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1. Rates of convergence for regression with the graph poly-Laplacian

   https://doi.org/10.1007/s43670-023-00075-5  

   Trillos, Nicolás García ; Murray, Ryan ; Thorpe, Matthew ( November 2023 , Sampling Theory, Signal Processing, and Data Analysis)

   In the (special) smoothing spline problem one considers a variational problem with a quadratic data fidelity penalty and Laplacian regularization. Higher order regularity can be obtained via replacing the Laplacian regulariser with a poly-Laplacian regulariser. The methodology is readily adapted to graphs and here we consider graph poly-Laplacian regularization in a fully supervised, non-parametric, noise corrupted, regression problem. In particular, given a dataset$$\{x_i\}_{i=1}^n$$${\left\{x_i\right\}}_{i=1}^n$and a set of noisy labels$$\{y_i\}_{i=1}^n\subset \mathbb {R}$$${\left\{y_i\right\}}_{i=1}^n\subset R$we let$$u_n{:}\{x_i\}_{i=1}^n\rightarrow \mathbb {R}$$$u_n:{\left\{x_i\right\}}_{i=1}^n\to R$be the minimizer of an energy which consists of a data fidelity term and an appropriately scaled graph poly-Laplacian term. When$$y_i = g(x_i)+\xi _i$$$y_i=g\left(x_i\right)+{\xi }_i$, for iid noise$$\xi _i$$${\xi }_i$, and using the geometric random graph, we identify (with high probability) the rate of convergence of$$u_n$$$u_n$togin the large data limit$$n\rightarrow \infty $$$n\to \infty$. Furthermore, our rate is close to the known rate of convergence in the usual smoothing spline model.

    

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2. Finding solutions with distinct variables to systems of linear equations over $$\mathbb {F}_p$$

   https://doi.org/10.1007/s00208-022-02391-y  

   Sauermann, Lisa ( April 2022 , Mathematische Annalen)

   Let us fix a primepand a homogeneous system ofmlinear equations$$a_{j,1}x_1+\dots +a_{j,k}x_k=0$$$a_{j,1}{x}_1+\cdots +a_{j,k}{x}_k=0$for$$j=1,\dots ,m$$$j=1,\cdots ,m$with coefficients$$a_{j,i}\in \mathbb {F}_p$$$a_{j,i}\in F_p$. Suppose that$$k\ge 3m$$$k\ge 3m$, that$$a_{j,1}+\dots +a_{j,k}=0$$$a_{j,1}+\cdots +a_{j,k}=0$for$$j=1,\dots ,m$$$j=1,\cdots ,m$and that every$$m\times m$$$m\times m$minor of the$$m\times k$$$m\times k$matrix$$(a_{j,i})_{j,i}$$${\left(a_{j,i}\right)}_{j,i}$is non-singular. Then we prove that for any (large)n, any subset$$A\subseteq \mathbb {F}_p^n$$$A\subseteq F_p^n$of size$$|A|> C\cdot \Gamma ^n$$$\left|A\right|>C·{\Gamma }^n$contains a solution$$(x_1,\dots ,x_k)\in A^k$$$(x_1,\cdots ,x_k)\in A^k$to the given system of equations such that the vectors$$x_1,\dots ,x_k\in A$$$x_1,\cdots ,x_k\in A$are all distinct. Here,Cand$$\Gamma $$$\Gamma$are constants only depending onp,mandksuch that$$\Gamma $\Gamma <p$. The crucial point here is the condition for the vectors$$x_1,\dots ,x_k$$$x_1,\cdots ,x_k$in the solution$$(x_1,\dots ,x_k)\in A^k$$$(x_1,\cdots ,x_k)\in A^k$to be distinct. If we relax this condition and only demand that$$x_1,\dots ,x_k$$$x_1,\cdots ,x_k$are not all equal, then the statement would follow easily from Tao’s slice rank polynomial method. However, handling the distinctness condition is much harder, and requires a new approach. While all previous combinatorial applications of the slice rank polynomial method have relied on the slice rank of diagonal tensors, we use a slice rank argument for a non-diagonal tensor in combination with combinatorial and probabilistic arguments.

    

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3. From approximate to exact integer programming

   https://doi.org/10.1007/s10107-024-02084-1  

   Dadush, Daniel ; Eisenbrand, Friedrich ; Rothvoss, Thomas ( April 2024 , Mathematical Programming)

   Approximate integer programming is the following: For a given convex body$$K \subseteq {\mathbb {R}}^n$$$K\subseteq R^n$, either determine whether$$K \cap {\mathbb {Z}}^n$$$K\cap Z^n$is empty, or find an integer point in the convex body$$2\cdot (K - c) +c$$$2·(K-c)+c$which isK, scaled by 2 from its center of gravityc. Approximate integer programming can be solved in time$$2^{O(n)}$$$2^{O\left(n\right)}$while the fastest known methods for exact integer programming run in time$$2^{O(n)} \cdot n^n$$$2^{O\left(n\right)}·n^n$. So far, there are no efficient methods for integer programming known that are based on approximate integer programming. Our main contribution are two such methods, each yielding novel complexity results. First, we show that an integer point$$x^* \in (K \cap {\mathbb {Z}}^n)$$$x^{\ast }\in (K\cap Z^n)$can be found in time$$2^{O(n)}$$$2^{O\left(n\right)}$, provided that theremaindersof each component$$x_i^* \mod \ell $$$x_i^{\ast }mod\ell$for some arbitrarily fixed$$\ell \ge 5(n+1)$$$\ell \ge 5(n+1)$of$$x^*$$$x^{\ast }$are given. The algorithm is based on acutting-plane technique, iteratively halving the volume of the feasible set. The cutting planes are determined via approximate integer programming. Enumeration of the possible remainders gives a$$2^{O(n)}n^n$$$2^{O\left(n\right)}{n}^n$algorithm for general integer programming. This matches the current best bound of an algorithm by Dadush (Integer programming, lattice algorithms, and deterministic, vol. Estimation. Georgia Institute of Technology, Atlanta, 2012) that is considerably more involved. Our algorithm also relies on a newasymmetric approximate Carathéodory theoremthat might be of interest on its own. Our second method concerns integer programming problems in equation-standard form$$Ax = b, 0 \le x \le u, \, x \in {\mathbb {Z}}^n$$$Ax=b,0\le x\le u,x\in Z^n$. Such a problem can be reduced to the solution of$$\prod _i O(\log u_i +1)$$${\prod }_{i}O(log{u}_i+1)$approximate integer programming problems. This implies, for example thatknapsackorsubset-sumproblems withpolynomial variable range$$0 \le x_i \le p(n)$$$0\le x_i\le p\left(n\right)$can be solved in time$$(\log n)^{O(n)}$$${(logn)}^{O\left(n\right)}$. For these problems, the best running time so far was$$n^n \cdot 2^{O(n)}$$$n^n·2^{O\left(n\right)}$.

    

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4. Lower Bounds Against Sparse Symmetric Functions of ACC Circuits: Expanding the Reach of #SAT Algorithms

   https://doi.org/10.1007/s00224-022-10106-8  

   Vyas, Nikhil ; Williams, R. Ryan ( November 2022 , Theory of Computing Systems)

   We continue the program of proving circuit lower bounds via circuit satisfiability algorithms. So far, this program has yielded several concrete results, proving that functions in$\mathsf {Quasi}\text {-}\mathsf {NP} = \mathsf {NTIME}[n^{(\log n)^{O(1)}}]$$\mathrm{Quasi}-\mathrm{NP}=\mathrm{NTIME}\left[n^{{\left(\log n\right)}^{O\left(1\right)}}\right]$and other complexity classes do not have small circuits (in the worst case and/or on average) from various circuit classes$\mathcal { C}$$C$, by showing that$\mathcal { C}$$C$admits non-trivial satisfiability and/or#SAT algorithms which beat exhaustive search by a minor amount. In this paper, we present a new strong lower bound consequence of having a non-trivial#SAT algorithm for a circuit class${\mathcal C}$$C$. Say that a symmetric Boolean functionf(x₁,…,x_n) issparseif it outputs 1 onO(1) values of${\sum }_{i} x_{i}$${\sum }_i{x}_i$. We show that for every sparsef, and for all “typical”$\mathcal { C}$$C$, faster#SAT algorithms for$\mathcal { C}$$C$circuits imply lower bounds against the circuit class$f \circ \mathcal { C}$$f\circ C$, which may bestrongerthan$\mathcal { C}$$C$itself. In particular:

   #SAT algorithms forn^k-size$\mathcal { C}$$C$-circuits running in 2ⁿ/n^ktime (for allk) implyNEXPdoes not have$(f \circ \mathcal { C})$$(f\circ C)$-circuits of polynomial size.

   #SAT algorithms for$2^{n^{{\varepsilon }}}$$2^{n^{\epsilon }}$-size$\mathcal { C}$$C$-circuits running in$2^{n-n^{{\varepsilon }}}$$2^{n-n^{\epsilon }}$time (for someε> 0) implyQuasi-NPdoes not have$(f \circ \mathcal { C})$$(f\circ C)$-circuits of polynomial size.

   Applying#SAT algorithms from the literature, one immediate corollary of our results is thatQuasi-NPdoes not haveEMAJ∘ACC⁰∘THRcircuits of polynomial size, whereEMAJis the “exact majority” function, improving previous lower bounds againstACC⁰[Williams JACM’14] andACC⁰∘THR[Williams STOC’14], [Murray-Williams STOC’18]. This is the first nontrivial lower bound against such a circuit class.

    

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5. Algorithms for covering multiple submodular constraints and applications

   https://doi.org/10.1007/s10878-022-00874-x  

   Chekuri, Chandra ; Inamdar, Tanmay ; Quanrud, Kent ; Varadarajan, Kasturi ; Zhang, Zhao ( June 2022 , Journal of Combinatorial Optimization)

   We consider the problem of covering multiple submodular constraints. Given a finite ground setN, a weight function$$w: N \rightarrow \mathbb {R}_+$$$w:N\to R_{+}$,rmonotone submodular functions$$f_1,f_2,\ldots ,f_r$$$f_1,f_2,\dots ,f_r$overNand requirements$$k_1,k_2,\ldots ,k_r$$$k_1,k_2,\dots ,k_r$the goal is to find a minimum weight subset$$S \subseteq N$$$S\subseteq N$such that$$f_i(S) \ge k_i$$$f_i\left(S\right)\ge k_i$for$$1 \le i \le r$$$1\le i\le r$. We refer to this problem asMulti-Submod-Coverand it was recently considered by Har-Peled and Jones (Few cuts meet many point sets. CoRR.arxiv:abs1808.03260Har-Peled and Jones 2018) who were motivated by an application in geometry. Even with$$r=1$$$r=1$Multi-Submod-Covergeneralizes the well-known Submodular Set Cover problem (Submod-SC), and it can also be easily reduced toSubmod-SC. A simple greedy algorithm gives an$$O(\log (kr))$$$O(log(kr\left)\right)$approximation where$$k = \sum _i k_i$$$k={\sum }_i{k}_i$and this ratio cannot be improved in the general case. In this paper, motivated by several concrete applications, we consider two ways to improve upon the approximation given by the greedy algorithm. First, we give a bicriteria approximation algorithm forMulti-Submod-Coverthat covers each constraint to within a factor of$$(1-1/e-\varepsilon )$$$(1-1/e-\epsilon )$while incurring an approximation of$$O(\frac{1}{\epsilon }\log r)$$$O(\frac{1}{ϵ}logr)$in the cost. Second, we consider the special case when each$$f_i$$$f_i$is a obtained from a truncated coverage function and obtain an algorithm that generalizes previous work on partial set cover (Partial-SC), covering integer programs (CIPs) and multiple vertex cover constraints Bera et al. (Theoret Comput Sci 555:2–8 Bera et al. 2014). Both these algorithms are based on mathematical programming relaxations that avoid the limitations of the greedy algorithm. We demonstrate the implications of our algorithms and related ideas to several applications ranging from geometric covering problems to clustering with outliers. Our work highlights the utility of the high-level model and the lens of submodularity in addressing this class of covering problems.

    

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