The maximum clique problem (MCP) is a fundamental problem in combinatorial optimization which finds important applications in real-word. This paper describes two new efficient branch-and-bound maximum clique algorithms NK-MaxClique and MMCQ, designed for solving MCP. We define some pruning conditions based on core numbers and vertex ordering to efficiently remove many of the search space. With respect to this ordering, the algorithms consider the vertices respectively to find the corresponding maximum clique in subproblems. Simulation results demonstrate that the algorithms outperform the previous wellknown algorithms for many instances when applied to DIMACS benchmark and random graphs.

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Euclidean Jordan algebra is a commonly used tool in designing interior-point algorithms for symmetric cone programs. In this paper, we present a full NesterovTodd (NT) step infeasible interior-point algorithm for horizontal linear complementarity problems over Cartesian product of symmetric cones. Since the algorithm uses only full-NT feasibility and centring steps, it has the advantage that no line searches are needed. The complexity result obtained here for symmetric cones using NT directions coincides with the best bound obtained for horizontal linear complementarity problems.