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Register a new userHigh Dimensional Discrete Integration over the Hypergrid
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Added by
Arya Mazumdar
Publication date
2018
fields
Informatics Engineering
and research's language is
English
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Recently Ermon et al. (2013) pioneered a way to practically compute approximations to large scale counting or discrete integration problems by using random hashes. The hashes are used to reduce the counting problem into many separate discrete optimization problems. The optimization problems then can be solved by an NP-oracle such as commercial SAT solvers or integer linear programming (ILP) solvers. In particular, Ermon et al. showed that if the domain of integration is ${0,1}^n$ then it is possible to obtain a solution within a factor of $16$ of the optimal (a 16-approximation) by this technique. In many crucial counting tasks, such as computation of partition function of ferromagnetic Potts model, the domain of integration is naturally ${0,1,dots, q-1}^n, q>2$, the hypergrid. The straightforward extension of Ermon et al.s method allows a $q^2$-approximation for this problem. For large values of $q$, this is undesirable. In this paper, we show an improved technique to obtain an approximation factor of $4+O(1/q^2)$ to this problem. We are able to achieve this by using an idea of optimization over multiple bins of the hash functions, that can be easily implemented by inequality constraints, or even in unconstrained way. Also the burden on the NP-oracle is not increased by our method (an ILP solver can still be used). We provide experimental simulation results to support the theoretical guarantees of our algorithms.
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Cut problems form one of the most fundamental classes of problems in algorithmic graph theory. For instance, the minimum cut, the minimum $s$-$t$ cut, the minimum multiway cut, and the minimum $k$-way cut are some of the commonly encountered cut problems. Many of these problems have been extensively studied over several decades. In this paper, we initiate the algorithmic study of some cut problems in high dimensions. The first problem we study, namely, Topological Hitting Set (THS), is defined as follows: Given a nontrivial $r$-cycle $zeta$ in a simplicial complex $mathsf{K}$, find a set $mathcal{S}$ of $r$-dimensional simplices of minimum cardinality so that $mathcal{S}$ meets every cycle homologous to $zeta$. Our main result is that this problem admits a polynomial-time solution on triangulations of closed surfaces. Interestingly, the optimal solution is given in terms of the cocycles of the surface. For general complexes, we show that THS is W[1]-hard with respect to the solution size $k$. On the positive side, we show that THS admits an FPT algorithm with respect to $k+d$, where $d$ is the maximum degree of the Hasse graph of the complex $mathsf{K}$. We also define a problem called Boundary Nontrivialization (BNT): Given a bounding $r$-cycle $zeta$ in a simplicial complex $mathsf{K}$, find a set $mathcal{S}$ of $(r+1)$-dimensional simplices of minimum cardinality so that the removal of $mathcal{S}$ from $mathsf{K}$ makes $zeta$ non-bounding. We show that BNT is W[1]-hard with respect to the solution size as the parameter, and has an $O(log n)$-approximation FPT algorithm for $(r+1)$-dimensional complexes with the $(r+1)$-th Betti number $beta_{r+1}$ as the parameter. Finally, we provide randomized (approximation) FPT algorithms for the global variants of THS and BNT.
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