1. What is the purpose of introducing a new variable in Algorithm 1?
The purpose of introducing a new variable in Algorithm 1, SimpleBoundedVariableAddition, is to maximize the number of remaining resolvents. The algorithm identifies the literal l max to add to the grid, which maximizes the reduction in formula size. This process involves searching for the literal that appears in the greatest number of clauses and adding it to the grid. The grid starts with dimension 1 x |F l |, where F l is the set of clauses containing l. The algorithm continues to add literals and remove clauses until the addition of a literal no longer increases the size of the formula reduction. This approach enables the algorithm to achieve large reductions in formula size and allows for the reuse of auxiliary variables in future variable introductions.
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2. What is the D r,k problem in grid packing coloring?
The D r,k problem asks whether a grid of radius r can be colored with k colors. It involves variables v i,c representing that grid location i has color c. The problem includes three types of clauses: At-Least-One-Color, At-Most-One-Distance, and Center-Clause. At-Least-One-Color ensures each tile is colored with a color between 1 and k. At-Most-One-Distance prevents two tiles with a distance less than or equal to the color from having the same color. Center-Clause is a symmetry-breaking optimization that has no effect on BVA. BVA reduces the size of formulas by a factor of 4 and induces a speedup on larger instances. Auxiliary variables introduced by BVA capture regions of grid tiles within a particular color, replacing At-Most-One-Distance clauses with smaller clauses involving the auxiliary variables.
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3. How does variable randomization affect BVA substitutions in D5,10?
Variable randomization significantly impacts the solve time of BVA substitutions in D5,10. The first variable addition has a disproportionate effect on solve time, with a single addition achieving a 6x speedup over the original formula. However, randomization before BVA does not affect size reduction but slows down the randomized formula with a single BVA step by 2 times compared to the original formula. This suggests that BVA's impact is derived from both size reduction and the structure of variable additions. Isolating the effect of a single replacement by allowing BVA to produce only one new auxiliary variable helps evaluate the impact of randomization on solve time. Table 1 illustrates the substantial impact of a single variable addition and its sensitivity to randomization.
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4. How does BVA handle ties between literals?
BVA handles ties between literals by choosing the literal that maximizes the number of remaining resolvents to be eliminated. If there is a tie, the original algorithm does not specify which literal to use. The original implementation provided by [15] breaks ties using the variable number in the original formula. Figure 4 demonstrates how breaking ties differently leads to different variable additions. However, since the original implementation used variable number to break ties and ordered variables from top-left to bottom-right, the variable additions it produces follow that structured pattern. When the variable order is randomized, the resulting region lacks structure and the formula takes longer to solve. In the D 5,10 packing problem, colors 9 and 10 are almost fully connected, and when BVA creates a variable for these pairwise constraints, all of the clauses are tied for the number of preserved resolvents. This is because every pair of color-10 variables appears in an at-most-one-distance clause. Therefore, the handling of ties in BVA is crucial for the efficiency and structure of the resulting formula.
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