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EFTA01139453 | EFTA01139481 6 Complexity Results for the Exponential Fitness Function ExF In this section we consider the case where the fitness of an individual at a vertex is an exponential function of the payoff, i.e., the fitness of an individual at vertex v is If f(v) =°c13 EuEEIZ P v7I ti) ) and we do not have density constraint. We first present the results, and describe how to obtain them. I. Result I. The qualitative problem can be solved in polynomial time. 2. Result 2. For the no resident reproduction case (i.e., when the fitness of a resident is set to 0), the quantitative approximation problem can be solved in polynomial time. 3. Result 3. For the resident reproduction case, we have the same complexity bounds as in the case where we have the LBF. Result I. Since the fitness is an exponential function and always positive, the fixation probability is positive if the replacement graph is connected, and otherwise the fixation probability is 0. Since whether a graph is connected or not can be checked in polynomial time, it follows that the qualitative problem can be solved in polynomial time. Result 2. In case the resident does not reproduce, and the fitness of the mutant is always positive due to an exponential function of the payoff, then for every vertex a mutant at the vertex reproduces to turn all its successor in the replacement graph as mutants. Hence given a vertex v. if the mutant arises at v, then all vertices that are reachable from v in the replacement graph become mutants with probability I. Given a vertex v, we say that v E All, iff all vertices in the graph are reachable from v in the replacement graph. Then the fixation probability is IAIII/In i.e., the probability that the mutant arises initially in any one of the vertices in All. Since reachability can be computed in polynomial time, it follows that the exact fixation probability (and hence also the approximation) can be computed in polynomial time. Theorem 16. For the general l&R model with the fitnessfunction is exponentialfunction of the payoff the following assertions hold: I. The qualitative problem can be solved in polynomial time. 2. For no resident reproduction, the exactfixation probability (hence also the quantitative approximation problem) can be computed in polynomial time. Result 3. Given Theorem 16 the only remaining problem is to consider the problem of quantitative approximation for the model with resident reproduction. We prove that the complexity results of Section 5 also hold when the fitness is an exponential function of the fitness. The PSPACE upper bound. The same argument for the PSPACE upper bound of Section 5.1 also holds when the entries of the payoff matrix are polynomial in the input, as an exponential size Markov chain can be constructed (the probabilities are obtained using the exponential fitness function), and hence we obtain the PSPACE upper bound. The PSPACE lower bound. The rest of the section is devoted to show how to modify the PSPACE lower bound of Section 5.2 to show that the lower bound is also applicable to the case where the fitness is exponential of the payoff (and there is no density constraint). Note that Remark I shows that density constraints can be encoded in LBF, but it does not show how to encode density constraint in ExF. In the sequel, we use "with high probability" to refer to that the probability is at least 1 — tp ) , where it is the size of input, and poly is a polynomial function. The key idea. The key idea is as follows: First step: First, we consider the problem with constant payoff along with density constraints and argue that the PSPACE hardness result holds even in the case where either mutants or residents fixate within an exponential number of steps with high probability. EFTA01139481
May 10, 2026 at 3:10 PM