Abstract
Given an n-dimensional convex body by a membership oracle in general, it is known that any polynomial-time deterministic algorithm cannot approximate its volume within ratio \((n/\log n)^n\). There is a substantial progress on randomized approximation such as Markov chain Monte Carlo for a high-dimensional volume, and for many #P-hard problems, while only a few #P-hard problems are known to yield deterministic approximation. Motivated by the problem of deterministically approximating the volume of a \(\mathcal{V}\)-polytope, that is a polytope with a small number of vertices and (possibly) exponentially many facets, this paper investigates the problem of computing the volume of a “knapsack dual polytope,” which is known to be #P-hard due to Khachiyan (1989). We reduce an approximate volume of a knapsack dual polytope to that of the intersection of two cross-polytopes, and give FPTASs for those volume computations. Interestingly, computing the volume of the intersection of two cross-polytopes (i.e., \(L_1\)-balls) is #P-hard, unlike the cases of \(L_{\infty }\)-balls or \(L_2\)-balls.
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- 1.
Precisely, they are concerned with a “well-rounded” convex body, after an affine transformation of a general finite convex body.
- 2.
Given \(\varvec{a} \in \mathbb {Z}_{>0}^n\) and \(b \in \mathbb {Z}_{>0}\), the problem is to compute \(|\{ \varvec{x} \in \{0,1\}^n \mid \sum _{i=1}^n a_i x_i \le b \}|\). Remark that it is computed in polynomial time when all the inputs \(a_i\) (\(i=1,\ldots ,n\)) and b are bounded by \(\mathrm {poly}(n)\), using a version of the standard dynamic programming for knapsack problem (see e.g., [7, 13]). It should be worth noting that [12, 24] needed special techniques, different from ones for optimization problems, to design FPTASs for the counting problem.
- 3.
See [23] for the duality of polytopes. In fact, \(P_{\varvec{a}}\) itself is not the dual of a knapsack polytope in a canonical form, but it is obtained by an affine transformation from a dual of knapsack polytope under some assumptions. Khachiyan [16] says that computing \(\mathrm {Vol}(P_{\varvec{a}})\) ‘is “polar” to determining the volume of the intersection of a cube and a halfspace.’ .
- 4.
If all \(a_i\) (\(i=1,\ldots ,n\)) are bounded by \(\mathrm {poly}(n)\), it is computed in polynomial time, so did the counting knapsack solutions. See also footnote 1 for counting knapsack solutions.
- 5.
Suppose you know that x is approximately 49 within 1% error. Then, you know that \(x+50\) is approximately 99 within 1% error. However, it is difficult to say \(50-x\) is approximately 1. Even when additionally you know that x does not exceed 50, \(50-x\) may be 2, 1, 0.1 or smaller than 0.001, meaning that the approximation ratio is unbounded.
- 6.
We will set \(\beta = 1-\dfrac{\epsilon }{2n\Vert \varvec{a}\Vert _1}\), later.
- 7.
Most of the proofs cannot be included due to the space limit.
- 8.
Remark that \(\mathrm {Vol}(C(\varvec{c},r)\cap C(\varvec{c}',r')) = r^n\mathrm {Vol}\left( C(\varvec{0},1) \cap C\!\left( \tfrac{(\varvec{c}-\varvec{c}')^+}{r}, \tfrac{r'}{r} \right) \right) \) holds for any \(\varvec{c}, \varvec{c}' \in \mathbb {R}^n\) and \(r,r' \in \mathbb {R}_{>0}\), where \((\varvec{c} - \varvec{c}')^+ = (|c_1-c_1'|,|c_2-c_2'|,\ldots ,|c_n-c_n'| )\).
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Acknowledgments
This work is partly supported by Grant-in-Aid for Scientific Research on Innovative Areas MEXT Japan “Exploring the Limits of Computation (ELC)” (No. 24106008, 24106005) and by JST PRESTO Grant Number JPMJPR16E4, Japan.
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Ando, E., Kijima, S. (2017). An FPTAS for the Volume of Some \(\mathcal{V}\)-polytopes—It is Hard to Compute the Volume of the Intersection of Two Cross-Polytopes. In: Cao, Y., Chen, J. (eds) Computing and Combinatorics. COCOON 2017. Lecture Notes in Computer Science(), vol 10392. Springer, Cham. https://doi.org/10.1007/978-3-319-62389-4_2
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