Alternating direction method of multipliers (ADMM) receives much attention in the field of optimization and computer science, etc. The generalized ADMM (G-ADMM) proposed by Eckstein and Bertsekas incorporates an acceleration factor and is more efficient than the original ADMM. However, G-ADMM is not applicable in some models where the objective function value (or its gradient) is computationally costly or even impossible to compute. In this paper, we consider the two-block separable convex optimization problem with linear constraints, where only noisy estimations of the gradient of the objective function are accessible. Under this setting, we propose a stochastic linearized generalized ADMM (called SLG-ADMM) where two subproblems are approximated by some linearization strategies. And in theory, we analyze the expected convergence rates and large deviation properties of SLG-ADMM. In particular, we show that the worst-case expected convergence rates of SLG-ADMM are $\mathcal{O}\left( {{N}^{-1/2}}\right)$ and $\mathcal{O}\left({\ln N} \cdot {N}^{-1}\right)$ for solving general convex and strongly convex problems, respectively, where N is the iteration number, similarly hereinafter, and with high probability, SLG-ADMM has $\mathcal{O}\left ( \ln N \cdot N^{-1/2} \right ) $ and $\mathcal{O}\left ( \left ( \ln N \right )^{2} \cdot N^{-1} \right ) $ constraint violation bounds and objective error bounds for general convex and strongly convex problems, respectively.

Given a graph G, the minimum Connected-k-Subgraph Cover problem (MinCkSC) is to find a minimum vertex subset C of G such that every connected subgraph of G on k vertices has at least one vertex in C. If furthermore the subgraph of G induced by C is connected, then the problem is denoted as MinCkSC$_{con}$. In this paper, we first present a PTAS for MinCkSC on an H-minor-free graph, where H is a graph with a constant number of vertices. Then, we design an $O((\omega+1)(2(k-1)(\omega+2))^{3\omega+3})|V|$-time FPT algorithm for MinCkSC$_{con}$ on a graph with treewidth $\omega$, based on which we further design an $O(2^{O(\sqrt{t}\log t)}|V|^{O(1)})$ time subexponential FPT algorithm for MinCkSC$_{con}$ on an H-minor-free graph, where t is an upper bound of solution size.

Let $T=(V,E)$ be a tree in which each edge is assigned a cost; let $\mathcal{P}$ be a set of source–sink pairs of vertices in V in which each source–sink pair produces a profit. Given a lower bound K for the profit, the K-prize-collecting multicut problem in trees with submodular penalties is to determine a partial multicut $M\subseteq E$ such that the total profit of the disconnected pairs after removing M from T is at least K, and the total cost of edges in M plus the penalty of the set of still-connected pairs is minimized, where the penalty is determined by a nondecreasing submodular function. Based on the primal-dual scheme, we present a combinatorial polynomial-time algorithm by carefully increasing the penalty. In the theoretical analysis, we prove that the approximation factor of the proposed algorithm is $(\frac{8}{3}+\frac{4}{3}\kappa+\varepsilon)$, where $\kappa$ is the total curvature of the submodular function and $\varepsilon$ is any fixed positive number. Experiments reveal that the objective value of the solutions generated by the proposed algorithm is less than 130% compared with that of the optimal value in most cases.

We consider a two-stage submodular maximization problem subject to a cardinality constraint and k matroid constraints, where the objective function is the expected difference of a nonnegative monotone submodular function and a nonnegative monotone modular function. We give two bi-factor approximation algorithms for this problem. The first is a deterministic $\left( {{1 \over {k + 1}}\left( {1 - {1 \over {{e^{k + 1}}}}} \right),1} \right)$-approximation algorithm, and the second is a randomized $\left( {{1 \over {k + 1}}\left( {1 - {1 \over {{e^{k + 1}}}}} \right) - \varepsilon ,1} \right)$-approximation algorithm with improved time efficiency.

In this paper, we analyze two types of refutations for Unit Two Variable Per Inequality (UTVPI) constraints. A UTVPI constraint is a linear inequality of the form: $a_{i}\cdot x_{i}+a_{j} \cdot x_{j} \le b_{k}$, where $a_{i},a_{j}\in \{0,1,-1\}$ and $b_{k} \in \mathbb{Z}$. A conjunction of such constraints is called a UTVPI constraint system (UCS) and can be represented in matrix form as: ${\bf A \cdot x \le b}$. UTVPI constraints are used in many domains including operations research and program verification. We focus on two variants of read-once refutation (ROR). An ROR is a refutation in which each constraint is used at most once. A literal-once refutation (LOR), a more restrictive form of ROR, is a refutation in which each literal ($x_i$ or $-x_i$) is used at most once. First, we examine the constraint-required read-once refutation (CROR) problem and the constraint-required literal-once refutation (CLOR) problem. In both of these problems, we are given a set of constraints that must be used in the refutation. RORs and LORs are incomplete since not every system of linear constraints is guaranteed to have such a refutation. This is still true even when we restrict ourselves to UCSs. In this paper, we provide NC reductions between the CROR and CLOR problems in UCSs and the minimum weight perfect matching problem. The reductions used in this paper assume a CREW PRAM model of parallel computation. As a result, the reductions establish that, from the perspective of parallel algorithms, the CROR and CLOR problems in UCSs are equivalent to matching. In particular, if an NC algorithm exists for either of these problems, then there is an NC algorithm for matching.

In this paper, the notion of locally algebraic intersection structure is introduced for algebraic L-domains. Essentially, every locally algebraic intersection structure is a family of sets, which forms an algebraic L-domain ordered by inclusion. It is shown that there is a locally algebraic intersection structure which is order-isomorphic to a given algebraic L-domain. This result extends the classic Stone’s representation theorem for Boolean algebras to the case of algebraic L-domains. In addition, it can be seen that many well-known representations of algebraic L-domains, such as logical algebras, information systems, closure spaces, and formal concept analysis, can be analyzed in the framework of locally algebraic intersection structures. Then, a set-theoretic uniformity across different representations of algebraic L-domains is established.