{"id":"78e059fe-e2dd-47b1-b543-d2ecb47d15ac","arxiv_id":"2507.02381","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":3,"one_line_summary":"The paper extends the average-gain drift model to a k-step multiple-gain bound and applies it to obtain expected first hitting time upper bounds for (mu+lambda) EA on three combinatorial problems.","lead":"This paper proposes a multiple-gain drift model and claims tighter running-time upper bounds for a (mu+lambda) evolutionary algorithm on knapsack, k-MAX-SAT, and convex-position TSP instances. The model is largely a restatement of standard k-step drift analysis, and the worst-case experimental verification fits a parameter from the same runs.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3's knapsack drift bound is not established: the proof multiplies the probability of an improvement by an unconditional case mixture and drops the 1/(n^2 e) factor, so Eq. (7) can exceed the true one-step drift on a single Case-1 state.","rationale":"The reader's weakest assumption identified the same load-bearing defect in Theorem 3: the drift lower bound mixes the two improvement cases with state-independent probabilities p1 and p2 and then multiplies by the unnormalized gain mixture, which is not a valid lower bound for every state. My concrete calculation confirms the failure is quantitative, not merely a missing justification: on a feasible Case-1 state the claimed bound is roughly twice the exact drift. Since Theorem 3 and Corollary 2 are the main claimed improvements over Neumann and Sutton [4], the central contribution is not established. The k-MAX-SAT and TSP sections do not rescue the claim, and the experimental validation fits klow from the same data that Equation (10) is checked against. I therefore retain the reader's REJECT verdict; no adjustment is needed.","tokens_in":23854,"tokens_out":15877,"duration_ms":183789,"concrete_test":"Recompute the one-step drift in Theorem 3 for the single Case-1 state n=100, q=2, mu=1, lambda=10, K=3, values (2,2,1,...,1), weights (1,1,2,...,2), x=(1,0,0,1,0,...,0), exactly following Algorithm 1's rejection rule. Enumerate all 1-bit and 2-bit mutations of this state and compute E[Y_t - Y_{t+1} | x_t = x]. Compare this value to the right-hand side of Eq. (7) using p2=3/(3n-2) and p1=1-p2. If the right-hand side exceeds the exact drift, Theorem 3's drift inequality G(t,1) >= h1(r_p) fails and the knapsack bound is not established.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claimed improvement over [4] rests on Theorem 3's lower bound G(t,1) >= (1 - exp(-lambda(p1+n(1-p1))/(mu n^2 e)))(p1 dmin + p2 vmin). The proof derives p4 = 1 - exp(...) and then multiplies p4 by p1 dmin + p2 vmin. This is invalid in two ways. First, p1 and p2 are global fractions of feasible solutions, but for a fixed parent x_t the case is determined by x_t: if x_t has a 1 in positions q+1..n, only the two-bit swap can improve; if it has none, only the one-bit flip can. A convex combination of the two cases cannot hold for every state. Second, p4 is the probability that at least one improving offspring is generated; unconditional expected gain is the sum over improving event types of probability times gain, not p4 times the unnormalized mixture. The p3 expression already contains 1/(n^2 e), but multiplying by (p1 dmin + p2 vmin) removes it. Concrete failure: n=100, q=2, mu=1, lambda=10, capacity 3, values (2,2,1,...,1), weights (1,1,2,...,2), x=(1,0,0,1,0,...,0). The only improving mutation is swapping the zero in the first q positions with the one in the last n-q positions, with probability at least 1/(n^2 e) per offspring. The true one-step drift is about 1-(1-1/(n^2 e))^10, roughly 0.00037, while the proof's bound with p2=3/(3n-2) and p1=1-p2 is about 0.00073. Eq. (7) is therefore not a valid drift lower bound, and the claimed improvement over [4] is unproven.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a 'multiple-gain' model, a k-step generalization of the average-gain drift technique, and uses it to derive upper bounds on the expected first hitting time of (μ+λ) EAs for the knapsack problem with favorably correlated weights, general k-MAX-SAT, and the TSP on point sets in convex position. It also reports experiments with 1000 runs per instance and correlation coefficients between the derived bounds and empirical hitting times. The paper claims improvements over the known knapsack bound of Neumann and Sutton and the TSP bound of Sutton et al.","tokens_in":24334,"tokens_out":17807,"duration_ms":176502,"significance":"The paper addresses a real gap in the runtime analysis of population-based EAs on combinatorial problems, and the multi-step gain framework is a reasonable idea that could be useful if made rigorous. Theorem 5 is a valid, though weak, closed-form bound: it relies on mutation probability 1/2 and gives O(2^n ln s/(λ Nopt)). The main claimed contributions for knapsack and TSP, however, are not established because the drift lower bound in Theorem 3 is false for a concrete state and the probability computation in Theorem 7 uses the wrong distribution for the number of 2-opt inversions. The experimental work is transparent about its protocol, but the worst-case validation is circular because klow is estimated from the same runs used to evaluate the bound. Overall the central claims are not yet reliable.","major_comments":[{"comment":"The drift lower bound in the proof of Theorem 3 is not valid, because p1 and p2 are global fractions of feasible solutions rather than probabilities over the current parent state, and the proof multiplies the probability of an improving offspring by the unweighted mixture p1 dmin + p2 vmin. For a fixed xt, the case is determined by whether xt has a 1-bit in positions q+1..n, so a convex combination of the Case 1 and Case 2 gains cannot be a lower bound for every state. Concretely, take n=100, q=2, μ=1, λ=10, capacity 3, values (2,2,1,...,1), weights (1,1,2,...,2), and xt=(1,0,0,1,0,...,0); here the only improving single mutation is the swap of a zero in the first two positions with the 1-bit at position 4, whose probability is at least 1/(n^2 e) per offspring, giving a true one-step drift of approximately 1-(1-1/(n^2 e))^{10} ≈ 0.00037, while the claimed bound in Eq. (7) is about 0.00073. Hence Eq. (7) can exceed the true drift on a single state, and the knapsack upper bound and its claimed improvements over [4] are not established.","section":"IV-B, Theorem 3, Eq. (7)"},{"comment":"The proofs of Theorems 1 and 2 have an optional-stopping gap. Theorem 1 applies Lemma 1 to the k-skeleton process {f(x_{kt})} with the original stopping time T0, but T0 is not adapted to the skeleton filtration because the event {T0 ≤ n} depends on f(x_t) for t that are not multiples of k; the integrability and finiteness conditions of the optional stopping theorem are also not verified. Theorem 2 invokes Theorem 1 directly ('According to Theorem 1'), so the gap propagates into the average-case bounds Eqs. (4), (7), (13), and (19). The issue may be repairable with a standard drift argument on the skeleton plus an overshoot term, but the manuscript does not provide such an argument.","section":"III-B, Theorems 1-2, Eqs. (2) and (4)"},{"comment":"In the proof of Theorem 7, E1 is defined as the event that mutation performs only one random 2-opt inversion, which has probability P(s=0)=e^{-λp} under the Poisson(λp) mutation, but the proof states the probability is λp/e^{λp}, which is instead P(s=1), i.e., two inversions. This wrong factor is used in the lower bound on the success probability p and therefore in Eq. (19), so the stated TSP bound is not established for general λp (the error is masked for λp=1 because both expressions equal 1/e). The conditional probabilities 1/(n−2) and 1/(n−3) used in the computation of g are also asserted without a distributional justification, and Theorem 8 inherits these issues.","section":"IV-D, Theorem 7, Eq. (19)"},{"comment":"The worst-case experimental validation is circular. In Section V-B the theoretical klow from Eq. (10) is replaced by the empirical average bk of the longest zero-gain intervals from 1000 runs (Eq. (5)), and this bk is then plugged into Eq. (11) to produce the 'theoretical worst-case upper bound' that is compared with Tmax from those same 1000 runs. This comparison does not test the predictive content of Theorems 4, 6, and 8; it only checks that an empirical plug-in quantity exceeds the empirical maximum. The paper's claim that the experiments verify the worst-case bounds is therefore not supported.","section":"V-B, Eqs. (5), (10), (11)"}],"minor_comments":[{"comment":"'attend' is used where 'attain' is meant in several places, e.g., the Abstract and Definition 3; the wording should be corrected throughout.","section":"Global"},{"comment":"Table II is titled 'RESULTS OF CORRELATION COEFFICIENT OF ALGORITHM 1 FOR THE k-MAX-SAT PROBLEM', but the experiment reported there concerns Algorithm 2.","section":"V-C, Table II"},{"comment":"The experimental setting states that Algorithms 1 and 2 were implemented in Matlab, but Algorithm 3 (TSP) is also benchmarked; the sentence should include it.","section":"V-A"},{"comment":"The proof of Theorem 7 uses S = {0,1,...,n}, whereas the text before Algorithm 3 states that f(x) ∈ {0,2,3,...,n} for n>5; these two descriptions of the fitness range should be reconciled.","section":"IV-D"},{"comment":"The intermediate expression for g in the proof of Theorem 7 and the closed form in Eq. (19) are not shown to be equal; adding the algebraic step would help the reader verify the calculation.","section":"IV-D, Eq. (19)"}],"recommendation":"reject","confidential_remarks":"The multiple-gain framework has potential, and a revised manuscript that fixes the drift argument in Theorem 3 and the inversion-count probability in Theorem 7, and that removes the circular worst-case validation, could be reconsidered; the current version is not publishable."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The core claim here is that a k-step generalization of the average gain model yields provable upper bounds for (μ+λ) EA on knapsack, k-MAX-SAT, and TSP. The framework is cleanly written, and the individual analyses are serious attempts. But the central knapsack result is not established, and the experimental validation is circular.\n\nThe stress-test counterexample for Theorem 3 is real. Take n=100, q=2, capacity 3, values (2,2,1,...,1), weights (1,1,2,...,2), and the parent x=(1,0,0,1,0,...,0). This parent can only improve via a two-bit swap, and the true one-step drift is about 1-(1-1/(n^2 e))^10, roughly 0.00037. The proof's bound, using the unweighted mixture p1 dmin + p2 vmin, is about 0.00073. So Eq. (7) exceeds the actual drift at this state. The gap is not cosmetic: p1 and p2 are global combinatorial fractions, but the case is determined by the current parent, and the expected gain conditional on an improvement must weight one-bit and two-bit improvements differently. Corollary 2's claimed log-factor improvement over Neumann and Sutton's knapsack bound falls with it.\n\nThe k-MAX-SAT result (Theorem 5) is correct but nearly trivial: with mutation probability 1/2, the algorithm is essentially random search, and the bound is the standard coupon-collector/random-sampling estimate. The TSP bound on convex-position instances may be valid, but comparing it to Sutton et al.'s planar Euclidean bound overclaims, since convex position is a restricted subclass.\n\nWhat the paper does well: the k-step gain definition is a reasonable pedagogical framing, Theorems 1–2 are straightforward martingale arguments (modulo minor optional-stopping details), and the authors are transparent about setting k=1 in the case studies. The related-work survey is useful. None of that rescues the main theorem.\n\nThe experiments are not confirmatory: klow is estimated from the same 1000 runs and then substituted into Formula (11), and α is estimated from the same dataset; the reported correlations mostly measure self-consistency of a fitted curve, not agreement between an independent prediction and observation.\n\nWho gets value? A reader interested in drift-analysis techniques for population-based EAs might skim the framework, but I would not rely on any of the stated bounds as proved. The paper deserves a serious referee because the flaws are specific and potentially repairable—if the drift argument in Theorem 3 can be fixed and the experiments de-circularized, the knapsack and TSP results could be worth publishing. As written, it should not be accepted.","headline":"A well-structured but ultimately under-supported paper: the multiple-gain model is drift analysis in disguise, and the central knapsack bound (Theorem 3) rests on an invalid drift inequality that the stress-test counterexample correctly exposes.","tokens_in":24824,"tokens_out":2418,"would_cite":false,"duration_ms":29504,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["68Q25","68W20","90C27"],"pacs":[],"model":"deepseek-v4-flash","headline":"One drift bound tames (µ+λ) EA on knapsack, SAT, TSP","keywords":["(µ+λ) evolutionary algorithm","multiple-gain model","expected first hitting time","running-time analysis","knapsack problem","k-MAX-SAT","traveling salesperson problem","average gain drift"],"falsifier":"Enumerate every feasible solution of a small knapsack instance with favorably correlated weights, compute the exact transition probabilities of Algorithm 1 from each state, and verify whether $G(t,1)\\ge(1-e^{-\\lambda(p_1+n(1-p_1))/(\\mu n^2 e)})(p_1d_{\\min}+p_2v_{\\min})$ holds for every state with $Y_t>0$; a single state that violates it, with $p_1,p_2$ computed as in Theorem 3, refutes the claimed bound. For a concrete test, take $n$ small (say 5), choose values so that $d_{\\min}>v_{\\min}$ (e.g., values $50,30,1,1,1$), and check the states whose last $n-q$ bits are all zero, where Case 2 dominates.","tokens_in":23651,"feed_emoji":"⏱️","tokens_out":14745,"duration_ms":146254,"temperature":0.7,"pith_summary":"This paper sets out to extend running-time analysis from simple (1+1) heuristics to population-based $(\\mu+\\lambda)$ evolutionary algorithms on combinatorial optimization problems. It replaces the single-generation average gain with an expected multiple-gain $G(t,k)$, the expected fitness drop of the best individual over $k$ generations, and proves two drift theorems that turn a lower bound on that gain into an upper bound on the expected first hitting time. The theorems are applied to three problems: knapsack with favorably correlated weights, general $k$-MAX-SAT, and the traveling salesperson problem on points in convex position. The paper claims average-case and worst-case upper bounds for all three, including a closed-form $k$-MAX-SAT expression and improvements over previously known knapsack and TSP bounds. A sympathetic reader would care because population-based EAs are widely used but have far fewer provable time bounds than the $(1+1)$ EA.","feed_headline":"One drift bound tames (µ+λ) EA on knapsack, SAT, TSP","feed_subtitle":"A gain-based drift framework converts population runs into provable expected-hitting-time upper bounds.","key_machinery":"The load-bearing object is the expected multiple-gain $G(t,k)$, a fitness-difference drift measured over $k$ generations instead of one; the average gain is the special case $k=1$. Theorems 1 and 2 are the levers: they construct a potential $g(f(x_t))=\\sum_{i=1}^{p}(r_i-r_{i-1})/h(r_i)$ whose expected decrease is at least one per $k$-step block, and then use the supermartingale optional-stopping lemma to convert that into a bound on $T_0=\\min\\{t:Y_t=0\\}$. The worst-case version also introduces $k_{\\mathrm{low}}$, the smallest $k$ for which the $k$-step gain covers the adjacent fitness gap, and the paper estimates it both from the drift formulas and, experimentally, as the average longest zero-gain stretch over repeated runs.","core_discovery":"The central claim is that the expected multiple-gain $G(t,k)=E(f(x_t)-f(x_{t+k})\\mid H_t)$ is sufficient to control the whole run. Theorem 2 shows that if $G(t,k)\\ge h(f(x_t))$ for a monotonically nondecreasing $h$, then the average-case expected first hitting time satisfies $E(T_0\\mid f(x_0)=r_L)\\le k\\sum_{i=1}^{L}(r_i-r_{i-1})/h(r_i)$; Theorem 1 and Corollary 1 give the worst-case form $E(T_0\\mid f(x_0))\\le k_{\\mathrm{low}}f(x_0)/\\alpha$ when the gain over $k$ generations covers the smallest adjacent fitness gap $\\alpha$. Using $k=1$ for simplicity, the paper derives the knapsack bound $E(T_0\\mid Y_0)\\le (Y_0-r_0)/((1-e^{-\\lambda(p_1+n(1-p_1))/(\\mu n^2 e)})(p_1d_{\\min}+p_2v_{\\min}))$, the $k$-MAX-SAT bound $E(T_0\\mid Y_0)\\le (\\sum_{x=1}^{s}1/x)/(1-e^{-\\lambda N_{\\mathrm{opt}}/2^n})$, and the convex-TSP bound $E(T_0\\mid f(x_0)=L)\\le \\frac{2}{1+g}(L+\\mu e^{\\lambda_p}C_n^2/(\\lambda\\lambda_p)\\sum_{x=1}^{L}1/x)$, which is $O((\\mu/\\lambda)n^2\\ln n+n)$. It further claims these improve the existing knapsack and TSP upper bounds in the stated parameter regimes.","pith_inferences":["Because the knapsack proof uses $p_1$ and $p_2$ as fixed fractions of the feasible-solution set and multiplies them directly by $d_{\\min}$ and $v_{\\min}$, the bound's validity depends on whether that state-independent mixture is truly a lower bound on $G(t,1)$ for every population state; a natural check is to enumerate all states for small $n$ and compare the exact conditional drift with the formu","A corrected version of the knapsack argument would likely condition on the actual improvement probabilities, which weight Case 2 improvements by roughly $n$ relative to Case 1; that would change the claimed rate in regimes where $d_{\\min}$ and $v_{\\min}$ differ substantially.","The same $G(t,k)$ machinery should transfer to other population EAs and other combinatorial objectives whenever the best individual's fitness admits a per-generation expected-improvement lower bound, suggesting that the three case studies are templates rather than isolated results.","A fully theoretical worst-case analysis would replace the experimentally estimated $k_{\\mathrm{low}}$ with a rigorous upper bound on the longest zero-gain stretch; doing so would make the worst-case theorems independent of measurement."],"forward_implications":["For the knapsack instance, Corollary 2 gives $O(\\mu n^2(r_L-r_0)/(\\lambda d_{\\min}))$ when $p_1=1$ and $O(\\mu n(r_L-r_0)/(\\lambda v_{\\min}))$ when $p_1=0$; at $\\mu=\\lambda=1$ these improve the earlier $O(n^2(\\ln n+p_{\\max}))$ bound by factors of $\\ln n$ and $n\\ln n$.","For every $k$-MAX-SAT instance, Corollary 3 gives $O(2^n(\\ln s+1)/(\\lambda N_{\\mathrm{opt}}))$, where $N_{\\mathrm{opt}}$ is the number of global optima, extending results that previously covered only specific random or small-$k$ instances.","For convex-position TSP, Corollary 4 gives $O((\\mu/\\lambda)n^2\\ln n+n)$, improving the earlier $O(n\\max\\{(\\mu/\\lambda)n^2,1\\})$ bound by $n/\\ln n$ when $(\\mu/\\lambda)n^2>1$.","The worst-case counterparts (Theorems 4, 6, 8) use the same drift machinery with $k_{\\mathrm{low}}$; the paper estimates $k_{\\mathrm{low}}$ from runs and reports correlation coefficients above 0.99 between the theoretical bounds and observed first hitting times on the three test instances.","The knapsack theorem contains the earlier $(1+1)$ EA bound as the special case $\\mu=1,\\lambda=1$, so the population-level result is a strict generalization rather than a separate analysis."],"supporting_citations":[{"why":"supplies the knapsack upper bound that Theorem 3 generalizes and claims to improve.","marker":"[4]"},{"why":"provides the average-gain model and the supermartingale stopping-time lemma on which the multiple-gain theorems are built.","marker":"[28]"},{"why":"motivates extending gain-based first-hitting-time estimation from numerical to combinatorial optimization.","marker":"[29]"},{"why":"gives the parameterized planar Euclidean TSP bound that Corollary 4 improves for convex instances.","marker":"[32]"},{"why":"prior running-time analysis of MAX-2-SAT, the specific-instance baseline for the general k-MAX-SAT result.","marker":"[31]"},{"why":"prior analysis of the (1+1) EA on random 3-CNF formulas, marking the gap to general k-MAX-SAT.","marker":"[10]"},{"why":"prior analysis of another GA variant on random 3-CNF, used as a k-MAX-SAT comparison point.","marker":"[30]"},{"why":"context for the (µ+λ) EA formulation and its known lower-bound behavior on pseudo-Boolean functions.","marker":"[46]"}],"fun_headline_variants":["Multiple-gain drift tightens EA bounds for knapsack, SAT, TSP","Multiple-gain model improves (µ+λ) EA runtime analysis on three problems","Tighter expected hitting times for (µ+λ) EA on knapsack, SAT, TSP","New drift model yields sharper EA bounds for combinatorial optimization","Gain-based drift improves EA runtime bounds for knapsack, SAT, TSP"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The knapsack bound rests on treating $p_1$ and $p_2$ as fixed combinatorial fractions of all feasible solutions and on the unproven inequality $G(t,1)\\ge(1-e^{-\\lambda(p_1+n(1-p_1))/(\\mu n^2 e)})(p_1d_{\\min}+p_2v_{\\min})$ holding for every population state; if that state-independent mixture is not a valid lower bound on the conditional drift, Theorem 3 collapses.","fun_headline_variants_meta":{"raw":{"variants":["Multiple-gain drift tightens EA bounds for knapsack, SAT, TSP","Multiple-gain model improves (µ+λ) EA runtime analysis on three problems","Tighter expected hitting times for (µ+λ) EA on knapsack, SAT, TSP","New drift model yields sharper EA bounds for combinatorial optimization","Gain-based drift improves EA runtime bounds for knapsack, SAT, TSP"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000978,"raw_usage":{"total_tokens":4274,"prompt_tokens":1183,"completion_tokens":3091,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":799,"completion_tokens_details":{"reasoning_tokens":2984}},"tokens_in":799,"tokens_out":3091,"duration_ms":24961,"temperature":1.0,"reasoning_tokens":2984,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T20:33:56.769294+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Enumerate every feasible solution of a small knapsack instance with favorably correlated weights, compute the exact transition probabilities of Algorithm 1 from each state, and verify whether $G(t,1)\\ge(1-e^{-\\lambda(p_1+n(1-p_1))/(\\mu n^2 e)})(p_1d_{\\min}+p_2v_{\\min})$ holds for every state with $Y_t>0$; a single state that violates it, with $p_1,p_2$ computed as in Theorem 3, refutes the claimed bound. For a concrete test, take $n$ small (say 5), choose values so that $d_{\\min}>v_{\\min}$ (e.g., values $50,30,1,1,1$), and check the states whose last $n-q$ bits are all zero, where Case 2 dominates.","supporting_citations":[{"cited_title":"Runtime analysis of evolutionary algorithms for the knapsack problem with favorably correlated weights,","cited_arxiv_id":null,"evidence_quote":"supplies the knapsack upper bound that Theorem 3 generalizes and claims to improve."},{"cited_title":"Runtime analysis for continuous (1+1) evolutionary algorithm based on average gain model (in Chinese),","cited_arxiv_id":null,"evidence_quote":"provides the average-gain model and the supermartingale stopping-time lemma on which the multiple-gain theorems are built."},{"cited_title":"First hitting time analysis of continuous evolutionary algorithms based on average gain,","cited_arxiv_id":null,"evidence_quote":"motivates extending gain-based first-hitting-time estimation from numerical to combinatorial optimization."},{"cited_title":"A parameterized runtime analysis of evolutionary algorithms for MAX-2-SAT,","cited_arxiv_id":null,"evidence_quote":"gives the parameterized planar Euclidean TSP bound that Corollary 4 improves for convex instances."},{"cited_title":"Runtime analysis of the (1+( λ, λ)) genetic algorithm on random satisfiable 3-CNF formulas,","cited_arxiv_id":null,"evidence_quote":"prior running-time analysis of MAX-2-SAT, the specific-instance baseline for the general k-MAX-SAT result."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"prior analysis of the (1+1) EA on random 3-CNF formulas, marking the gap to general k-MAX-SAT."},{"cited_title":"An experimental method to estimate running time of evolutionary algorithms for continuous optimization,","cited_arxiv_id":null,"evidence_quote":"prior analysis of another GA variant on random 3-CNF, used as a k-MAX-SAT comparison point."},{"cited_title":"Crossover can simulate bounded tree search on a fixed- parameter tractable optimization problem,","cited_arxiv_id":null,"evidence_quote":"context for the (µ+λ) EA formulation and its known lower-bound behavior on pseudo-Boolean functions."}],"review_version":1}