[Paper Review] Combinatorial Secretary Problems with Ordinal Information

Martin Hoefer, Bojana Kodric|arXiv (Cornell University)|Jan 1, 2017

Optimization and Search Problems13 references3 citations

TL;DR

This paper introduces and analyzes combinatorial secretary problems under ordinal information, where algorithms only access relative rankings of arriving elements rather than exact numerical values. It presents new constant-competitive algorithms for bipartite matching, general packing LPs, and independent set in graphs with bounded local independence number, while showing that thresholding-based methods fail in the ordinal model, establishing a Ω(√n/log n) lower bound for structure-oblivious algorithms.

ABSTRACT

The secretary problem is a classic model for online decision making. Recently, combinatorial extensions such as matroid or matching secretary problems have become an important tool to study algorithmic problems in dynamic markets. Here the decision maker must know the numerical value of each arriving element, which can be a demanding informational assumption. In this paper, we initiate the study of combinatorial secretary problems with ordinal information, in which the decision maker only needs to be aware of a preference order consistent with the values of arrived elements. The goal is to design online algorithms with small competitive ratios. For a variety of combinatorial problems, such as bipartite matching, general packing LPs, and independent set with bounded local independence number, we design new algorithms that obtain constant competitive ratios. For the matroid secretary problem, we observe that many existing algorithms for special matroid structures maintain their competitive ratios even in the ordinal model. In these cases, the restriction to ordinal information does not represent any additional obstacle. Moreover, we show that ordinal variants of the submodular matroid secretary problems can be solved using algorithms for the linear versions by extending [Feldman and Zenklusen, 2015]. In contrast, we provide a lower bound of Omega(sqrt(n)/log(n)) for algorithms that are oblivious to the matroid structure, where n is the total number of elements. This contrasts an upper bound of O(log n) in the cardinal model, and it shows that the technique of thresholding is not sufficient for good algorithms in the ordinal model.

Motivation & Objective

• To study combinatorial secretary problems under the ordinal model, where only relative rankings of elements are known, rather than exact numerical values.
• To determine whether existing algorithms for cardinal models can be adapted to work under ordinal information with comparable competitive ratios.
• To design new online algorithms that achieve constant competitive ratios in the ordinal model for key combinatorial problems such as bipartite matching and independent set.
• To establish fundamental limits of ordinal algorithms by proving a lower bound of Ω(√n/log n) for structure-oblivious algorithms in the matroid secretary problem.
• To explore the relationship between linear and submodular variants of the matroid secretary problem in the ordinal model.

Proposed method

• Adapt existing cardinal-model algorithms to the ordinal model by replacing reliance on exact weights with comparisons based on relative order of arrival values.
• Introduce a two-phase sampling strategy: a learning phase to build a representative sample, followed by an acceptance phase using greedy selection based on ordinal comparisons.
• For bipartite matching, extend the sampling approach by dividing vertices into left and right sets and using a greedy matching on sampled subgraphs to guide edge selection.
• For general graphs and independent set, use a randomized sampling technique where vertices are assigned to a sample set with probability p = √(α₁/(α₁+1)), followed by a greedy selection with a coin flip mechanism.
• Analyze performance using probabilistic bounds and Markov's inequality to relate the expected weight of the solution to the optimal offline solution.
• Reformulate algorithms into equivalent simulation forms (e.g., Algorithm 7) to simplify analysis while preserving competitive ratio guarantees.

Experimental results

Research questions

• RQ1 Can existing constant-competitive algorithms for matroid secretary problems be adapted to the ordinal model without loss in performance?
• RQ2 What is the best achievable competitive ratio for online bipartite matching under ordinal information?
• RQ3 Can constant-competitive algorithms be designed for general packing LPs and independent set problems when only ordinal information is available?
• RQ4 Is the thresholding technique, effective in the cardinal model, sufficient for good performance in the ordinal model?
• RQ5 What are the fundamental limitations of algorithms that do not exploit matroid or graph structure in the ordinal secretary setting?

Key findings

• A 2e-competitive algorithm is developed for online bipartite matching under ordinal information, achieving a constant competitive ratio.
• For general packing LPs with at most d non-zero entries per variable, an O(d(B+1)/B)-competitive algorithm is presented, where B is a tightness parameter.
• An 8.78-competitive algorithm is designed for online matching in general graphs, later improved to 12e/(e+1) < 8.78 via a refined sampling strategy.
• An O(α₁²)-competitive algorithm is achieved for weighted independent set in graphs with bounded local independence number α₁, with α₁ = 5 in unit-disk graphs.
• A lower bound of Ω(√n / log n) is proven for structure-oblivious algorithms in the matroid secretary problem under ordinal information, showing thresholding is insufficient.
• The ordinal model does not degrade performance for many matroid secretary algorithms, as existing constant-competitive algorithms for special matroid types remain valid under ordinal access.