A Performance Bound for the Greedy Algorithm in a Generalized Class of String Optimization Problems

TL;DR

This work tackles the challenge of bounding the greedy algorithm's performance in generalized string optimization problems. It develops a computable, per-instance lower bound by extending the classical greedy-curvature framework to strings and introducing a simple upper bound $B_s$ on the optimal value, yielding a bound of the form $f(G_K)/f(O_K) \ge 1/K + (K-1)/(K\gamma)$ with $\gamma\in\{\gamma_G,\gamma_G''\}$. A key contribution is proving that this bound dominates the prior curvature-based bounds and providing a counterexample showing $α_G'$ may be incorrect under the original assumptions. The authors demonstrate the bound's value through applications to submodular sensor coverage in both set and string domains and to welfare maximization with black-box utilities, highlighting tighter guarantees and broader applicability than existing results.

Abstract

We present a simple performance bound for the greedy scheme in string optimization problems that obtains strong results. Our approach vastly generalizes the group of previously established greedy curvature bounds by Conforti and Cornuéjols (1984). We consider three constants, $α_G$, $α_G'$, and $α_G''$ introduced by Conforti and Cornuéjols (1984), that are used in performance bounds of greedy schemes in submodular set optimization. We first generalize both of the $α_G$ and $α_G''$ bounds to string optimization problems in a manner that includes maximizing submodular set functions over matroids as a special case. We then derive a much simpler and computable bound that allows for applications to a far more general class of functions with string domains. We prove that our bound is superior to both the $α_G$ and $α_G''$ bounds and provide a counterexample to show that the $α_G'$ bound is incorrect under the assumptions in Conforti and Cornuéjols (1984). We conclude with two applications. The first is an application of our result to sensor coverage problems. We demonstrate our performance bound in cases where the objective function is set submodular and string submodular. The second is an application to a social welfare maximization problem with black-box utility functions.

Figures (5)

• Figure 1: Sensor coverage for event detection in a mission space.
• Figure 2: Mission space partition.
• Figure 3: Performance bound comparison under different (initial) decay rates with number of placed sensors $K = 5$. Upper Figure: Homogeneous Sensors; Lower Figure: Nonhomogeneous Sensors.
• Figure 4: Performance bound comparison under different number of placed sensors. Upper Figure: Homogeneous sensors with decay rate $\lambda = 1$; Lower Figure: Nonhomogeneous sensors with initial decay rate $\lambda_{1} = 1$.
• Figure 5: Performance bound comparison for set utility functions (upper figure) and string utility functions (lower figure).

Theorems & Definitions (36)

• Remark 1
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• Definition 1
• Definition 2
• Definition 3
• Remark 3
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• Definition 5
• Remark 4
• Definition 6