Regular Switching Components
Résumé
We consider a problem of Discrete Tomography which consists in reconstructing a lattice set S ⊂ Z² with given horizontal and vertical X-rays (in other words, with prescribed number of points in each row and column). Without complementary assumption, the problem can be solved in polynomial time [1, 2]. Many variants add the constraint to find a solution in a chosen class C. Among others, the problem is NP-complete for the class HV of HV-convex lattice sets [3] and it becomes polynomial for the class HV4 of the HV-convex polyominoes [4]. Twenty years after these results, its complexity remains unknown for the class P of the convex lattice sets (in other words two-dimensional lattice polytopes). The difficulty of this problem comes from combinatorial structures called switching components. Given the border of a solution (its feet), switching components are finite sequences of points (p i) 1≤i≤2l with the property that either the points with odd indices, or the points with even indices are in a solution S. This binary choice is encoded in a boolean variable associated with the switching component. Then the convexity constraints are simply encoded by clauses (2-clauses for HV-convexity and 3-clauses for convexity). The purpose of the paper is to investigate the properties of the switching components and their relations induced by convexity. We divide the switching components in two classes: regular if their turning angle is constant , irregular otherwise. We prove that the 4-connected regular switching components have equal boolean variable. It leads to merge them in extended switching components. If all switching components are regular, we prove that the extended switching components are all independent (then the number of solutions with the considered feet is 2 power n where n is the number of extended switching components) and that they are geometrically ordered.
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