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In communication complexity, we encounter the complexity measure $\chi(f)$ for $f : \{0,1\}^{2n} \to \{0,1\}$ which is the minimal number of $f$-monochromatic rectangles needed to tile the $2^n \times 2^n$ matrix describing $f$ (where here $f$-monochromatic means that the individual rectangles are either all $0$ or all $1$).

I am wondering what is known about the computational complexity of $\chi(f)$. I would imagine that it cannot be computed in polynomial time for example (and maybe a minimal tiling cannot be verified in polynomial time) but I could not find any references addressing these issues. Is it known to be, say, NP-hard?

Thank you for reading and I'm sorry if I have missed some obvious reference, I am brand new to thinking about communication complexity.

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  • $\begingroup$ By "rectangle" here do you really mean "submatrix"? ("Rectangle" has the connotation of being contiguous, but I doubt that's what you intend.) $\endgroup$
    – Neal Young
    Feb 7 at 18:39
  • $\begingroup$ I don't mean to imply that it is contiguous. By a rectangle in a set $X \times Y$, I just mean a subset of the form $A \times B$ for $A \subset X, B \subset Y$ (this seems to be standard in the communication complexity literature). $\endgroup$
    – user101010
    Feb 7 at 18:55
  • $\begingroup$ Yes, that makes sense. $\endgroup$
    – Neal Young
    Feb 7 at 19:59

1 Answer 1

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When viewing the Boolean matrix as the bipartite adjacency matrix of a bipartite graph, the problem of determining $\chi_1(f)$, that is, partitioning all the $1$s of the matrix into monochromatic rectangles, is the biclique partition problem, which is NP-complete (also for bipartite graphs). Approximation hardness results are also known.

  • Doina Bein, Linda Morales, Wolfgang W. Bein, Charles O. Shields Jr., Z. Meng, Ivan Hal Sudborough: Clustering and the Biclique Partition Problem. HICSS 2008: 475
  • Parinya Chalermsook, Sandy Heydrich, Eugenia Holm, Andreas Karrenbauer: Nearly Tight Approximability Results for Minimum Biclique Cover and Partition. ESA 2014: 235-246

Update 09/02/2022: the above is an answer to a related problem, and does not answer the OP's question. 🤔

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  • $\begingroup$ Thank you very much for this! I have one technical question related to your answer. If $G_f$ is the bipartite graph described by $f$, then as you point out $\chi(f) = \text{bp}(G_f) + \text{bp}(\overline{G_f})$ (where $\text{bp}$ is the bipartite partition number - which as you mention is NP-complete to compute, even for bipartite graphs). I don't understand how knowing $\chi(f)$ tells us $\text{bp}(G_f)$. Are $\text{bp}(G_f)$ and $\text{bp}(\overline{G_f})$ related in a simple way? $\endgroup$
    – user101010
    Feb 9 at 2:23
  • $\begingroup$ Thanks, I missed this. So the above is only a partial answer, or maybe an answer to a closely related question. I'll update my answer accordingly, and look again into the literature. $\endgroup$ Feb 9 at 20:30
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    $\begingroup$ Perhaps, somewhat related: the gap between $\mathrm{bp}(G)$ and $\mathrm{bp}(\overline{G})$ can be large: there are graphs $G$ with $\mathrm{bp}(G)=t$ but $\mathrm{bp}(\overline{G})\geq t^{\Omega(\log t)}$. This is a consequence of a current progress concerning tilling numbers; see notes at web.vu.lt/mif/s.jukna/boolean/tilling-gaps.html. $\endgroup$
    – Stasys
    Feb 11 at 11:00

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