Multi-Objective Combinatorial Optimization (MOCO) is fundamental to the development and optimization of software systems. We propose five novel parallel algorithms for solving MOCO problems exactly and efficiently. Our algorithms rely on off-the-shelf solvers to search for exact Pareto-optimal solutions, and they parallelize the search via collaborative communication, divide-and-conquer, or both. We demonstrate the feasibility and performance of our algorithms by experiments on three case studies of software-system designs. A key finding is that one algorithm, which we call FS-GIA, achieves substantial (even super-linear) speedups that scale well up to 64 cores. Furthermore, we analyze the performance bottlenecks and opportunities of our parallel algorithms, which facilitates further research on exact, parallel MOCO.

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We present a method and an associated system, called Math-Check, that embeds the functionality of a computer algebra system (CAS) within the inner loop of a conflict-driven clause-learning SAT solver. SAT+CAS systems, a la MathCheck, can be used as an assistant by mathematicians to either counterexample or finitely verify open universal conjectures on any mathematical topic (e.g., graph and number theory, algebra, geometry, etc.) supported by the underlying CAS system. Such a SAT+CAS system combines the efficient search routines of modern SAT solvers, with the expressive power of CAS, thus complementing both. The key insight behind the power of the SAT+CAS combination is that the CAS system can help cut down the search-space of the SAT solver, by providing learned clauses that encode theory-specific lemmas, as it searches for a counterexample to the input conjecture (just like the T in DPLL(T)). In addition, the combination enables a more efficient encoding of problems than a pure Boolean representation. In this paper, we leverage the graph-theoretic capabilities of an opensource CAS, called SAGE. As case studies, we look at two long-standing open mathematical conjectures from graph theory regarding properties of hypercubes: the first conjecture states that any matching of any ddimensional hypercube can be extended to a Hamiltonian cycle; and the second states that given an edge-antipodal coloring of a hypercube, there always exists a monochromatic path between two antipodal vertices. Previous results have shown the conjectures true up to certain low-dimensional hypercubes, and attempts to extend them have failed until now. Using our SAT+CAS system, MathCheck, we extend these two conjectures to higher-dimensional hypercubes. We provide detailed performance analysis and show an exponential reduction in search space via the SAT+CAS combination relative to finite brute-force search.

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