Consensus-Halving: Does It Ever Get Easier?

In the $\varepsilon$-Consensus-Halving problem, a fundamental problem in fair division, there are $n$ agents with valuations over interval $[0,1]$, and goal is to divide into pieces assign label "$+$" or "$-$" each piece, such that every agent values total amount of almost equally. The was recently proven by Filos-Ratsikas Goldberg [2019] be first "natural" complete for computational class PPA, answering decade-old open question. this paper, we examine extent which becomes easy solve, if one restricts valuation functions. To end, provide following contributions. First, obtain strengthening PPA-hardness result [Filos-Ratsikas Goldberg, 2019], case when have piecewise uniform only two blocks. We via new reduction, fact conceptually much simpler than corresponding 2019]. Then, consider single-block (uniform) parameterized polynomial time algorithm solving any $\varepsilon$, as well polynomial-time $\varepsilon=1/2$. Finally, an important application our techniques hardness generalization Consensus-Halving, Consensus-$1/k$-Division [Simmons Su, 2003]. particular, prove $\varepsilon$-Consensus-$1/3$-Division PPAD-hard.