Table 2.

Algorithm 2 Decycle: REMOVE ALL CYCLES FROM A GRAPH
Require: A directed graph , including:
a function (e) that specifies the successors to node e, and
a function (e) that specifies the predecessors to node e
a function f(e) that specifies frequency of epitope e in the population
Require: Functions STRONGLY_CONNECTED_COMPONENTS and SHORTEST_PATH,  both are provided by NetworkX software package41
1: repeat
2:  J ← STRONGLY_CONNECTED_COMPONENTS ()
	▹ J is a list of all components; each component is a list of nodes in
3:  J ← J − {j ∈ J, such that |j| = 1}
	▹ Discard all single-node components – no cycles there!
4:  for all j ∈ J do	▹ For each component
5:   repeat
6:    Choose (a, b) ∈ j	▹ Randomly choose two nodes from the selected component
7:    C ← CYCLEFROMTWONODES(, a, b)
8:    if C ≠ ∅ then
9:     (ea, eb) ← WEAKEDGEINCYCLE(, C)
10:     Remove edge (ea, eb) from
11:    until C = ∅
12: until J = ∅	▹ is acyclic; we are done.
13: procedure CYCLEFROMTWONODES(, a, b)
14:  Pab ← SHORTEST_PATH(, a, b)
15:  Pba ← SHORTEST_PATH(, a, b)
16:  if either call to SHORTEST_PATH fails then
17:   C ← ∅
18:  else
19:   C ← Pab + Pba	▹ Merge two paths into a cycle
20:  return C
21: procedure WEAKEDGEINCYCLE(, C)
22:  Write C as a list of nodes [e1, …, ek] and edges [(e1, e2), (e2, e3), …, (ek, e1)]
23:  for all (ei, ej) in [(e1, e2), (e2, e3), …, (ek, e1)] do
24:   vij ← f(ei) + f(ej)	▹ v is heuristic “value” of edge
25:   vij ← vij + f (ei) if |(ei)| = 1	▹ Add value if cutting edge would isolate ei
26:   vij ← vij + f (ej) if |(ej)| = 1	▹ Add value if cutting edge would isolate ej
27:  Let io, jo ← argmin vij
28:  return(,)	▹ return lowest-value edge in cycle

Algorithm 3 Cocktail: FIND (AND ITERATIVELY REFINE) A SET OF m ANTIGENS
Require: Directed Acyclic Graph , including
a function f (e) that specifies frequency of epitope e in the population
Require: Function EPIGRAPH(, f) that returns a sequence q, corresponding to a path through the graph that maximizes
▷See Algorithm 1
1:  ← ∅
2: f*(e) ← f (e) for all epitopes e ∈	▷Initialize
3: for n = 1 … m do	▷Sequential solution
4:  qn ← EPIGRAPH(, f*)	▷Compute next antigen sequence qn
5:   ←  ∪ {qn}	▷Add qn to vaccine
6:  for e ∈ (qn) do	▷Now that e is in the vaccine
7:   f*(e) ← 0	▷No credit for including e in subsequent antigens
	▷At this point, f*(e) = 0 for all e ∈  ()
8: repeat	▷Iterative Refinement (optional)
9:  for n = 1 … m do
10:   ←− {qn}	▷Remove sequence qn from vaccine
11:   for e ∈  () −  (qn) do	▷With e not in the vaccine anymore,
12:    f*(e) ← f(e)	▷f*(e) gives credit for including e in subsequent antigens
13:   qn ← EPIGRAPH(, f*)	▷Compute replacement for old sequence qn
14:   ←  ∪ {qn}	▷Add qn to vaccine
15:   f*(e) ← 0 for all e ∈  (qn)
16: until no change in q1, …, qm
17: return cocktail of m sequences:  = {q1, …, qm}

Algorithm 4 TTV: TAILORED THERAPEUTIC VACCINE
Require: Sequence set  = {s1, …, sN},
Require: Directed Acyclic Graph,
Require: Function EPIGRAPH(, f) that returns a sequence q, corresponding to a path through the graph that maximizes
	▷See Algorithm 1
1: Define u(s, e) = 1 if epitope e appears in sequence s	▷i.e., if e ∈ (s)
2: f(e) ← (1/N)	▷Frequency of epitope e in sequence set
3: qo ← EPIGRAPH(, f)	▷qo is centroid of the full data set
4: Randomly select m − 1 sequences from ; call them q1, …, qm−1.
5: repeat
6:  Initialize: 1 = … = m−1 = ∅
7:  for i = 1…N do	▷Loop over sequences
8:   for n = 1…m−1 do	▷Loop over centroids
9:	▷Coverage of sequence si by antigens {qo, qn}
	▷Here, u({qo, qn}, e) = max[u(qo,. e), u(qn, e)]
10:   n ← argmaxn′cin′
11:   n ← n ∪ {si}	▷Put si into cluster n
12:  for n = 1…m − 1 do	▷Loop over clusters
13:   f (n)(e)(1/N) ←	▷Frequency of e in sequence cluster n
14:   f (n)(e) ← 0 for all e ∈ (qo)	▷No credit for epitopes already covered by qo
15:   qn ← EPIGRAPH(, f (n))	▷New centroid for n
16: until convergence
17: return tailored therapeutic vaccine: qo, q1, …, qm−1