Paying for Fractional Error: Tight Constant-Round Approximation of Covering Problems with Short Broadcasts
We present a general framework for deterministic constant-round approximation of covering problems in sparse networks in the broadcast CONGEST model, where every vertex broadcasts the same $O(\log n)$-bit message to all its neighbors. The framework combines two independent tools. The first is a solver that computes a $(1+δ)$-approximate solution of any local covering linear program with bounded positive integer costs on every class that excludes a fixed minor. The second is a rounding principle: a rounding rule that saves a fixed fraction of its own output, for every feasible input, pays for a solver error fixed in advance, so that an approximate fractional solution yields an exact coefficient. For ordinary domination, we achieve a $2a+1$ approximation against the fractional optimum $γ_f(G)$ for every fixed rational $a\ge1$, whenever every minor has a ratio of edges to vertices of at most $a$. On planar graphs this is seven, matching the lower bound of Hilke, Lenzen, and Suomela. The coefficients are also optimal at bounded treewidth and pathwidth, for $K_t$-minor-free graphs with $3\le t\le9$, on triangle-free planar graphs (five), and on planar graphs of large girth, where it approaches the optimal three. On every fixed surface the exact bound is $7γ(G)$, and we show that a bound of $7γ_f(G)$ is impossible on some fixed surfaces even sequentially. For total domination, on graphs without isolated vertices and under the same minor-density assumption, the exact coefficient is $2a$ against the fractional optimum, including the optimal six on planar graphs. Red-blue domination, and hence set cover and hitting set on planar incidence graphs, has the optimal limiting coefficient four. Our method also gives randomized constant-round and deterministic $O(\log^* n)$-round $(1+\varepsilon)$-approximation schemes, with the randomized cost guarantee holding in expectation.