Partitioning a bipartite graph into proportional factors

Let G=(V,E) be a bipartite graph, and [math]c_1,\dots c_k[/math] positive reals whose sum is 1. Can E always be partitioned into k parts [math]E_1,\dots,E_k,[/math] so that for every [math]v \in V[/math] and every [math]i \in \{1,\dots,k\}[/math] we have [math]\lfloor c_i d_E(v) \rfloor \leq d_{E_i}(v) \leq \lceil c_i d_E(v) \rceil[/math]?

Remarks

This question was asked by Correa and Goemans . They proved a weaker bound [math]\lfloor c_i d_E(v) \rfloor-2 \leq d_{E_i}(v) \leq \lceil c_i d_E(v) \rceil+2[/math]. An alternative proof was found by Feige and Singh , who also showed that if only upper bounds or only lower bounds are considered, then [math]\pm 2[/math] can be replaced by [math]\pm 1[/math].

If [math]c_i=1/k[/math] for every i, then the statement follows from total unimodularity, see de Werra . If k=2, then it follows from the theorem on Orientation with degree bounds. The case when G is regular is also easy, because we can partition E into regular subgraphs satisfying the condition.