Bayes-Split-Edge: Bayesian Optimization for Constrained Collaborative Inference in Wireless Edge Systems

Safaeipour, Fatemeh Zahra (ORCID:0009000854612148); Chakareski, Jacob (ORCID:0000000324289518); Hashemi, Morteza (ORCID:0000000292536608)

doi:10.1145/3769102.3770629

Let $$G$$ be a multigraph. A subset $$F$$ of $E(G)$ is an edge cover of $$G$$ if every vertex of $$G$$ is incident to an edge of $$F$$. The cover index, $$\xi(G)$$, is the largest number of edge covers into which the edges of $$G$$ can be partitioned. Clearly $$\xi(G) \le \delta(G)$$, the minimum degree of $$G$$. For $$U\subseteq V(G)$$, denote by $E^+(U)$ the set of edges incident to a vertex of $$U$$. When $|U|$ is odd, to cover all the vertices of $$U$$, any edge cover needs to contain at least $(|U|+1)/2$ edges from $E^+(U)$, indicating $$ \xi(G) \le |E^+(U)|/ ((|U|+1)/2)$$. Let $$\rho_c(G)$$, the co-density of $$G$$, be defined as the minimum of $|E^+(U)|/((|U|+1)/2)$ ranging over all $$U\subseteq V(G)$$, where $$|U| \ge 3$$ and $|U|$ is odd. Then $$\rho_c(G)$$ provides another upper bound on $$\xi(G)$$. Thus $$\xi(G) \le \min\{\delta(G), \lfloor \rho_c(G) \rfloor \}$$. For a lower bound on $$\xi(G)$$, in 1978, Gupta conjectured that $$\xi(G) \ge \min\{\delta(G)-1, \lfloor \rho_c(G) \rfloor \}$$. Gupta himself verified the conjecture for simple graphs, and Cao et al. recently verified this conjecture when $$\rho_c(G)$$ is not an integer. In this paper, we confirm the conjecture when the maximum multiplicity of $$G$$ is at most two or $$ \min\{\delta(G)-1, \lfloor \rho_c(G) \rfloor \} \le 6$$. The proof relies on a newly established result on edge colorings. The result holds independent interest and has the potential to significantly contribute towards resolving the conjecture entirely.

More Like this