Network　administrators　usually　realize　network　functions　in　data　plane　programs.　They　employ　the　network-wide　program　deployment　that　decomposes　input　programs　into　match-action　tables　(MATs)　while　deploying　each　MAT　on　a　specific　switch.　Since　MATs　may　be　deployed　on　different　switches,　existing　solutions　propose　the　inter-switch　coordination　that　uses　the　per-packet　header　space　to　deliver　crucial　packet　processing　information　among　switches.　However,　such　coordination　incurs　non-trivial　per-packet　byte　overhead,　leading　to　end-to-end　performance　degradation.　We　propose　Hermes,　a　framework　that　aims　to　minimize　the　per-packet　byte　overhead.　The　key　idea　is　to　formulate　network-wide　program　deployment　as　a　mixed-integer　programming　(MIP)　problem　with　the　objective　of　minimizing　the　per-packet　byte　overhead.　Also,　Hermes　offers　a　greedy-based　heuristic　that　solves　the　problem　in　a　near-optimal　and　timely　manner.　We　have　implemented　Hermes　on　Tofino　switches.　Compared　to　existing　frameworks,　Hermes　decreases　the　per-packet　byte　overhead　by　156bytes　while　preserving　end-to-end　performance　in　terms　of　flow　completion　time　and　goodput.　©　2024　IEEE.