Abstract In this paper, we apply the Rank-Sparsity Matrix Decomposition to the planted Maximum Quasi-Clique Problem (MQCP). This problem has the planted Maximum Clique Problem (MCP) as a special case. The maximum clique problem is NP-hard. A Quasi-clique or $$\gamma $$ γ -clique is a dense graph with the edge density of at least $$\gamma $$ γ , $$\gamma \in (0, 1]$$ γ ∈ ( 0 , 1 ] . The maximum quasi-clique problem seeks to find such a subgraph with the largest cardinality in a given graph. Our method of choice is the low-rank plus sparse matrix splitting technique. We present a theoretical basis for when our convex relaxation problem recovers the planted maximum quasi-clique. We have derived a new bound on the norm of the dual matrix that certifies the recovery using $$l_{\infty , 2}$$ l ∞ , 2 norm. We have showed that when certain conditions are met, our convex formulation recovers the planted quasi-clique exactly. The numerical experiments we have performed corroborate our theoretical findings.

The minimum weight-dominating set (MWDS) problem is NP-hard and has a lot of applications in the real world. Several metaheuristic methods have been developed for solving the problem effectively, but suffering from high CPU time on large-scale instances. In this paper, we design an effective hybrid memetic algorithm (HMA) for the MWDS problem. First, the MWDS problem is formulated as a constrained 0-1 programming problem and is converted to an equivalent unconstrained 0-1 problem using an adaptive penalty function. Then, we develop a memetic algorithm for the resulting problem, which contains a greedy randomized adaptive construction procedure, a tabu local search procedure, a crossover operator, a population-updating method, and a path-relinking procedure. These strategies make a good tradeoff between intensification and diversification. A number of experiments were carried out on three types of instances from the literature. Compared with existing algorithms, HMA is able to find high-quality solutions in much less CPU time. Specifically, HMA is at least six times faster than existing algorithms on the tested instances. With increasing instance size, the CPU time required by HMA increases much more slowly than required by existing algorithms.

In this paper, we propose a “multistart-type” algorithm for solving the max-k-cut problem. Central to our algorithm is an auxiliary function we propose. We formulate the max-k-cut problem as an explicit mathematical form, which allows us to use an easy implementable local search. The construction of the auxiliary function requires a local maximizer of the max-k-cut problem. If the best local maximizer obtained is used in the construction of the auxiliary function, then the local maximization of the auxiliary function leads to a better maximizer of the max-k-cut problem. This proves to be a good strategy to escape from the current local optima and to search a broader solution space. Indeed, we have shown, both numerically and theoretically, that the maximization of the auxiliary function by the local search method can escape successfully from previously converged discrete local maximizers by taking increasing values of a parameter. Computational results on many test instances with different sizes and densities show that the proposed algorithm is efficient and stable to find approximate global solutions for the max-k-cut problems. Although we have presented results for k ≥ 2, the robustness of our algorithm is shown for k = 2 by comparisons with a number of recent methods. A number of theoretical results are also presented, which justify the design of our algorithm.

Floorplanning in very large scale integrated-circuit (VLSI) design is the first phase in the process of designing the physical layout of a chip. This makes the floorplanning problem of paramount importance, since it determines the performance, size, yield, and reliability of VLSI chips . From the computational point of view, the VLSI floorplanning is an NP-hard problem. In this paper, we present a hybrid simulated annealing algorithm (HSA) for nonslicing VLSI floorplanning. The HSA uses a new greedy method to construct an initial B*-tree, a new operation on the B*-tree to explore the search space, and a novel bias search strategy to balance global exploration and local exploitation. Experimental results on Microelectronic Center of North Carolina (MCNC) benchmarks show that the HSA can quickly produce optimal or nearly optimal solutions for all the tested problems.