Holant problems capture a class of Sum-of-Product computations such as counting matchings. It is inspired by holographic algorithms and is equivalent to tensor networks, with counting CSP being a special case. A complexity classification for Holant problems is more difficult to prove, not only because it logically implies a classification for counting CSP, but also due to the deeper reason that there exist more intricate polynomial time tractable problems in the broader framework. We discover a new family of constraint functions L which define polynomial time computable counting problems. These do not appear in counting CSP, and no newly discovered tractable constraints can be symmetric. This family has a delicate support structure related to error-correcting codes. Local holographic transformations are fundamental in its tractability. We prove a complexity dichotomy theorem for all Holant problems defined by any real valued constraint function set on Boolean variables and contains two 0-1 pinning functions. Previously, dichotomy for the same framework was only known for symmetric constraint functions. The set L supplies the last piece of tractability. We also prove a dichotomy for a variant of counting CSP as a technical component toward this Holant dichotomy.
Consider a discrete time infection process in a graph G=(V,E) where we start with a set of initially infected vertices S⊆V(G) at time τ=0 and, for τ≥1, a vertex v∈V(G) gets infected if and only if v has at least its threshold value t(v) of infected neighbors in the previous time step τ−1. The infection ends when no more vertices can get infected. We study a variant of the problem (k,l)-Influence Immunization Bounding [9], in which we have to decide if it is possible to immunize (remove) l vertices from G such that, at the end of the process, no more than k≥|S| vertices of G are infected. Most of our results are restricted to the case when k=|S|, which we call Total Restriction. We show that this problem is W[2]-hard parameterized by l even in split graphs and bipartite graphs, and is NP-complete even in bipartite planar graphs with maximum degree three. We show how to solve the total restriction cases in polynomial time if t(v)=1 for any v∈V(G)∖S or if G is a tree with any threshold function. If G is a complete graph, we show polynomial-time algorithms to compute a minimum set of vertices needed to immunize to restrict the infection to at most k vertices, for any k≥|S| and any threshold function t, both for our variant of the problem and for the original problem from [9] in which only vertices from V(G)∖S can be immunized.
We consider a language together with the subword relation, binary relations that relate the lengths of two words, and regular predicates. For such structures, we consider the extension of first-order logic by threshold and modulo counting quantifiers. Depending on the language, the used predicates, and the fragment of the logic, we determine four new combinations that yield decidable theories. These results extend earlier ones where only the language of all words without the length relations and fragments of first-order logic were considered. (c) 2026 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
In a multi-echelon supply chain, to mitigate stockout risks and holding costs, the optimal inventory strategy can be transformed into an optimization problem that maximizes the difference between two DR-submodular functions. Specifically, let x denote the inventory level at each node (an integer variable), where g(x) represents the revenue from inventory meeting demand (DR-submodular, i.e., the marginal revenue of inventory diminishes as the inventory level increases), and h(x) denotes inventory holding costs (also DR-submodular). Maximizing g-h thus effectively mitigates stockout risks and cuts down inventory holding costs. In this paper, we design two algorithms for the problem of maximizing the difference of two DR-submodular function g-h (DDRS) on the integer lattice, where g(x) is monotone and non-negative DR-submodular, and h(x) is monotone and non-negative submodular with curvature ch. We first propose a one-pass streaming algorithm for the unconstrained problem. On this basis, we analyze the algorithm's performance and prove that it provides verifiable approximation guarantees. Second, we design a threshold streaming algorithm for the cardinality-constrained problem. Finally, numerical experiments are conducted to verify the superiority of the proposed algorithms in terms of solution efficiency and approximation accuracy, further corroborating the effectiveness of the theoretical analysis and the practical application value of the algorithms. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
In this paper, we study the two-machine job shop scheduling problem with two competing agents. The goal is to minimize a given objective function for one agent, subject to a given upper bound of another objective function associated with the second agent. We prove the NP-hardness of this problem with respect to various criteria and settings. With respect to the makespan for both agents, we first exhibited polynomial time algorithms for several special cases. We then developed mathematical models, a branch and bound algorithm, and a genetic algorithm, followed by an intensive experimental simulation to study their performance. (c) 2026 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.
