How can I handle blown cover in a high-stakes game as a GM?

I will be running a high-stakes Eclipse Phase game in the coming weeks and I find myself anticipating a severe problem.

The scenario (The Devotees, for any EP players) is based on infiltrating a super violent human-trafficking cartel’s asteroid, investigating an existential threat to humanity, dealing with it, and getting off station alive.

While the PCs are entirely capable of accomplishing this task in a variety of ways, it’s likely they will fail some task somewhere at some point. Here are a few examples off the top of my head:

  • While hacking security, a failure might label them as infiltrators that should be dealt with.
  • Failing to lie about their true motivations to a critical NPC could get them slagged.
  • Killing an NPC that found them out could put a bounty on their head.

The PCs could definitively handle a few firefights, but not hundreds of armed gang members out for their blood. A completely blown cover seems to mean TPK and not much else.

Considering that the opponents would not hesitate to blow their heads off, how can I ensure the game doesn’t end up in an unsatisfying TPK in their cover is blown?

I’ve thought of a few options that mesh well with the scenario, but I’d like more:

  • The cartel is divided into 3 factions that hate each other, blowing their cover for one wouldn’t necessarily affect the others.
  • Giving the PCs a chance to act tough enough to discourage retaliation (“Yeah I blew his head off, what are you gonna do about it?”)
  • Retconning the mistake (I’d REALLY rather avoid this as it would remove the high-stakes, but it’s preferable to “everyone’s dead, see you next game!”)
  • Running a second game right after with “erasure squad” characters (aka: Can’t have an existential threat if everything on the habitat has been reduced to dust by a conveniently placed tactical nuclear weapon.)

Worst Case running time of the Minimum Vertex Cover Approximation algorithm

Considering this factor $ 2$ minimum vertex cover approximation algorithm :

Repeat while there is an edge:

Arbitrarily pick an uncovered edge $ e=(u,v)$ and add $ u$ and $ v$ to the solution. Delete $ u$ and $ v$ from the graph. Finally output the candidate cover.

I want to find the worst case running time of this algorithm. Since in a fully connected graph we have $ O(n^2)$ edges, then the loop will run at most $ O(n^2)$ times, I guess here I am not sure what the maximal number of delete operations could be or perhaps for less than $ O(n^2)$ edges there would be some scenario with a large number of delete operations.

Any insights appreciated.

Checking if a $k$-subset of a graph is a vertex cover in time $O(kn)$

Given a graph $ G=(V,E)$ with $ |V|=n,|E|=m$ . I am reading a $ \textit{brute force}$ solution to determining whether each candidate vertex cover of size $ k \leq n$ is a vertex cover. The graph does not have loops.

As per page 2 of the notes here we have:

  • 1) $ C(n,k) = O(n^k)$ $ k$ -subsets of $ V$
  • 2) $ O(kn)$ time to check whether a subset is a vertex cover

I am considering 2).

What I would think of would be to take each edge in our graph, for which there is a maximum of $ {n \choose 2} = O(n^2)$ , and check the candidate subset to see if at least one of the vertices is an endpoint of the edge, which would take $ O(kn^2)$ checks.

I am not sure how the author arrived at $ O(kn)$ operations, any insights appreciated.

Minimum Clique Cover – Mixed Integer Programming

I have a general (undirected) graph with a set of nodes, a set of edges, and a weight for each edge. I want to find a minimum clique cover of the graph, that is, a partition of the graph into the smallest number of cliques. I also want to maximize the sum of edge weights over the cliques. I want to use an integer programming approach for this problem.

Can any one give me some hints or some references that use mixed integer linear programming for the (maximum weight) minimum clique partition?

Thank you very much.

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Does cover affect melee attacks?

In what circumstances would cover affect melee attacks?

If three characters are lined up as shown in the diagram, and the bottom-right man in armor makes a melee attack against the frog creature in the upper-left (through the woman’s square, and around the corner), does the frog have half-cover from the attack?

Melee Cover Question

Would removing the “total cover” part of a Paladin’s Divine Sense unbalance the feature?

Inspired by this question: What good is the paladin's Divine Sense?

I am running Lost Mines of Phandelver to introduce a couple of people to D&D (just me and my two players; I use DMPCs to pad out the party), and one of my players is a Paladin. During their time in Thundertree, he used Divine Sense a few times to try to detect the zombies throughout the town (there’s a sign that says “Beware! Zombies” or something, so this was a completely in-character thing to do).

Since, as the linked question mentions, it’s a very niche ability in D&D 5e, I wasn’t that familiar with it; I remembered that it detects undead and fiends (forgot celestials at the time, but it’s irrelevant to this case) within 60 feet, but forgot about the total cover thing, so we proceeded without the total cover part of the ability. It was very useful for sniffing out those zombies hiding in the buildings.

Only later (because of a question on here, I think – this was a while ago), I realised that total cover was part of how it worked, and informed my player that, what happened happened, but going forward we’d have to bear total cover in mind. His response was basically “ok, you’re the DM, but doesn’t that kinda make this ability useless?”; my only response was about detecting invisible fiends or undead, which he accepted gracefully, but I doubt we’ll be seeing much more use out of it…

They’re now in Wave Echo Cave (our sessions are quite infrequent), but have not encountered anything undead yet, and will eventually progress onto Out of the Abyss, which has many fiends, and in light of the question I link to at the top, I am starting to wonder whether I should go back on myself and allow my player’s Paladin to use Divine Sense like how he used it in Thundertree. I think it will be more fun if he can actually use his ability outside of its otherwise rare niche uses.

This question is about the impacts on balance. What issues would arise from me ruling to ignore the total cover rule of Divine Sense? Would the ability become overpowered or otherwise unbalanced compared to other classes?