Boolean lexicographic optimization: algorithms & applications

Abstract

Multi-Objective Combinatorial Optimization (MOCO) problems find a wide range of practical application problems, some of which involving Boolean variables and constraints. This paper develops and evaluates algorithms for solving MOCO problems, defined on Boolean domains, and where the optimality criterion is lexicographic. The proposed algorithms build on existing algorithms for either Maximum Satisfiability (MaxSAT), Pseudo-Boolean Optimization (PBO), or Integer Linear Programming (ILP). Experimental results, obtained on problem instances from haplotyping with pedigrees and software package dependencies, show that the proposed algorithms can provide significant performance gains over state of the art MaxSAT, PBO and ILP algorithms. Finally, the paper also shows that lexicographic optimization conditions are observed in the majority of the problem instances from the MaxSAT evaluations, motivating the development of dedicated algorithms that can exploit lexicographic optimization conditions in general MaxSAT problem instances.

Speeding up the Constraint-Based Method in Difference Logic

“…no redundant invariants are generated, conditional invariants are compatible with initial transitions) that can be encoded via Farkas' Lemma or via our novel difference logic encoding presented in the previous section. By making some of these conditions soft with the use of appropriate weights as in [31], we can order the five possible outputs from most desirable (I) to least desirable (V). For example, the optimal solution gives output (III) only if no solution exists that gives results (II) or (I).…”

Improving SAT-Based Weighted MaxSAT Solvers

“…They are inspired in [15]. The idea is to harden all soft clauses with weight bigger than the sum of the weights of the clauses not sent to the WPM plus the clauses returned in ϕ res .…”

Improvements to Core-Guided Binary Search for MaxSAT

“…If no soft clause needs to be relaxed and |subC| = 1, then subC = {< R s , λ s , ν s , µ s >} and λ s is updated to ν s (line 11). Otherwise, all the required soft clauses are relaxed, an entry for the new core C s is added to C, which aggregates the information of the previous cores in subC, and each C i ∈ subC is removed from C (lines [13][14][15][16].…”

“…if ϕC ∩ ϕS = ∅ and |subC| = |{< Rs, λs, νs, ǫs >}| = 1 then 15 λs ← Refine({wj }r j ∈R S , νs) given by K 1 = Ci∈C µ i . However, after merging disjoint cores, if the SAT solver outcome is again SAT, it can happen that K 2 = Ci∈C µ i > K 1 .…”

“…The effect of the more accurate bounds maintained by BCD2 and the additional techniques is even more significant on weighted partial MaxSAT industrial instances. A second experiment, see Figure 2, shows the results for 100 upgradeability instances, 32 timetabling instances [3] and 500 haplotyping with pedigrees instances [15]. Observe that the haplotyping with pedigrees instances are divided in 7 sets (A, B, C, D, E, F, G).…”