On minimization of the number of branches in branch-and-bound algorithms for the maximum clique problem

WeGotYouCovered: The Winning Solver from the PACE 2019 Challenge, Vertex Cover Track

“…We now explain the main configuration that we use in our experimental setup. In the following, MoMC runs the MoMC clique solver by Li et al [28] on the complement of the input graph; RMoMC applies reductions to the input graph exhaustively, and then runs MoMC on the complement of the resulting kernel; LSBnR applies reductions exhaustively, then runs local search to obtain a high-quality solution on the kernel which is used as a initial bound in the branchand-reduce algorithm that is run on the kernel; BnR applies reductions and then runs the branch-and-reduce algorithm on the kernel (no local search is used to improve an initial bound); FullA is the full algorithm as described in the previous section, using a short time limit of one second and a long time limit of thirty seconds. Tables 1 and 2 give an overview of the instances that each of the solver solved, including the kernel size, and the minimum vertex cover size for those instances solved by any of the four algorithms.…”

“…The long-standing canonical algorithms for finding the maximum clique are the MCS algorithm by Tomita et al [39] and the bit-parallel algorithms of San Segundo et al [32,33]. However, recently Li et al [28] introduced the MoMC algorithm, which uses incremental MaxSAT logic to achieve speed ups of up to 1 000 over MCS. Experiments by Batsyn et al [4] show that MCS can be sped up significantly by giving an initial solution found through local search.…”

“…To win the PACE 2019 Implementation Challenge, we deployed a portfolio of solvers, using techniques from the literature on all three problems. These include data reduction rules and branch-and-reduce for the minimum vertex cover problem [2], iterated local search for the maximum independent set problem [3], and a state-of-the-art branch-andbound maximum clique solver [28].…”

Robustness to spontaneous emission of a variational quantum algorithm

“…When ν 1 (below the percolation threshold [28]), the UD-graph defined by the atoms can be partitioned in many subgraphs and finding the MIS is easy, as it amounts to summing up the contributions of all the smaller subgraphs. In that regime, there exist efficient classical algorithms that can find the MIS in polynomial time [29]. In the opposite regime, ν 1, all the vertices are connected and we trivially have M IS = 1.…”

Testing a Heuristic Algorithm for Finding a Maximum Clique on DIMACS and Facebook Graphs

“…The exact algorithms such as the MCQ algorithm [3], MCR [4], MCS [5], Max-CliqueDyn [6], MaxCliquePara (MCP) [7], BBmax-Clique [8], FastMaxClique (FMC) [9], parallel maximum clique (PMC) algorithm [10], an implementation of a MotzkinStraus-based iterative clique-finding algorithm for GPUs [11], among many others, have proved their applicability in solving MCP. One of the currently best exact approaches was proposed in the paper [12].…”