Integer Flow Research Articles

Integer flows on triangularly connected signed graphs

AbstractA triangle‐path in a graph is a sequence of distinct triangles in such that for any with , and if . A connected graph is triangularly connected if for any two nonparallel edges and there is a triangle‐path such that and . For ordinary graphs, Fan et al. characterize all triangularly connected graphs that admit nowhere‐zero 3‐flows or 4‐flows. Corollaries of this result include the integer flow of some families of ordinary graphs, such as locally connected graphs due to Lai and some types of products of graphs due to Imrich et al. In this paper, Fan's result for triangularly connected graphs is further extended to signed graphs. We proved that a flow‐admissible triangularly connected signed graph admits a nowhere‐zero 4‐flow if and only if it is not the wheel associated with a specific signature. Moreover, this result is sharp since there are infinitely many unbalanced triangularly connected signed graphs admitting a nowhere‐zero 4‐flow but no 3‐flow.

Block-based state-expanded network models for multi-activity shift scheduling

This paper presents new mixed-integer linear programming formulations for multi-activity shift scheduling problems (MASSP). In these formulations, the rules governing shift feasibility are encoded in block-based state-expanded networks in which nodes are associated with states and arcs represent assignments of blocks of work or break periods inducing state transitions. A key advantage of these formulations is that for the anonymous MASSP in which all employees are considered as equal only a single network with integer flow variables is needed as long as the network encodes all shift composition rules. A challenging aspect is that the networks can become very large, yielding huge models that are hard to solve for large problem instances. To address this challenge, this paper proposes two exact modeling techniques that substantially reduce the size of the model instances: First, it introduces a set of aggregate side constraints enforcing that an integer flow solution can be decomposed into paths representing feasible shifts. Second, it proposes to decouple the shift composition from the assignment of concrete activities to blocks of work periods, thereby removing a large amount of symmetry from the original model. In a computational study with two MASSP instance sets from the literature dealing with shift scheduling problems, we demonstrate the effectiveness of these techniques for reducing the both size of the model instances and the solution time: We are able to solve all instances, including more than 70 previously open instances, to optimality–the vast majority of them in less than 30 min on a notebook computer.

Lattice of integer flows and the poset of strongly connected orientations for regular matroids

A 2010 result of Amini provides a way to extract information about the structure of the graph from the geometry of the Voronoi polytope of the lattice of integer flows (which determines the graph up to two-isomorphism). Specifically, Amini shows that the face poset of the Voronoi polytope is isomorphic to the poset of strongly connected orientations of subgraphs. This answers a question raised by Caporaso and Viviani, and Amini also proves a dual result for integer cuts. In this paper we generalize Amini's result to regular matroids; in this context the theorem for integer cuts becomes a direct consequence of the theorem for integer flows, by making duality explicit as matroid duality.

Characterizing the radius of influence during pumping tests using the absolute critical drawdown criterion: Cases of integer flow dimensions

Determining the radius of influence r0 of wells during pumping tests is critical for the characterization of aquifers and the management of groundwater. However, because a convenient analytical interpretative framework is lacking, this is a very difficult task during routine investigations. Practicing hydrogeologists have resorted to using semi-empirical equations developed by certain authors. Most studies aiming to characterize the radius of influence are based on radial flow models. In this study, we propose to investigate, from an analytical standpoint, the radius of influence equation for integer flow dimensions (n=1,2,3), using both Barker’s generalized radial flow model and Theis’ radial flow model. The current approach may thus be considered valid for those hydrogeological contexts (fractured or granular aquifer media) that produce the specified flow dimension. The radius of influence is defined as the maximum distance from the pumping well at which the drawdown reaches its critical value of detectability: the absolute critical drawdown criterion sc. The radius of influence is a theoretical, non-intrinsic and variable parameter that reflects the ability of drawdown recording systems to measure very small variations. Our investigations show that the radius of influence equation can be generalized as follows: r0=Ctγ where the coefficients C and γ depend not only on the flow dimension parameter n, but also on the criterion sc, the pumping flow rate Q, the hydraulic conductivity K and the aquifer thickness b. The specificities of the radius of influence equation for each flow dimension are also discussed. Finally, results obtained from this analytical approach are verified against numerical simulations.

Finding all minimum cost flows and a faster algorithm for the K best flow problem

This paper addresses the problem of determining all optimal integer solutions of a linear integer network flow problem, which we call the all optimal integer flow (AOF) problem. We derive an O ( F ( m + n ) + m n + M ) time algorithm to determine all F many optimal integer flows in a directed network with n nodes and m arcs, where M is the best time needed to find one minimum cost flow. We remark that stopping Hamacher’s well-known method for the determination of the K best integer flows (Hamacher, 1995) at the first sub-optimal flow results in an algorithm with a running time of O ( F m ( n log n + m ) + M ) for solving the AOF problem. Our improvement is essentially made possible by replacing the shortest path sub-problem with a more efficient way to determine a so-called proper zero cost cycle using a modified depth-first search technique. As a byproduct, our analysis yields an enhanced algorithm to determine the K best integer flows that runs in O ( K n 3 + M ) . Besides, we give lower and upper bounds for the number of all optimal integer and feasible integer solutions. Our bounds are based on the fact that any optimal solution can be obtained by an initial optimal tree solution plus a conical combination of incidence vectors of all induced cycles with bounded coefficients.

Novel direct algorithm for computing simultaneous all-level reliability of multistate flow networks

Different types of networks, such as, Internet of Things, social networks, wireless sensor networks, transportation networks, and 4g/5G serve to benefit and help our daily lives. The multistate flow network (MFN) is used to model the network structures and applications. The level d reliability, Rd, of the MFN is the success probability of sending at least d units of integer flow from the nodes 1 to n and is a popular index for designing, managing, controlling, and evaluating MFNs. The traditional indirect algorithms must have all d-MPs (special connected vectors) or (d-1)-MCs (special disconnected vectors) first, and then use inclusion-exclusion technique (IET) or sum-of-disjoint product (SDP) in terms of found d-MPs or (d-1)-MCs to calculate Rd. The above four procedures are all NP-Hard and #P-Hard and cannot calculate Rd for d = 1, 2, …, dMAX simultaneously, that is, they can only calculate R1, R2, …, and RdMAX sequentially. Thus, in this study a novel algorithm is proposed to calculate the Rd directly for all d simultaneously, eliminating the need of using the above four procedures. The time complexity and demonstration of the proposed algorithm were analyzed with suitable examples. Furthermore, an experiment was conducted on 12 benchmark networks to validate the proposed algorithm.

Koszul algebras and flow lattices

We provide a homological algebraic realization of the lattices of integer cuts and integer flows of graphs. To a finite 2-edge-connected graph Γ with a spanning tree T, we associate a finite dimensional Koszul algebra AΓ,T. Under the construction, planar dual graphs with dual spanning trees are associated Koszul dual algebras. The Grothendieck group of the category of finitely-generated AΓ,T modules is isomorphic to the Euclidean lattice ZE(Γ), and we describe the sublattices of integer cuts and integer flows on Γ in terms of the representation theory of AΓ,T. The grading on AΓ,T gives rise to q-analogs of the lattices of integer cuts and flows; these q-lattices depend non-trivially on the choice of spanning tree. We give a q-analog of the matrix-tree theorem, and prove that the q-flow lattice of (Γ1,T1) is isomorphic to the q-flow lattice of (Γ2,T2) if and only if there is a cycle preserving bijection from the edges of Γ1 to the edges of Γ2 taking the spanning tree T1 to the spanning tree T2. This gives a q-analog of a classical theorem of Caporaso-Viviani and Su-Wagner.

Rotation snark, Berge-Fulkerson conjecture and Catlin’s 4-flow reduction

It is conjectured by Berge and Fulkerson that every bridgeless cubic graph has six perfect matchings such that each edge is contained in exactly two of them. An infinite family R, of cyclically 5-edge-connected rotation snarks, was discovered in [European J. Combin. 2021] by Máčajová and Škoviera. In this paper, the Berge-Fulkerson conjecture is verified for the family R, and furthermore, a sup-family of R. Catlin’s contractible configuration and Tutte’s integer flow are applied here as the key methods.

Berge–Fulkerson coloring for some families of superposition snarks

It is conjectured by Berge and Fulkerson that every bridgeless cubic graph has six perfect matchings such that each edge is contained in exactly two of them. This conjecture has been verified for many families of snarks with small (≤5) cyclic edge-connectivity. An infinite family, denoted by SK, of cyclically 6-edge-connected superposition snarks was constructed in [European J. Combin. 2002] by Kochol. In this paper, the Berge–Fulkerson conjecture is verified for the family SK, and, furthermore, some larger families containing SK. This is the first paper about the Berge–Fulkerson conjecture for superposition snarks and cyclically 6-edge-connected snarks. Tutte’s integer flow and Catlin’s contractible configuration are applied here as the key methods.

A Branch-and-Bound Algorithm for Building Optimal Data Gathering Tree in Wireless Sensor Networks

In this paper, we consider the maximum lifetime data gathering tree (MLDT) problem in sensor networks. A data gathering tree is a spanning tree rooted at a specified sink so that every node can send its messages to the sink along the tree. The lifetime of a tree is defined as the minimum lifetime among nodes where each node’s lifetime is determined by its initial energy and transmission load. The MLDT problem is NP-hard, and the state-of-the-art solution formulates a decision version of the problem as an integer linear program (ILP) and then solves it by conducting binary search over all possible lifetimes. In this paper, we first give an ILP for the optimization problem rather than its decision version, and then show that using ILP solvers to solve these programs could be highly inefficient. We then propose a branch-and-bound algorithm that incorporates two novel features. First, the bounding method takes into account integer flows, and contains a new set of constraints. Second, a special set of edges are deleted to reduce the number of subproblems generated by the branching process. Numerical simulations on randomly generated networks show that the proposed algorithm outperforms existing algorithms in terms of the number of solved problem instances in a fixed amount of time. Summary of Contribution: We study the maximum lifetime data gathering tree (MLDT) problem in the context of wireless sensor network. MLDT is a fundamental problem in both computer science and operations research. Since sensor nodes are often resource limited, the data gathering tree must be carefully constructed to prolong the network lifetime. In this paper, we first give an integer linear program for the optimization problem rather than its decision version, and then show that using ILP solvers to solve these programs could be highly inefficient. We then propose a branch and bound algorithm that incorporates two novel features.

Solving Robust Variants of Integer Flow Problems with Uncertain Arc Capacities

This paper deals with robust optimization and network flows. Several robust variants of integer flow problems are considered. They assume uncertainty of network arc capacities as well as of arc unit costs (where applicable). Uncertainty is expressed by discrete scenarios. Since the considered variants of the maximum flow problem are easy to solve, the paper is mostly concerned with NP-hard variants of the minimum-cost flow problem, thus proposing an approximate algorithm for their solution. The accuracy of the proposed algorithm is verified by experiments.

Integer Flows and Modulo Orientations of Signed Graphs

This paper studies the fundamental relations among integer flows, modulo orientations, integer-valued and real-valued circular flows, and monotonicity of flows in signed graphs. A (signed) graph is...

State-Expanded Network Models for Multi-Activity Shift Scheduling

This paper presents new formulations for multi-activity shift scheduling problems (MASSP). In these formulations, the rules governing the feasibility of shifts are encoded in state-expanded networks in which nodes are associated with states and arcs represent assignments inducing state transitions. These networks are embedded into a mixed-integer linear programming model as network flow components. A key advantage of the formulation is that for the anonymous MASSP in which all employees are considered as equal only a single network with integer flow variables is needed as long as the network encodes all shift composition rules. A challenging aspect of this type of formulation is that the networks can become very large, yielding huge model instances that are hard to solve for large problem instances. In order to address this issue, this paper proposes two exact techniques: First, it introduces a set of aggregate side constraints implicitly enforcing that an integer flow solution can be decomposed into feasible shifts. Second, it proposes to decouple the shift composition from the assignment of concrete activities to blocks of work periods. This results in two model components, both based on state-expanded networks, that are coupled to each other using linking constraints. In a computational study with two sets of MASSP instances from the literature, we demonstrate the effectiveness of these techniques for reducing the both size of the model instances and the computation time: We are able to solve all instances, including more that 70 previously open problem instances, to optimality -- the vast majority of them in less than 30 minutes on a notebook computer.

A unified approach to construct snarks with circular flow number 5

AbstractThe well‐known 5‐flow Conjecture of Tutte, stated originally for integer flows, claims that every bridgeless graph has circular flow number at most 5. It is a classical result that the study of the 5‐flow Conjecture can be reduced to cubic graphs, in particular to snarks. However, very few procedures to construct snarks with circular flow number 5 are known. In the first part of this paper, we summarise some of these methods and we propose new ones based on variations of the known constructions. Afterwards, we prove that all such methods are nothing but particular instances of a more general construction that we introduce into detail. In the second part, we consider many instances of this general method and we determine when our method permits to obtain a snark with circular flow number 5. Finally, by a computer search, we determine all snarks having circular flow number 5 up to 36 vertices. It turns out that all such snarks of order at most 34 can be obtained by using our method, and that the same holds for 96 of the 98 snarks of order 36 with circular flow number 5.

Avoiding unnecessary demerging and remerging of multi‐commodity integer flows

AbstractResource flows may merge and demerge at a network node. Sometimes several demerged flows may be immediately merged again, but in different combinations compared to before they were demerged. However, the demerging is unnecessary in the first place if the total resources at each of the network nodes involved remains unchanged. We describe this situation as “unnecessary demerging and remerging (UDR)” of flows, which would incur unnecessary operations and costs in practice. Multi‐commodity integer flows in particular will be considered in this paper. This deficiency could be theoretically overcome by means of fixed‐charge variables, but the practicality of this approach is restricted by the difficulty in solving the corresponding integer linear program (ILP). Moreover, in a problem where the objective function has many cost elements, it would be helpful if such operational costs are optimized implicitly. This paper presents a heuristic branching method within an ILP solver for removing UDR without the use of fixed‐charge variables. We use the concept of “flow potentials” (different from “flow residuals” for max‐flows) guided by which underutilized arcs are heuristically banned, thus reducing occurrences of UDR. Flow connection bigraphs and flow connection groups (FCGs) are introduced. We prove that if certain conditions are met, fully utilizing an arc will guarantee an improvement within an FCG. Moreover, a location sub‐model is given when the former cannot guarantee an improvement. More importantly, the heuristic approach can significantly enhance the full fixed‐charge model by warm‐starting. Computational experiments based on real‐world instances have shown the usefulness of the proposed methods.

Heuristic solutions to robust variants of the minimum-cost integer flow problem

This paper deals with robust optimization applied to network flows. Two robust variants of the minimum-cost integer flow problem are considered. Thereby, uncertainty in problem formulation is limited to arc unit costs and expressed by a finite set of explicitly given scenarios. It is shown that both problem variants are NP-hard. To solve the considered variants, several heuristics based on local search or evolutionary computing are proposed. The heuristics are experimentally evaluated on appropriate problem instances.

Probabilistic tree-based representation for solving minimum cost integer flow problems with nonlinear non-convex cost functions

The minimum cost flow problem (MCFP) is the most generic variation of the network flow problem which aims to transfer a commodity throughout the network to satisfy demands. The problem size (in terms of the number of nodes and arcs) and the shape of the cost function are the most critical factors when considering MCFPs. Existing mathematical programming techniques often assume the cost functions to be linear or convex. Unfortunately, the linearity and convexity assumptions are too restrictive for modelling many real-world scenarios. In addition, many real-world MCFPs are large-scale, with networks having a large number of nodes and arcs. In this paper, we propose a probabilistic tree-based genetic algorithm (PTbGA) for solving large-scale minimum cost integer flow problems with nonlinear non-convex cost functions. We first compare this probabilistic tree-based representation scheme with the priority-based representation scheme, which is the most commonly-used representation for solving MCFPs. We then compare the performance of PTbGA with that of the priority-based genetic algorithm (PrGA), and two state-of-the-art mathematical solvers on a set of MCFP instances. Our experimental results demonstrate the superiority and efficiency of PTbGA in dealing with large-sized MCFPs, as compared to the PrGA method and the mathematical solvers.

Integrated train timetabling and locomotive assignment

Train timetabling and locomotive assignment are often performed separately in a sequential manner. One obvious disadvantage of such hierarchical planning process is that it often results in poor coordination between the train schedule and the locomotive schedule. This paper focuses on modeling and solving an integrated train timetabling and locomotive assignment problem. To solve this integrated problem, we first construct a three-dimensional state-space-time network in which a state is used to indicate which train a locomotive is serving. We then formulate the problem as a minimum cost multi-commodity network flow problem with incompatible arcs and integer flow restrictions. We present a Lagrangian relaxation heuristic for solving this network flow problem. We conduct a computational study to test the effectiveness of our Lagrangian relaxation heuristic, compare the performance of our heuristic with that of two benchmark solution methods, and report the benefits obtained by integrating train timetabling and locomotive assignment decisions.

Shortest circuit covers of signed graphs

A shortest circuit cover F of a bridgeless graph G is a family of circuits that covers every edge of G and is of minimum total length. The total length of a shortest circuit cover F of G is denoted by SCC(G). For ordinary graphs (graphs without sign), the subject of shortest circuit cover is closely related to some mainstream areas, such as, Tutte's integer flow theory, circuit double cover conjecture, Fulkerson conjecture, and others. For signed graphs G, it is proved recently by Máčajová, Raspaud, Rollová and Škoviera that SCC(G)≤11|E| if G is s-bridgeless, and SCC(G)≤9|E| if G is 2-edge-connected. In this paper this result is improved as follows,SCC(G)≤|E|+3|V|+z where z=min⁡{23|E|+43ϵN−7,|V|+2ϵN−8} and ϵN is the negativeness of G. The above upper bound can be further reduced if G is 2-edge-connected with even negativeness.

Modulo orientations with bounded independence number

A mod (2p+1)-orientation D is an orientation of G such that dD+(v)≡dD−(v)(mod2p+1) for any vertex v∈V(G). Extending Tutte’s integer flow conjectures, it was conjectured by Jaeger that every 4p-edge-connected graph has a mod (2p+1)-orientation. However, this conjecture has been disproved in Han et al. (2018) recently. Infinite families of 4p-edge-connected graphs (for p≥3) and (4p+1)-edge-connected graphs (for p≥5) with no mod (2p+1)-orientation are constructed in Han et al. (2018). In this paper, we show that every family of graphs with bounded independence number has only finitely many contraction obstacles for admitting mod (2p+1)-orientations, contrasting to those infinite families. More precisely, we prove that for any integer t≥2, there exists a finite family F=F(p,t) of graphs that do not have a mod (2p+1)-orientation, such that every graph G with independence number at most t either admits a mod (2p+1)-orientation or is contractible to a member in F. This indicates that the problem of determining whether every k-edge-connected graph with independence number at most t admits a mod (2p+1)-orientation is computationally solvable for fixed k and t. In particular, the graph family F(p,2) is determined, and our results imply that every 8-edge-connected graph G with independence number at most two admits a mod 5-orientation.