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Given a k-CNF formula and an integer s, we study algorithms that obtain s solutions to the formula that are maximally dispersed. For s=2, the problem of computing the diameter of a k-CNF formula was initiated by Creszenzi and Rossi, who showed strong hardness results even for k=2. Assuming SETH, the current best upper bound [Angelsmark and Thapper '04] goes to 4n as k→∞. As our first result, we give exact algorithms for using the Fast Fourier Transform and clique-finding that run in O(2^((s−1)n)) and O(s^2|Ω_F|^(ω⌈s/3⌉)) respectively, where |Ω_F| is the size of the solution space of the formula F and ω is the matrix multiplication exponent. As our main result, we re-analyze the popular PPZ (Paturi, Pudlak, Zane '97) and Schöning's ('02) algorithms (which find one solution in time O∗(2^(ε_k n)) for εk≈1−Θ(1/k)), and show that in the same time, they can be used to approximate the diameter as well as the dispersion (s>2) problems. While we need to modify Schöning's original algorithm, we show that the PPZ algorithm, without any modification, samples solutions in a geometric sense. We believe that this property may be of independent interest. Finally, we present algorithms to output approximately diverse, approximately optimal solutions to NP-complete optimization problems running in time poly(s)O(^(2εn)) with ε<1 for several problems such as Minimum Hitting Set and Feedback Vertex Set. For these problems, all existing exact methods for finding optimal diverse solutions have a runtime with at least an exponential dependence on the number of solutions s. Our methods find bi-approximations with polynomial dependence on s.
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A Dynamic Programming Framework for Generating Approximately Diverse and Optimal Solutions
We develop a general framework, called approximately-diverse dynamic programming (ADDP) that can be used to generate a collection of k≥2 maximally diverse solutions to various geometric and combinatorial optimization problems. Given an approximation factor 0≤c≤1, this framework also allows for maximizing diversity in the larger space of c-approximate solutions. We focus on two geometric problems to showcase this technique: 1. Given a polygon P, an integer k≥2 and a value c≤1, generate k maximally diverse c-nice triangulations of P. Here, a c-nice triangulation is one that is c-approximately optimal with respect to a given quality measure σ. 2. Given a planar graph G, an integer k≥2 and a value c≤1, generate k maximally diverse c-optimal Independent Sets (or, Vertex Covers). Here, an independent set S is said to be c-optimal if |S|≥c|S′| for any independent set S′ of G. Given a set of k solutions to the above problems, the diversity measure we focus on is the average distance between the solutions, where d(X,Y)=|XΔY|. For arbitrary polygons and a wide range of quality measures, we give poly(n,k) time (1−Θ(1/k))-approximation algorithms for the diverse triangulation problem. For the diverse independent set and vertex cover problems on planar graphs, we give an algorithm that runs in time 2^(O(k.δ^(−1).ϵ^(−2)).n^O(1/ϵ) and returns (1−ϵ)-approximately diverse (1−δ)c-optimal independent sets or vertex covers. Our triangulation results are the first algorithmic results on computing collections of diverse geometric objects, and our planar graph results are the first PTAS for the diverse versions of any NP-complete problem. Additionally, we also provide applications of this technique to diverse variants of other geometric problems.
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- Award ID(s):
- 1910873
- PAR ID:
- 10568358
- Publisher / Repository:
- Arxiv
- Date Published:
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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There has been a long-standing interest in computing diverse solutions to optimization problems. In 1995 J. Krarup [28] posed the problem of finding k-edge disjoint Hamiltonian Circuits of minimum total weight, called the peripatetic salesman problem (PSP). Since then researchers have investigated the complexity of finding diverse solutions to spanning trees, paths, vertex covers, matchings, and more. Unlike the PSP that has a constraint on the total weight of the solutions, recent work has involved finding diverse solutions that are all optimal. However, sometimes the space of exact solutions may be too small to achieve sufficient diversity. Motivated by this, we initiate the study of obtaining sufficiently-diverse, yet approximately-optimal solutions to optimization problems. Formally, given an integer k, an approximation factor c, and an instance I of an optimization problem, we aim to obtain a set of k solutions to I that a) are all c approximately-optimal for I and b) maximize the diversity of the k solutions. Finding such solutions, therefore, requires a better understanding of the global landscape of the optimization function. Given a metric on the space of solutions, and the diversity measure as the sum of pairwise distances between solutions, we first provide a general reduction to an associated budget-constrained optimization (BCO) problem, where one objective function is to optimized subject to a bound on the second objective function. We then prove that bi-approximations to the BCO can be used to give bi-approximations to the diverse approximately optimal solutions problem. As applications of our result, we present polynomial time approximation algorithms for several problems such as diverse c-approximate maximum matchings, shortest paths, global min-cut, and minimum weight bases of a matroid. The last result gives us diverse c-approximate minimum spanning trees, advancing a step towards achieving diverse c-approximate TSP tours. We also explore the connection to the field of multiobjective optimization and show that the class of problems to which our result applies includes those for which the associated DUALRESTRICT problem defined by Papadimitriou and Yannakakis [35], and recently explored by Herzel et al. [26] can be solved in polynomial timore » « less
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