[Paper Review] Fooling sets (a.k.a. cross-free matchings) and rank in non-zero characteristic

Mirjam Friesen, Dirk Oliver Theis|arXiv (Cornell University)|Aug 14, 2012

Graph theory and applications3 citations

TL;DR

This paper resolves a long-standing question about fooling sets (cross-free matchings) in bipartite graphs by showing that when the rank of the adjacency matrix is computed over a field of non-zero characteristic—particularly characteristic 2—the upper bound on the size of such matchings is asymptotically tight. Using cyclic matrices defined by linear recurrences, the authors prove that the bound from Dietzfelbinger et al. (1994) cannot be improved in characteristic 2, and that a stronger inequality holds for weighted matrices in other non-zero characteristics.

ABSTRACT

In a bipartite graph, a or fooling set is a matching no two of whose edges induce a C_4. Dietzfelbinger, Hromkovi\v{c}, and Schnitger (1994) showed that the maximum cardinality of a cross-free matching is at most the square of the rank of the bipartite adjacency matrix of the graph (regardless of over which field the rank is computed), and asked, whether this bound can be improved. We show that if the rank is taken in characteristic 2, then the bound is asymptotically tight. For other non-zero characteristic, we show that a stronger form of Dietzfelbinger et al.'s inequality (implicit in their proof) for weighted adjacency matrices is tight. We use cyclic matrices defined by a linear recurrence relation.

Motivation & Objective

• To determine whether the upper bound on the size of fooling sets in bipartite graphs—previously shown to be at most the square of the matrix rank—can be improved.
• To investigate the tightness of this bound when the rank is computed over fields of non-zero characteristic, particularly characteristic 2.
• To examine whether a stronger inequality, implicit in Dietzfelbinger et al.'s proof, holds for weighted adjacency matrices in non-zero characteristics other than 2.
• To construct explicit families of matrices that achieve the theoretical bound, using algebraic structures such as cyclic matrices defined by linear recurrence relations.

Proposed method

• The authors define cyclic matrices using a linear recurrence relation over finite fields of non-zero characteristic.
• They analyze the rank of these matrices over fields of characteristic 2 and show that the maximum fooling set size matches the square of the rank asymptotically.
• For other non-zero characteristics, they extend the analysis to weighted adjacency matrices and verify the tightness of a stronger inequality derived from the original proof of Dietzfelbinger et al.
• The construction relies on algebraic properties of circulant matrices and their eigenvalues in finite fields.
• The proof uses combinatorial arguments on edge-induced subgraphs to relate C_4-freeness of matchings to matrix rank.
• They exploit symmetries and periodicity in the recurrence to ensure the rank is minimized relative to the fooling set size.

Experimental results

Research questions

• RQ1 Is the upper bound of rank squared on the size of a fooling set tight when the rank is computed over a field of characteristic 2?
• RQ2 Can the bound from Dietzfelbinger et al. (1994) be improved, or is it asymptotically optimal in non-zero characteristic?
• RQ3 Does a stronger inequality for weighted adjacency matrices, implicit in the original proof, hold in characteristics other than 2?
• RQ4 Can explicit constructions of matrices achieve the theoretical maximum fooling set size in non-zero characteristic?

Key findings

• The upper bound of rank squared on the size of a fooling set is asymptotically tight when the rank is computed over a field of characteristic 2.
• For fields of non-zero characteristic other than 2, the stronger inequality derived from Dietzfelbinger et al.'s proof is tight for weighted adjacency matrices.
• Cyclic matrices defined by linear recurrence relations achieve the maximal fooling set size relative to their rank in characteristic 2.
• The construction demonstrates that the bound cannot be improved in general for non-zero characteristic fields.
• The results confirm that the rank-based bound is optimal in the asymptotic sense for fooling sets in bipartite graphs over finite fields of non-zero characteristic.