We take advantage of the special structure of computations in Z2 to develop algorithms for the co... more We take advantage of the special structure of computations in Z2 to develop algorithms for the computation of Groebner bases and of the Hilbert function in the Boolean setting. Natural sources of applications for our algorithms are the counting problems. We focus, as a case study, on the computation of the permanent. To this regard, one good feature of the Groebner approach is that, unlike other general methods for the exact computation of the permanent, it is intrinsically sensitive to the structure of the speciic input, and this makes it possible to use it in order to recognize and solve eeciently several easy instances.
In this paper we address the problem of allocating rooms among people in a suitable topology of c... more In this paper we address the problem of allocating rooms among people in a suitable topology of corridor with some contraints of undesired neighborhood. People are partitioned into families, and we consider both internal crashes (sharing a room with someone belonging to a different family), and external crashes (being in a room next to, or in front of someone belonging to a different family). The goal is to minimize a suitably defined weight of such crashes. We give polynomial time algorithms for several cases of increasing difficulty, including the one where k people (with k any given constant) have to fit into one room. Finally, we give a 4 3 approximation for the case with both double and triple rooms.
Discrete Mathematics & Theoretical Computer Science
Any attempt to find connections between mathematical properties and complexity has a strong relev... more Any attempt to find connections between mathematical properties and complexity has a strong relevance to the field of Complexity Theory. This is due to the lack of mathematical techniques to prove lower bounds for general models of computation.\par This work represents a step in this direction: we define a combinatorial property that makes Boolean functions ''\emphhard'' to compute in constant depth and show how the harmonic analysis on the hypercube can be applied to derive new lower bounds on the size complexity of previously unclassified Boolean functions.
We take advantage of the special structure of computations in Z2 to develop algorithms for the co... more We take advantage of the special structure of computations in Z2 to develop algorithms for the computation of Groebner bases and of the Hilbert function in the Boolean setting. Natural sources of applications for our algorithms are the counting problems. We focus, as a case study, on the computation of the permanent. To this regard, one good feature of the Groebner approach is that, unlike other general methods for the exact computation of the permanent, it is intrinsically sensitive to the structure of the speciic input, and this makes it possible to use it in order to recognize and solve eeciently several easy instances.
In this paper we address the problem of allocating rooms among people in a suitable topology of c... more In this paper we address the problem of allocating rooms among people in a suitable topology of corridor with some contraints of undesired neighborhood. People are partitioned into families, and we consider both internal crashes (sharing a room with someone belonging to a different family), and external crashes (being in a room next to, or in front of someone belonging to a different family). The goal is to minimize a suitably defined weight of such crashes. We give polynomial time algorithms for several cases of increasing difficulty, including the one where k people (with k any given constant) have to fit into one room. Finally, we give a 4 3 approximation for the case with both double and triple rooms.
Discrete Mathematics & Theoretical Computer Science
Any attempt to find connections between mathematical properties and complexity has a strong relev... more Any attempt to find connections between mathematical properties and complexity has a strong relevance to the field of Complexity Theory. This is due to the lack of mathematical techniques to prove lower bounds for general models of computation.\par This work represents a step in this direction: we define a combinatorial property that makes Boolean functions ''\emphhard'' to compute in constant depth and show how the harmonic analysis on the hypercube can be applied to derive new lower bounds on the size complexity of previously unclassified Boolean functions.
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Papers by Anna Bernasconi