Title: Prophet Secretary and Matching: the Significance of the Largest Item
Domain: Computer Science

Question: # Design and Analysis of Improved Online Algorithms for Prophet Secretary and Prophet Secretary Matching Problems | | ## Objectives | | ### Prophet Secretary Problem | Design an algorithm with a higher competitive ratio to break the previous 0.675 barrier. | | ### Prophet Secretary Matching Problem | Develop an algorithm to surpass the 1−1/e (approximately 0.632) competitive ratio barrier.

Background: # Prophet Inequality and Prophet Matching | | - The study of prophet inequality dates back to the 1970s from optimal stopping theory. It involves $n$ items with independent random values arriving in an adversarial order. Each item's value distribution $F_i$ is known upfront, but the actual value $v_i$ is only revealed upon arrival. The algorithm must decide immediately whether to accept an item, with the goal of maximizing the expected value of the accepted item against the prophet's expected maximum value. The optimal competitive ratio for this problem is $\frac{1}{2}$. | - Prophet matching extends the prophet inequality to bipartite graphs with edge weights from known distributions. One set of vertices is known upfront, while the other arrives online. Upon the arrival of an online vertex, the algorithm decides whether to match it to an offline vertex. The classic prophet inequality is a special case of this model with one offline vertex. A tight $\frac{1}{2}$ competitive algorithm was provided by Feldman et al. in 2015, with further generalizations to settings where all vertices arrive online. | | # Prophet Secretary Problem | | - The prophet secretary problem combines elements of the prophet inequality and the secretary problem. Elements are drawn from known independent distributions and arrive in a uniformly random order. The goal is to design algorithms that achieve the highest possible competitive ratio against the prophet's expected maximum value. | - The study of prophet secretary was initiated by Esfandiari et al., who designed a $1-1/e \approx 0.632$ competitive algorithm and provided an upper bound of $0.75$. Subsequent works have aimed to close the gap between lower and upper bounds, with the state-of-the-art lower bound being $0.672$ and the upper bound $0.723$. | | # Prophet Secretary Matching Problem | | - Less progress has been made on the prophet secretary matching problem. Ehsani et al. provided a $1-1/e$ competitive algorithm, which remained the best result for the general case until recent advancements in special cases. Beating the $1-1/e$ barrier for the general case of prophet secretary matching remains a significant open question in the online algorithms community.

Definitions: ## Competitive Ratio | Measures the performance of online algorithms, defined as the ratio of the algorithm's expected gain to the prophet's expected maximum gain. | ## Blind Strategies | Algorithms that rely only on the distribution of the maximum value, ignoring fine-grained distributional information of individual items' values. | ## Activation Rates | A new perspective introduced to design algorithms, representing the probability of accepting an item based on its identity, value, and arrival time, independent of future items.

Methodology: ## Activation-Based Algorithms | | * **Changing Perspective** : Shift from designing acceptance probabilities to selecting activation rates. When an item arrives, decide whether to activate it based on its identity, value, and arrival time, independent of the subset of future items. Accept the item if it is the first activated one. | * **Poisson Process Approximation** : Regard activation events as a Poisson process. By analyzing the probability of activating items, simplify the dependence of different items' strategies and design activation rates to match their contribution to the prophet benchmark. | | ## Significance of the Largest Item | | * **Focusing on the Largest Item** : Identify the item with the highest probability of being selected by the prophet (the largest item) and design a special strategy for it, as it plays a crucial role in the problem. | * **Special Activation Strategy** : Set the activation probability of the largest item as a piece-wise constant function rather than a simple activation rate. This allows better control over its activation and improves the competitive ratio. | | ## Multistage Algorithm for Matching | | * **Multistage Design** : Propose a three-stage algorithm for prophet secretary matching. The first two stages are non-adaptive, similar to the step activation rates algorithm for prophet secretary. The third stage is adaptive, considering the randomness of the instance and the algorithm's execution in the first two stages. | * **Adjusting Activation Rates** : In different stages, adjust the activation rates of edges. For example, in the first stage, use higher activation rates, and in subsequent stages, adjust them based on the situation to balance exploration and exploitation. | | ## Hybrid Algorithm | | * **Combining Algorithms** : To address different scenarios, design multiple algorithms and combine them into a hybrid algorithm to achieve a better competitive ratio. | * **Randomized Selection** : Randomly select between different algorithms with certain probabilities. This leverages the advantages of each algorithm in different situations, enhancing the overall competitive performance.

Answer: Use the continuous‐time formulation of prophet secretary. There are n items with non-negative values v_1,\dots,v_n drawn independently from F_1,\dots,F_n. WLOG we (i) discretize each F_i (continuous can be \varepsilon-approximated), and (ii) assume disjoint supports via infinitesimal tie-breaking so the maximum is unique. Let p_i^v=\Pr[v_i=v]. | | Each item i arrives at time t_i\sim\mathrm{Unif}[0,1] independently. Upon arrival we observe (i,v_i,t_i) and must accept or reject irrevocably; at most one item can be accepted. Performance is measured against the prophet \mathbb{E}[\max_i v_i]. An algorithm is \Gamma-competitive if its expected value is \ge \Gamma\cdot \mathbb{E}[\max_i v_i]. | | Equivalence to random order. The continuous model is equivalent to sampling a uniform random permutation and revealing items in that order; we work in continuous time because it yields cleaner integrals. | | Activation-Based Online Algorithms | | We represent any algorithm by acceptance/activation probabilities g_i^v(t)\in[0,1]: when item i with value v arrives at time t, we activate it with probability g_i^v(t) and accept the first activated item. | | It is convenient to parameterize g via activation rates a_i^v(t)\ge0: | g_i^v(t)\;=\;a_i^v(t)\cdot \exp\!\Big(-\!\int_0^t A_i(x)\,dx\Big), | \qquad | A_i(t)\stackrel{\rm def}{=}\mathbb{E}_{v\sim F_i}[a_i^v(t)]. | Under this form (a nonhomogeneous Poisson view): | • Probability item i activates before \theta: 1-\exp\!\big(-\!\int_0^\theta A_i(t)\,dt\big). | • Probability we accept item i with value v: | \int_0^1 p_i^v\,a_i^v(t)\cdot \exp\!\Big(-\!\int_0^t \sum_j A_j(x)\,dx\Big)\,dt. | | Let | \rho_i^v\stackrel{\rm def}{=} \prod_{j\ne i}\Pr[v_j< v], \qquad | x_i^v\stackrel{\rm def}{=} p_i^v\,\rho_i^v, | so \sum_{i,v} x_i^v=1. A sufficient condition for \Gamma-competitiveness is | \int_0^1 p_i^v\,g_i^v(t)\prod_{j\ne i}\Big(1-\!\int_0^t \mathbb{E}_{u\sim F_j}[g_j^u(x)]\,dx\Big)\,dt\;\ge\; \Gamma\,x_i^v | \quad(\forall i,v). | \tag{★} | | Warm-ups | • Constant Activation Rates a_i^v(t)=\rho_i^v for all t is (1-1/e)-competitive (classical). | • Two-stage Step Rates: split [0,1] at \beta and let pre/post rates sum to 2\rho_i^v; under a standard small-items condition this yields 0.694. | | Main Idea: Treat the Largest Item Separately | | Let the largest item i_0=\arg\max_i \sum_v x_i^v with x_0=\sum_v x_{i_0}^v. We focus on the item the prophet is most likely to take and give it a special (probability-based) policy, while all other items use rate-based step policies with invariants that depend on i_0’s characteristics. This “one large item + many small” split is key to beating blind-strategy barriers. | | Algorithm: Step Activation Except One Large Item | | Parameters \beta_0\le \beta_1\le \beta_2 (chosen by computer-assisted optimization). | • Largest item i_0 (use probabilities, not rates): | g_{i_0}^v(t)= | \begin{cases} | z_{i_0}^v,& t\in[0,\beta_0),\\ | 0,& t\in[\beta_0,\beta_2),\\ | 2\rho_{i_0}^v - z_{i_0}^v,& t\in[\beta_2,1], | \end{cases} | \quad z_{i_0}^v=(2\rho_{i_0}^v-1)_+. | • Other items i\ne i_0 (use rates): | a_i^v(t)= | \begin{cases} | z_i^v,& t\in[0,\beta_1),\\ | 2\rho_i^v - z_i^v,& t\in[\beta_1,1], | \end{cases} | \quad \text{with } z_i^v\in\big[(2\rho_i^v-1)+,\;\rho_i^v\big] | and the invariant | \sum{i\ne i_0}\sum_v p_i^v z_i^v | \;=\; | \min\{\,h_2(x_0)-h_0,\;1-x_0\,\}, | \quad | h_0=\sum_v p_{i_0}^v z_{i_0}^v, | where h_2(\cdot) upper-bounds \sum_{i,v}(2x_i^v-p_i^v)_+ given a largest mass x_0. (Intuition: we cap “early pressure” so that global hazard \sum_j A_j(t) has a clean two/three-phase profile, enabling tight integrals.) | | Explicit contribution of the largest item | | Let | P_{\mathrm{acc}}^{(i_0,v)} | | \int_0^1 p_{i_0}^v\, g_{i_0}^v(t)\, | \exp\!\Big(-\!\int_0^t \sum_j A_j(x)\,dx\Big)\,dt. | Because g_{i_0}^v(t) is zero on [\beta_0,\beta_2) and equals z_{i_0}^v or 2\rho_{i_0}^v-z_{i_0}^v on the other segments, we obtain the lower bound | P_{\mathrm{acc}}^{(i_0,v)} | \;\ge\; | p_{i_0}^v \Big[ | z_{i_0}^v\!\!\int_0^{\beta_0}\!\! e^{-H(t)}dt | +\big(2\rho_{i_0}^v\!-\!z_{i_0}^v\big)\!\!\int_{\beta_2}^{1}\!\! e^{-H(t)}dt | \Big], | \quad | H(t)=\int_0^t \sum_j A_j(x)\,dx, | and we show \sum_j A_j(t) is piecewise constant: | \sum_j A_j(t)= | \begin{cases} | h_0+h_{\mathrm{ot}}, & t\in[0,\beta_0),\\ | h_{\mathrm{ot}}, & t\in[\beta_0,\beta_1),\\ | 1, & t\in[\beta_1,\beta_2),\\ | 2-h_0, & t\in[\beta_2,1], | \end{cases} | with h_{\mathrm{ot}}=\min\{h_2(x_0)-h_0,\,1-x_0\}. This yields closed-form exponentials for each segment and, after summing over v and adding the “other items” contribution (handled analogously via rates), establishes the competitive lower bound. | | Main Result | | Theorem 1. There exists a 0.688-competitive algorithm for the prophet secretary problem. | Discussion. The previous best was 0.672 and blind strategies are capped by 0.675; our algorithm breaks this barrier by (i) shifting from acceptance probabilities to activation rates, (ii) explicitly conditioning on item identity (vs. blind), and (iii) special treatment of the largest item with invariants guiding the rest. Parameters (\beta_0,\beta_1,\beta_2) are chosen by computer-assisted search over the closed-form bounds above. | | ⸻ | | Prophet Secretary Matching | | We extend the activation framework to bipartite matching under random arrival of online vertices and independent types. Let x_{(i,u,v)} denote the mass that edge (u,v_i=v) is the prophet’s winner; \sum_{i,u,v} w_{uv} x_{(i,u,v)} upper-bounds the prophet’s expected matching weight. | | Multistage Activation-based Matching (MAM) | | Three phases with cut points \beta_0=0.05,\ \beta_1=0.75. | • Phase 1: a_{(i,u,v)}(t)=(2\rho_{(i,u,v)}-1)_+. | • Phase 2: a_{(i,u,v)}(t)=\rho_{(i,u,v)}. | • Phase 3: redistribute residual activation mass toward still-unmatched u’s; proposals to u are accepted iff u remains free. The per-vertex hazard \sum_i A_i^u(t) becomes piecewise constant, enabling tight integrals. This yields | \Pr[\text{match }(u,v_i=v)] \;\ge\; \Gamma_{\mathrm{MAM}}(x_u)\cdot x_{(i,u,v)}. | | Constant-Probability with a Reserved Edge (CPR) | | A complementary scheme that keeps constant proposals but defers activation for each u’s most important online index i_u until a threshold \alpha(x_u). | | Hybrid and Main Guarantee | | Theorem 17. The hybrid that runs MAM with probability 0.8 and CPR with probability 0.2 is at least 0.641-competitive for prophet secretary matching: | \Pr[\text{match }(u,v_i=v)]\ \ge\ \big(0.8\,\Gamma_{\mathrm{MAM}}(x_u)+0.2\,\Gamma_{\mathrm{CPR}}(x_u)\big)\,x_{(i,u,v)}\ \ge\ 0.641\,x_{(i,u,v)}. | Milestone. This surpasses the 1-1/e\approx 0.632 barrier for the first time. |

Checklist: 1. The answer should introduce the continuous-time formulation of the prophet secretary problem, where each item arrives at a random time in [0,1].  
2. It should explain that distributions can be discretized and supports assumed disjoint, so the maximum item is unique.  
3. The equivalence between the continuous-time model and random order arrival should be clearly stated.  
4. The activation-based framework must be described: algorithms are represented by activation probabilities g_i^v(t), derived from activation rates a_i^v(t).  
5. The acceptance probability formula involving integrals of activation rates over time must be included.  
6. The definitions of ρ_i^v (probability all other items are smaller) and x_i^v = p_i^v·ρ_i^v should be provided, with ∑ x_i^v = 1.  
7. The sufficient competitiveness condition (inequality (★)) connecting acceptance probability integrals to Γ·x_i^v should be mentioned.  
8. Warm-up strategies should be presented: constant activation rates yielding 1-1/e and two-stage step rates improving to ~0.694 under a small-item condition.  
9. The key idea of treating the largest item separately must be described, with i₀ = argmax ∑ x_i^v and the “one large + many small” philosophy.  
10. The special step activation algorithm should be detailed: three phases for the largest item with parameters (β₀,β₁,β₂) and invariants for other items.  
11. The piecewise constant hazard function Σ_j A_j(t) across the different intervals should be presented, enabling closed-form integrals.  
12. The contribution of the largest item should be explicitly lower bounded, combining the early and late segments of g_{i₀}^v(t).  
13. The main result should be stated: existence of a 0.688-competitive algorithm for prophet secretary, surpassing the blind-strategy barrier (0.675) and prior best (0.672).  
14. The improvements should be attributed to three factors: moving from acceptance probabilities to activation rates, conditioning on item identity, and special treatment of the largest item.  
15. The extension to prophet secretary matching must be covered, defining x_{(i,u,v)} and how it upper-bounds the prophet’s expected matching weight.  
16. The Multistage Activation-based Matching (MAM) algorithm should be described, with phases, hazards, and per-vertex integrals.  
17. The Constant-Probability with Reserved edge (CPR) scheme should be introduced as a complementary strategy.  
18. The hybrid algorithm combining MAM and CPR should be presented, with Theorem 17 guaranteeing at least 0.641-competitiveness.  
19. The significance should be emphasized: 0.641 breaks the classic 1-1/e ≈ 0.632 barrier for prophet secretary matching.