Robust minimum cost flow problem under consistent flow constraints

The robust minimum cost flow problem under consistent flow constraints (RobMCF$$\equiv $$ ≡ ) is a new extension of the minimum cost flow (MCF) problem. In the RobMCF$$\equiv $$ ≡ problem, we consider demand and supply that are subject to uncertainty. For all demand realizations, however, we require that the flow value on an arc needs to be equal if it is included in the predetermined arc set given. The objective is to find feasible flows that satisfy the equal flow requirements while minimizing the maximum occurring cost among all demand realizations. In the case of a finite discrete set of scenarios, we derive structural results which point out the differences with the polynomial time solvable MCF problem in networks with integral demands, supplies, and capacities. In particular, the Integral Flow Theorem of Dantzig and Fulkerson does not hold. For this reason, we require integral flows in the entire paper. We show that the RobMCF$$\equiv $$ ≡ problem is strongly $$\mathcal {NP}$$ NP -hard on acyclic digraphs by a reduction from the (3, B2)-Sat problem. Further, we demonstrate that the RobMCF$$\equiv $$ ≡ problem is weakly $$\mathcal {NP}$$ NP -hard on series-parallel digraphs by providing a reduction from Partition. If in addition the number of scenarios is constant, we propose a pseudo-polynomial algorithm based on dynamic programming. Finally, we present a special case on series-parallel digraphs for which we can solve the RobMCF$$\equiv $$ ≡ problem in polynomial time.

Matchtigs: minimum plain text representation of kmer sets

Kmer-based methods are widely used in bioinformatics, which raises the question of what is the smallest practically usable representation (i.e. plain text) of a set of kmers. We propose a polynomial algorithm computing a minimum such representation (which was previously posed as a potentially NP-hard open problem), as well as an efficient near-minimum greedy heuristic. When compressing genomes of large model organisms, read sets thereof or bacterial pangenomes, with only a minor runtime increase, we decrease the size of the representation by up to 60% over unitigs and 27% over previous work. Additionally, the number of strings is decreased by up to 97% over unitigs and 91% over previous work. Finally we show that a small representation has advantages in downstream applications, as it speeds up queries on the popular kmer indexing tool Bifrost by 1.66x over unitigs and 1.29x over previous work.

Polynomial and Pseudopolynomial Procedures for Solving Interval Two-Sided (Max, Plus)-Linear Systems

Max-plus algebra is the similarity of the classical linear algebra with two binary operations, maximum and addition. The notation Ax = Bx, where A, B are given (interval) matrices, represents (interval) two-sided (max, plus)-linear system. For the solvability of Ax = Bx, there are some pseudopolynomial algorithms, but a polynomial algorithm is still waiting for an appearance. The paper deals with the analysis of solvability of two-sided (max, plus)-linear equations with inexact (interval) data. The purpose of the paper is to get efficient necessary and sufficient conditions for solvability of the interval systems using the property of the solution set of the non-interval system Ax = Bx. The main contribution of the paper is a transformation of weak versions of solvability to either subeigenvector problems or to non-interval two-sided (max, plus)-linear systems and obtaining the equivalent polynomially checked conditions for the strong versions of solvability.

Development of doctrinal model for state financial security management and forecasting its level

A doctrinal model of state financial security management in the context of globalization changes has been developed. The model is formed at five levels (doctrinal, conceptual, strategic, programmatic, planned), contains a logical continuum of mission, priorities in the financial sector and the level of technological innovation, influencing factors and a system of actions aimed at achieving goals. This model accumulates a set of solutions aimed at adapting to transformational processes in the economy associated with new needs of states, globalization processes in the world financial space, technology development, new challenges and threats. As a result of the study, forecasting is carried out and the effectiveness of the results of modifying approaches to managing the financial security of the state using a polynomial algorithm for extrapolating the parameters of stochastic systems is proved. A polynomial correlation-regression model is presented, the input data of which were specific indicators of the effectiveness of innovative development of the state, perception of corruption and debt dependence. In fact, this is a set of those indicators at which the strategic directions of strengthening the financial security of the state are directed in the context of globalization changes. The generalized values of the state of financial security of the state, determined on the basis of the developed polynomial correlation-regression model, are obtained, as well as the absolute and relative amounts of error indicate the accuracy of the forecasts obtained. So, the mean level of error is 0.005 %, which means that the totality of these indicators can characterize the state of financial security of the state. Accordingly, this model is useful in the process of predicting the results of modifying approaches to the formation of the financial security of the state

Satisfiability in Boolean Logic (SAT problem) is polynomial?

We find a polynomial algorithm to solve SAT problem in Boolean Logic

Flow Increment through Network Expansion

The network expansion problem is a very important practical optimization problem when there is a need to increment the flow through an existing network of transportation, electricity, water, gas, etc. In this problem, the flow augmentation can be achieved either by increasing the capacities on the existing arcs, or by adding new arcs to the network. Both operations are coming with an expansion cost. In this paper, the problem of finding the minimum network expansion cost so that the modified network can transport a given amount of flow from the source node to the sink node is studied. A strongly polynomial algorithm is deduced to solve the problem.

LIST COLORING OF BLOCK GRAPHS AND COMPLETE BIPARTITE GRAPHS

List coloring is a vertex coloring of a graph where each vertex can be restricted to a list of allowed colors. For a given graph G and a set L(v) of colors for every vertex v, a list coloring is a function that maps every vertex v to a color in the list L(v) such that no two adjacent vertices receive the same color. It was first studied in the 1970s in independent papers by Vizing and by Erdős, Rubin, and Taylor. A block graph is a type of undirected graph in which every biconnected component (block) is a clique. A complete bipartite graph is a bipartite graph with partitions V 1, V 2 such that for every two vertices v_1∈V_1 and v_2∈V_2 there is an edge (v 1, v 2). If |V_1 |=n and |V_2 |=m it is denoted by K_(n,m). In this paper we provide a polynomial algorithm for finding a list coloring of block graphs and prove that the problem of finding a list coloring of K_(n,m) is NP-complete even if for each vertex v the length of the list is not greater than 3 (|L(v)|≤3).

Using linear algebra in decomposition of Farkas interpolants

The use of propositional logic and systems of linear inequalities over reals is a common means to model software for formal verification. Craig interpolants constitute a central building block in this setting for over-approximating reachable states, e.g. as candidates for inductive loop invariants. Interpolants for a linear system can be efficiently computed from a Simplex refutation by applying the Farkas’ lemma. However, these interpolants do not always suit the verification task—in the worst case, they can even prevent the verification algorithm from converging. This work introduces the decomposed interpolants, a fundamental extension of the Farkas interpolants, obtained by identifying and separating independent components from the interpolant structure, using methods from linear algebra. We also present an efficient polynomial algorithm to compute decomposed interpolants and analyse its properties. We experimentally show that the use of decomposed interpolants in model checking results in immediate convergence on instances where state-of-the-art approaches diverge. Moreover, since being based on the efficient Simplex method, the approach is very competitive in general.

INTERVAL EDGE-COLORINGS OF TREES WITH RESTRICTIONS ON THE EDGES

An edge-coloring of a graph $G$ with consecutive integers $c_1,\ldots,c_t$ is called an interval t-coloring, if all colors are used, and the colors of edges incident to any vertex of $G$ are distinct and form an interval of integers. A graph $G$ is interval colorable, if it has an interval t-coloring for some positive integer $t$. In this paper, we consider the case, where there are restrictions on the edges of the tree and provide a polynomial algorithm for checking interval colorability that satisfies those restrictions.