Does the running game open up the passing game or vice versa?

The answer to your question is yes.

I think when a defense respects a passing threat, it opens up a lot of space for the offense.

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I think when a defense respects a passing threat, it opens up a lot of space for the offense.

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Correct answer.

When a defense does not respect your passing game, they stack the box and stuff your running game. As we have seen before too many times with Miles.

Running helps open up the passing attack
AND
Passing helps open up the running attack

The trick (not a difficult one) is to recognize which one needs to be opened up! If a team comes out stacking the box then the run has already opened up the passing attack and the offense need to be good enough to take advantage of that.

At LSU I've always heard that the running game opens up the passing game. Yet, I've seen the passing game open up the running game for other schools. Like others have posted; play to your strength and open up your weakness. It just seems like the Coach is still not sure what the weakness is. He and his staff get paid a hell of a lot more than me to know strength versus weakness, but I-formation on 1st, 2nd, and 3rd down seams weak.

Both, it's a good example of the economic concept of the law of diminishing marginal returns. As you use more and more of one option (run or pass) then the marginal value of that option (yards per carry or yards per attempt) declines, and therefore the marginal value of the other option (pass or run) increases relative to the first.

In context of football it has to do with how the defense reacts to your gameplan. If you run the ball a lot, the defense stacks the box with 8-9 defenders. If you then pass the ball then you're often able to get 1-on-1 matchups on the outside, which significantly favor the offense and lead to greater yards per passing attempt (more completions for larger gains). The defense then has the hard choice of pulling out of the box (and leaving itself vulnerable to your running attack) or staying in (and doing the same to your passing attack).

On the flip side, if you're succeeding in the passing game and the defense tries to bring in additional defensive backs to cover your receivers, they leave fewer defenders in the box to stop your running game. If you then switch to running, then you catch them in the same dilemma.

To be successful, an offense has to do two things:
1. Control the line of scrimmage: If you can't control the line of scrimmage, then the defense can key in on your strength however they want. If you're losing the battle at the LOS, the defense can keep 6-7 in the box and still stop the run reliably. Your team falls behind and you find you can't pass on them either. Conversely, if you can run reliably without using more than six blockers (and ideally just your five linemen), you can bring in more skill position players and create better spacing.
2. Figure out what it does best and start with that: Whatever it is that your team is better at, start with that so you force the defense to adjust to it.

LSU has struggled this year because we routinely fail at #1 and doesn't yet know what #2 is. I think we're starting to figure out #2, and that will be the key. LSU is at its best when we have Brandon Harris in the game and we're able to throw and pass out of relatively "spread"-ish looks. (Not necessarily "the spread offense" but using 3 WRs to create spacing.) Harris's arm and the spacing created by the skill position players do a lot to take pressure off the offensive line and allow us to win battles at the LOS.

I sure do like the three wide set for us! It opens up running lanes, yet allows a lot of passing options too!

The ability to throw lasers to the far side on comeback routes and & 30-yard darts while rolling out to your left opens up EVERYTHING. There was a reason when he broke containment that there were no defenders in the immediate area in most instances...they needed EVERYONE to defend his arm.

He will consistently be able to run for 8-12 yds in this scenario if he chooses. His arm allows him NOT to have to choose this option every time. If he does, he is fast and surprisingly strong for 185 lbs.

On a side note, the dump off passes, passes to the fullback, tailback & wide receiver screens are all extremely encouraging going forward. The threat of these lead me to believe Leonard Fournette or the other backs will have more 100 yd games in the very near future.

It would depend on how the opposing defense came out lined up.