of a shortest postman tour of $E(G)$. Using this result an efficient algorithm for constructing shortest circuit covers for graphs that possess two disjoint spanning trees is obtained. It is also deduced that if H is a $2m$-edge connected graph, $m \geqq 2$, then there exists a circuit cover of $E(H)$ of length at most $|E(H)|+\min \{ {{| E(H) |} / {(2m + 1)}}, | V(H) | - 1 \}$ and that if G has a nowhere zero 4-flow, then there exists a circuit cover of $V(G)$ of length at most $2|V(G)| -2$. Finally, it is shown that the equivalence between shortest circuit covers and postman tours may be extended to binary matroids that possess a nowhere zero $\mathbb{Z} _2 ^2$-flow.

Undirected distances and the postman-structure of graphs

We present some properties of the distance function and of shortest paths in ±1- weighted undirected graphs. These extend some basic results, e.g., on matchings, on the Chinese postman problem, and on plane multicommodity flows. Furthermore, distances turn out to be efficient tools to generalize the matching-structure of graphs to a structure related to subgraphs having only parity constraints on their degrees (these are called joins or postman sets), a problem posed by Lovász and Plummer. The special cases include the generalization of the matching-structure of graphs to the weighted case. The main result of the paper is a good characterization (linear in the number of edges), conjectured by A. Frank, of the minimum weights of paths from a fixed vertex of an undirected graph without negative circuits. This result contains the well-known minimax theorems on minimum “odd joins” and maximum packings of “odd cuts” (namely, Lovász's theorem on half integer packings and its sharpening by Seymour and later by Frank and Tardos) and strengthens them by constructing a “canonical” maximum packing of odd cuts with favourable properties. This packing of odd cuts turns out then to be characteristic for the structure of minimum odd joins. Using these, a Gallai-Edmonds type structural description of minimum odd joins is worked out. (The generalization of the Kotzig-Lovàsz canonical partition will appear in a forthcoming paper.) Briefly, distances in ±1-weighted graphs make it possible for us to treat some properties of matchings themselves in a more compact way, and to generalize them providing new results on some other interesting special cases of ±1-weighted graphs as well. This technique is worked out in the present paper.

Algorithmic Generation of Protocol Conformance Tests

With the recent expansion of data communications networks, computers and terminals from different manufacturers must be interconnected. Before being connected to the network, these interacting elements should be tested to ensure that they conform to the network's protocol specifications. We describe an algorithmic method (which is based on unique input/output sequences and Rural Postman tours) for generating conformance-test sequences that require minimum cost (i.e., run time) and completely cover the state transitions defined by the protocol specification. The technique, which has been implemented as the POSTMAN software package, has been widely used in ATT for error recovery; and for window-flow-control procedures (see Sherif and Uyar1).

Protocol Modeling for Conformance Testing: Case Study for the ISDN LAPD Protocol

In this paper, we present a generic approach for modeling a communications protocol with state transitions and window and timer mechanisms, and for generating conformance test sequences automatically. Based on this model, minimum-cost (i.e., minimum run time) conformance test sequences can be generated by using a method based on unique input/output sequences and the Rural Chinese Postman tours. As a case study, this method is applied to the Integrated Services Digital Network link-access protocol on the D channel.

Four problems on graphs with excluded minors

The four problems we consider are the Chinese postman, odd cut, co-postman, and odd circuit problems. Seymour's characterization of matroids having the max-flow min-cut property can be specialized to each of these four problems to show that the property holds whenever the graph has no certain excluded minor. We develop a framework for characterizing graphs not having these excluded minors and use the excluded minor characterizations to solve each of the four optimization problems. In this way, a constructive proof of Seymour's theorem is given for these special cases. We also show how to solve the Chinese postman problem on graphs having no four-wheel minor, where the max-flow min-cut property need not hold.

On the Windy Postman Problem on eulerian graphs

The Windy Postman Problem (WPP) is defined as follows: Given an undirected connected graphG = (V, E) and costscij andcji for each edgeij ∈E (wherecij is the cost of traversing edgeij fromi toj), find a windy postman tour of minimum cost. Here, by a windy postman tour, we mean a closed directed walk which is an orientation of a closed walk inG containing each edge ofE at least once.

The Capacitated Canadian Postman Problem

Routing problems can be classified according to where the demand is located on the network: on nodes, on arcs or on both. The travelling salesman problem (tsp) is an example in which a vehicle must visit every node of the network and return to the original node at minimum cost. If a capacity restraint is added to the vehicle and the total demand exceeds the capacity of one vehicle, a vehicle routing problem (vrp) is obtained. When the demand to be satisfied lies on the arcs of the network, the equivalent to the tsp is the Chinese postman problem (cpp). This problem consists of covering, at minimum cost, all the arcs of a network with a route starting and ending at a specified originating point. If the generation of more than one route to serve the demand is necessary because of a capacity constraint, the result is termed the capacitated Chinese postmand problem (ccpp). This paper adapts the mathematical formulation of the ccpp to the context of the Canada post corporation. No capacity constraints on the route are present in this context, but there is a time limit on each route. Furthermore, this time limit includes the travel time between a central depot and the beginning and end of the route. A general heuristic is presented to solve the problem under certain hypotheses. To solve the sequencing problem, a euler's tour problem is defined with penalties assigned to each change of arc; this penalty corresponds to street crossing and the change of streets when passing from one arc to another. This problem has many other real-life applications, including mail distribution, snow removal and salt application, garbage collection, rural school transportation and parking meter collection.

POSTMAN ROUTING PROBLEM IN A HIERARCHICAL NETWORK

This paper presents a technique for obtaining a very good solution to the Chinese Postman Problem (CPP) in a directed, hierarchical network. This type of problem arises in a network where it is required that a group of arcs be traversed prior to another group. Arcs of the same hierarchy are grouped together and the CPP is applied to each group to obtain the optimal routing strategy for that hierarchy. Sometimes all the arcs in a particular hierarchy may not be connected and the CPP can not be applied until it is converted to a directly connected network. A procedure is presented for selecting arcs from another hierarchy to include in the unconnected hierarchy in order to make it connected.

New lower bounds for the Capacitated Arc Routing Problem

AbstractThe Capacitated Arc Routing Problem (CARP), a generalization of the arc routing problems such as the Chinese Postman Problem (CPP), is a very practical distribution management problem which has many real‐world applications. This paper briefly reviews two existing lower bounding procedures—the Matching Lower Bound and the Node Scanning Lower Bound, then introduces a new bounding technique to provide tighter lower bounds on the problem solutions.

A new algorithm for the directed chinese postman problem

The directed Chinese Postman Problem can be transformed into a minimum cost flow problem over a derived network, or be transformed into a transportation problem in which the coefficients are determined by a shortest path problem over an extended network. Based on the Complementary Slackness Theorem, this paper will present an algorithm for the directed Chinese Postman Problem, which can be applied directly to the original network and, compared with the existing algorithms, has a better computational complexity O( kn 2) where k depends on the structure of the network. The algorithm is extended to the directed m-CPP case.

On Four Problems in Graph Theory

The four problems considered are: the Chinese postman problem, the co-postman problem, the odd cut problem, and the odd circuit problem. Relationships are developed between these problems using results from dual matroids and blocking clutters. Connections with Gomory’s group problem are shown. The notion of representations of these binary group problems on augmented graphs is developed along with a discussion of the class of augmented graphs having the same solution set. After some blocking and duality results, we give forbidden augmented minors for problems of one type (e.g., Chinese postman) to be also a second type of problem (e.g., odd cut). Some results are given on b-regular problems and are used in the forbidden augmented minor characterizations.

Postman tour on a graph with precedence relation on arcs

AbstractSince the introduction of the Chinese Postman Problem (CPP), many variations on the same theme have been developed. In this paper we examine still another variation. The arcs of the graph are partitioned and a precedence relation defined, specifying the order in which the elements of the partition have to be traversed. We first examine the conditions for a feasible solution to the problem. Next, we specify the graph properties of the precedence partition that insure a polynomial complexity solution of O(N5), where N is the number of nodes in the original graph. When the precedence relation on sets of arcs is general, we prove that the problem of finding the minimum length of feasible postman tour is NP‐complete.