ABSTRACTModulo flow is a powerful tool in the study of flows in both ordinary graphs and signed graphs. For ordinary graphs, Tutte showed that a graph admits a nowhere‐zero ‐flow if and only if it admits a nowhere‐zero ‐flow. However, such equivalence does not hold any more for signed graphs. Mačajova and Škoviera [SIAM Journal of Discrete Mathematics 31 (2017) 1937–1952] proved that every flow‐admissible signed graph with a nowhere‐zero ‐flow admits a nowhere‐zero 4‐flow. DeVos et al. [Journal of Combinatorial Theory Series B 149 (2021) 198–221] proved that every flow‐admissible signed graph admits a nowhere‐zero 11‐flow by converting certain special nowhere‐zero ‐flows into integer flows, and as a key step, they showed that every signed graph with a nowhere‐zero ‐flow admits a nowhere‐zero 5‐flow. In this paper, we study how to convert ‐flows and ‐flows into integer flows by proving the following two results: (1) every flow‐admissible signed graph with a nowhere‐zero ‐flow admits a nowhere‐zero 8‐flow; (2) every bridgeless signed graph with a nowhere‐zero ‐flow admits a nowhere‐zero 7‐flow. Combining known results, it follows that every flow‐admissible signed graph with a nowhere‐zero ‐flow admits a nowhere‐zero ‐flow for each integer .

ABSTRACTIn this article, we extend the duality relation between face colorings and integer flows of graphs on orientable surfaces in Tutte's flow theory to both orientable and nonorientable surfaces and study Bouchet's 6‐flow conjecture from point of embeddings of graphs on surfaces. Consequently, we verify Bouchet's conjecture for a family of embedded graphs, which have a crosscap‐contractible circuit.

Path optimization is a fundamental concern across various real-world scenarios, ranging from traffic congestion issues to efficient data routing over the Internet. The traffic assignment problem (TAP) is a classic continuous optimization problem in this field. This study considers the integer traffic assignment problem (ITAP), a discrete variant of TAP. ITAP involves determining optimal routes for commuters in a city represented by a graph, aiming to minimize congestion while adhering to integer flow constraints on paths. This restriction makes ITAP an NP-hard problem. While conventional TAP prioritizes repulsive interactions to minimize congestion, this work also explores the case of attractive interactions, related to minimizing the number of occupied edges. We present and evaluate multiple algorithms to address ITAP, including a message passing algorithm, a greedy approach, simulated annealing, and relaxation of ITAP to TAP, including the message passing algorithm in Yeung, Saad, and Wong [Proc. Natl. Acad. Sci. USA 110, 13717 (2013)0027-842410.1073/pnas.1301111110], a greedy approach, simulated annealing, and relaxation of ITAP to TAP. Inspired by studies of random ensembles in the large-size limit in statistical physics, comparisons between these algorithms are conducted on large sparse random regular graphs with a random set of origin-destination pairs. Our results indicate that while the simplest greedy algorithm performs competitively in the repulsive scenario, in the attractive case the message-passing-based algorithm and simulated annealing demonstrate superiority. We then investigate the relationship between TAP and ITAP in the repulsive case. We find that, as the number of paths increases, the solution of TAP converges toward that of ITAP, and we investigate the speed of this convergence. Depending on the number of paths, our analysis leads us to identify two scaling regimes: In one the average flow per edge is of order one, and in another, the number of paths scales quadratically with the size of the graph, in which case the continuous relaxation solves the integer problem closely.

Abstract A 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.

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

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

Finding all minimum cost flows and a faster algorithm for the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e636" altimg="si10.svg"><mml:mi>K</mml:mi></mml:math> best flow problem

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

Koszul algebras and flow lattices

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

Berge–Fulkerson coloring for some families of superposition snarks

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.

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.

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

Abstract The 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.

Abstract Resource 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.

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