Maximum n-times Coverage for Vaccine Design

We introduce the maximum n-times coverage problem that selects k overlays to maximize the summed coverage of weighted elements , where each element must be covered at least n times . We also define the min-cost n-times coverage problem where the objective is to select the minimum set of overlays such that the sum of the weights of elements that are covered at least n times is at least τ . Maximum n-times coverage is a generalization of the multi-set multi-cover problem , is NP-complete , and is not submodular . We introduce two new practical solutions for n-times coverage based on integer linear programming and sequential greedy optimization . We show that maximum n-times coverage is a natural way to frame peptide vaccine design , and find that it produces a pan-strain COVID-19 vaccine design that is superior to 29 other published designs in predicted population coverage and the expected number of peptides displayed by each individual ’ s HLA molecules . 1 INTRODUCTION . In the maximum n-times coverage problem , a set of overlays is selected to cover elements zero or more times , where each overlay is predetermined to cover one or more elements . Each element is assigned a weight that reflects its importance . The objective of the problem is to maximize the sum of the weights of elements that are covered at least n times by at most k overlays . When the coverage of elements by overlays is determined by a machine learning method , the result summarizes machine learning results with a compact solution that is shaped by the element weights employed . Our introduction of weighted elements and required n-times coverage creates a new class of problem without a known solution or complexity bound for approximate solutions . The closest problem to the maximum n-times coverage problem is the multi-set multi-cover problem which does not assign weights to the elements and thus can be formulated as the Covering Integer Program ( CIP ) problem ( Srinivasan , 1999 ) . Thus like CIP , maximum n-times coverage is NP-complete . A log ( n ) time approximation algorithm for CIP can violate coverage constraints ( Dobson , 1982 ; Kolliopoulos , 2003 ; Kolliopoulos & Young , 2001 ) . Deletion-robust submodular maximization protects against adversarial deletion ( Bogunovic et al. , 2018 ; Mirzasoleiman et al. , 2017 ) and robust submodular optimization protects against objective function uncertainty ( Iyer , 2019 ) , but neither guarantees that each element is covered n times . In this work , we investigate properties of the maximum n-times coverage problem , provide a practical solution to the problem , and use the solution for machine learning based vaccine design . We show that maximum n-times coverage is not submodular , and introduce NTIMES-ILP and MARGINALGREEDY , efficient algorithms for solving the n-times coverage problem on both synthetic data and real vaccine design . In our framing of vaccine design , an element is a specific collection of HLA alleles ( a genotype ) , weights are the frequency of genotypes in the population , n is the desired number of peptides displayed by each individual , and an overlay is a peptide that is predicted to be displayed by each genotype a specified number of times . The solution of the maximum n-times coverage problem allows us to find a set of overlays that maximizes the sum of element weights ( population coverage ) . We show that framing vaccine design as maximum n-times coverage produces a solution that produces superior predicted population coverage when compared to 29 previous published vaccines for COVID-19 with less than 150 peptides . 2 THE MAXIMUM n-TIMES COVERAGE PROBLEM 2.1 MULTI-SET MULTI-COVER PROBLEM . In the standard SET COVER and MAXIMUM COVERAGE problems , we are given a set U of |U| elements ( also known as the universe ) and a collection S = { S1 , S2 , . . . , Sm } of m subsets of U such that ⋃ i Si = U . The goal in the SET COVER problem is to select a minimal-cardinality set of subsets from S such that their union covers U . The MULTI-SET MULTI-COVER ( MSMC ) problem is a generalization of the SET COVER problem , where multi-sets are sets in which an element can appear more than once . The objective of the MSMC problem is to determine the minimum number of multisets ( a multi-set can be chosen multiple times ) such that each element i is covered at least bi times . It can be formulated into Covering Integer Program ( CIP ) problem ( Srinivasan , 1999 ) : Definition 1 . ( Covering Integer Program , CIP ) Given A ∈ Rn×m+ , b ∈ Rn+ , w ∈ Rm+ , d ∈ Rm+ , a CIP P = ( A , b , w , d ) seeks to minimize wTx , subject to Ax ≥ b , x ∈ Zm+ , and x ≤ d. Here Aij represents the number of times i-th element appears in the j-th multi-set . The w is set to be all 1 in MSMC . The constraints x ≤ b are called multiplicity constraints which limit the number of times a single multi-set can be reused , and they generally make covering problems much harder as natural linear programming ( LP ) relaxation can have an unbounded integrality gap ( Chuzhoy & Naor , 2006 ) . Dobson ( 1982 ) provides a combinatorial greedy H ( maxj ∑ i Aij ) -approximation algorithm ( H ( t ) stands for t-th harmonic number ) but multiplicity constraints can be dealt with effectively only in the ( 0,1 ) case , and thus this algorithm can be as bad as polynomial . Kolliopoulos & Young ( 2001 ) ; Kolliopoulos ( 2003 ) gave a tighter-bound solution that can obtain O ( log n ) -approximation . 2.2 MAXIMUM n-TIMES COVERAGE We introduce the maximum n-times coverage problem , a variant of the MSMC problem that accounts for multiple coverage of each element while also assigning weights to different elements . We are given a set X with l elements { X1 , X2 , . . . , Xl } each associated with a non-negative weight w ( Xi ) , and a set of m overlays A = { A1 , A2 , . . . , Am } . Each overlay Aj covers each element i in X an element specific number of times cj ( Xi ) , which is similar to a multi-set . When cj ( Xi ) = 0 the element Xi is not covered by overlay Aj . We use a very strict multiplicity constraint such that each overlay can be used only once . Given a subset of overlays O ⊆ A , the total number of times an element Xi is covered by O is the sum of cj ( Xi ) for each overlay j in O : C ( Xi|O ) = ∑ j∈O cj ( Xi ) ( 1 ) We define the n-times coverage function fn ( O ) as the sum of weights of elements in X that are covered at least n times by O . Figure 1 shows an example computation of the n-times coverage function . fn ( O ) = l∑ i=1 w ( Xi ) 1 { C ( Xi|O ) ≥n } = l∑ i=1 w ( Xi ) 1 { ∑ j∈O cj ( Xi ) ≥n } ( 2 ) The objective of the MAXIMUM n-TIMES COVERAGE problem is to select a set of k overlays O ⊆ A such that fn ( O ) is maximized . This can be formulated as the maximization of the monotone set function fn ( O ) under cardinality constraint k : O∗ = argmax O⊆A , |O|≤k fn ( O ) ( 3 ) We define MIN-COST n-TIMES COVERAGE as the minimum set of overlays such that the sum of the weights of elements covered at least n times is ≥ τ . We assume A provides sufficient n-times coverage for fn ( A ) ≥ τ . We define the n-TIMES SET COVER problem as the special case of MIN-COST n-TIMES COVERAGE where τ = ∑ i w ( Xi ) . MIN-COST n-TIMES COVERAGE O∗ = argmin O |O| s.t . fn ( O ) ≥ τ ( 4 ) n-TIMES SET COVER O∗ = argmin O |O| s.t . fn ( O ) = fn ( A ) ( 5 ) Theorem 1 . The n-times coverage function fn ( O ) is not submodular . The proof can be found in Appendix A . Since the n-times coverage problem is not submodular , we can not take advantage of the proven near-optimal performance of the greedy algorithm for the SUBMODULAR MAXIMUM COVERAGE problem . Thus we seek new solutions . 3 THE NTIMES-ILP SOLUTION FOR MAXIMUM n-TIMES COVERAGE We first formulate the maximum-n times coverage problem as an integer linear program ( ILP ) . We associate a binary variable ai with each overlay Ai such that ai = 1 if Ai ∈ O∗ and otherwise ai = 0 . Thus the cardinality constraint can be written as |O| = ∑m i=1 ai ≤ k , and the number of times an element Xj is covered by O is C ( Xj |O ) = ∑ i∈O ci ( Xj ) = ∑m i=1 aici ( Xj ) . The main challenge is to encode the objective function fn ( O ) = ∑l j=1 w ( Xj ) 1 { ∑m i=1 aici ( Xj ) ≥n } in a linear fashion . We replace the step function 1 { ∑mi=1 aici ( Xj ) ≥n } with a variable tj such that for each element Xj : tj = 1 ⇐⇒ m∑ i=1 aici ( Xj ) ≥ n and tj = 0 ⇐⇒ m∑ i=1 aici ( Xj ) < n ( 6 ) We enforce the conditions in ( 6 ) with the Big M method ( Sherali et al. , 2013 ) , where we include the following inequalities for each element Xj ∈ X : −M ( 1− tj ) ≤ m∑ i=1 aici ( Xj ) − n+ ϵ ≤Mtj , ( 7 ) We set ϵ ∈ ( 0 , 1 ) and choose M to be a large number such as 10,000 which is larger than the term∑m i=1 aici ( Xj ) − n+ ϵ . Here we prove that tj = 1 ⇐⇒ ∑m i=1 aici ( Xj ) ≥ n. Proof . When tj = 1 , inequality ( 7 ) becomes 0 ≤ ∑m i=1 aici ( Xj ) − n + ϵ ≤ M . If M is a large number , we can ignore the right inequality and rearrange to get ∑m i=1 aici ( Xj ) ≥ n − ϵ. Since∑m i=1 aici ( Xj ) and n are both integers and ϵ ∈ ( 0 , 1 ) , we find ∑m i=1 aici ( Xj ) ≥ n. All these steps can be taken in the reverse order to prove that ∑m i=1 aici ( Xj ) ≥ n =⇒ tj = 1 , given that tj is a binary variable . We also prove that tj = 0 ⇐⇒ ∑m i=1 aici ( Xj ) < n. Proof . We start by showing the forward direction . If tj = 0 , then inequality ( 7 ) becomes −M ≤∑m i=1 aici ( Xj ) − n+ ϵ ≤ 0 . As M is large , −M is a large negative number , so we can ignore the left inequality . Hence , ∑m i=1 aici ( Xj ) ≤ n− ϵ < n. In the reverse direction , if ∑m i=1 aici ( Xj ) < n , because both the quantities ∑m i=1 aici ( Xj ) and n are integers , for any ϵ ∈ ( 0 , 1 ) , ∑m i=1 aici ( Xj ) ≤ n− ϵ . Rearranging this we get ∑m i=1 aici ( Xj ) − n+ ϵ ≤ 0 , which forces tj as a binary variable to be equal to 0 from inequality ( 7 ) . Therefore , the objective function to maximize becomes fn ( O ) = ∑l j=1 w ( Xj ) 1 { C ( Xj |O ) ≥n } =∑l j=1 tjw ( Xj ) , which is linear as a sum over l terms of a binary variable multiplied by a constant . The complete NTIMES-ILP formulation of n-times coverage is : • Variables – a1 , ... , am representing presence of each overlay A1 , ... , Am in final solution – t1 , ... , tl representing if each element X1 , ... , Xl has been covered at least n times • Constraints – ∑m i=1 ai ≤ k ( the maximum total number of overlays allowed in the final subset ) – −M ( 1− tj ) ≤ ∑m i=1 aici ( Xj ) − n+ ϵ ≤Mtj for each j ∈ { 1 , ... , l } • Objective to maximize – fn ( O ) = ∑l j=1 w ( Xj ) 1 { C ( Xj |O ) ≥n } = ∑l j=1 tjw ( Xj ) If we want to enforce non-redundancy constraints such that pairs of overlays that violate certain distance criteria are not both chosen , we can include additional constraints at + ar ≤ 1 for every pair of overlays ( At , Ar ) that we don ’ t want both to be included in the final subset . 4 THE MARGINALGREEDY ALGORITHM FOR MAXIMUM n-TIMES COVERAGE Although NTIMES-ILP can produce near-optimal solutions on problems with reasonable size , it may become intractable for problems where hundreds of thousands of elements and/or overlays are involved , such as certain variants of the vaccine design problem . Thus , we seek a polynomial time algorithm that provides good solutions to the maximum n-times coverage problem . A naive greedy solution is problematic when the n-times objective is directly approached . This is a consequence of potential early bad overlay choices by a greedy approach that can cause it to fail later to find overlays with sufficient n-times coverage . In addition , during greedy optimization available overlay choices may not provide differential marginal gain to avoid ties and random overlay selection . The MARGINALGREEDY algorithm is specifically designed to avoid early bad choices that will lead to failure by marginally approaching the n-times coverage objective . MARGINALGREEDY preserves marginal gains by employing look-ahead tie-breaking that assists in selecting overlays that benefit longer term objectives . Algorithm 1 MARGINALGREEDY algorithm ( for MIN-COST n-TIMES COVERAGE ) Input : Weights of the elements in X : w ( X1 ) , w ( X2 ) , . . . , w ( Xl ) , ground set of overlaysA where each overlay j in A covers Xi for cj ( Xi ) times , target minimum # times being covered ntarget , coverage cutoff for different n : τ1 , τ2 , . . . , τntarget , beam size b Initialize beam B0 ← { ∅ } , t = 0 for n = 1 , . . . , ntarget do Using set function fn ( S ) = ∑l i=1 w ( Xi ) 1 { ∑ j∈S cj ( Xi ) ≥n } as objective function . repeat Kt ← ∅ ▷ Kt is the set of candidate solutions for O ∈ Bt do ▷ Beam search for a ∈ A \O do Kt ← Kt ∪ { O ∪ { a } } ▷ Add the candidate solution { O ∪ { a } } to Kt Bt+1 ← { Top b candidate solutions in Kt as scored by fn ( S ) } t← t+ 1 mt ← median score of candidate solutions in the beam Bt until mt ≥ τn ( in ntarget-th cycle the termination condition is maxS∈Bt fn ( S ) ≥ τtarget ) ( for MAXIMUM n-TIMES COVERAGE with cardinality constraint k , there is additional termination condition t > k ) O∗ = argmaxS∈Bt fntarget ( S ) Output : The final selected subset of overlays O∗ MARGINALGREEDY optimizes n-times coverage with a sequence of greedy optimization cycles where the n-th cycle optimizes the coverage function fn ( S ) . We establish coverage starting at n = 1 and incrementally increase the n-times criteria to ntarget . Thus early overlay selections are guided by less stringent cycle-specific coverage objectives . A set of coverage cutoffs { τ1 , τ2 , . . . , τtarget } is used as the termination condition for each greedy optimization cycle , and when not specified , we assume τ1 = τ2 = · · · = τtarget by default . We use beam search to keep track of top b candidate solutions at each iteration . In general the larger the beam size , the closer the result is to the true optimal . However , there is a tradeoff between beam size and running time . In our results we choose the largest beam size that maintains a practical running time as we describe below . The full algorithm is given in Algorithm 1 . A similar algorithm can be used to solve the n-TIMES SET COVER problem in which τtarget = fntarget ( A ) . For the MAXIMUM n-TIMES COVERAGE problem with cardinality constraint k , the optimization terminates when t > k. When beam search is not used to reduce computation time ( b = 1 ) , we have extended the algorithm to break ties during the n-times coverage iteration by looking ahead to ( n+ 1 ) -times coverage . We call this extension look ahead tie-breaking . Another advantage of MARGINALGREEDY is that it is capable of guaranteeing high coverage for nt < ntarget by controlling the cycle-specific coverage cutoffs τt . This is desired in vaccine design where wider population coverage is also important and we want to make sure that almost 100 % of the population will be covered at least once .

Summary: Update: I have read the author feedback and other reviews. Based on the more detailed comparison to existing methods, I have updated several of my scores.  ---  In this work, the authors propose a variant of the set covering problem which aim to formalize peptide-based vaccine design as an optimization problem. The authors first formulate the problem and then draw connections to other set covering problems; they then show that the problem is NP-complete. They then propose a greedy algorithm to perform the optimization. A set of experiments suggests the proposed approach outperforms the formulations considered by the authors.