We consider the problem of covering multiple submodular constraints. Given a finite ground set
Approximate integer programming is the following: For a given convex body
 NSFPAR ID:
 10502164
 Publisher / Repository:
 Springer Science + Business Media
 Date Published:
 Journal Name:
 Mathematical Programming
 ISSN:
 00255610
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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Abstract N , a weight function ,$$w: N \rightarrow \mathbb {R}_+$$ $w:N\to {R}_{+}$r monotone submodular functions over$$f_1,f_2,\ldots ,f_r$$ ${f}_{1},{f}_{2},\dots ,{f}_{r}$N and requirements the goal is to find a minimum weight subset$$k_1,k_2,\ldots ,k_r$$ ${k}_{1},{k}_{2},\dots ,{k}_{r}$ such that$$S \subseteq N$$ $S\subseteq N$ for$$f_i(S) \ge k_i$$ ${f}_{i}\left(S\right)\ge {k}_{i}$ . We refer to this problem as$$1 \le i \le r$$ $1\le i\le r$MultiSubmodCover and it was recently considered by HarPeled and Jones (Few cuts meet many point sets. CoRR.arxiv:abs1808.03260 HarPeled and Jones 2018) who were motivated by an application in geometry. Even with$$r=1$$ $r=1$MultiSubmodCover generalizes the wellknown Submodular Set Cover problem (SubmodSC ), and it can also be easily reduced toSubmodSC . A simple greedy algorithm gives an approximation where$$O(\log (kr))$$ $O(log(kr\left)\right)$ 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 for$$k = \sum _i k_i$$ $k={\sum}_{i}{k}_{i}$MultiSubmodCover that covers each constraint to within a factor of while incurring an approximation of$$(11/e\varepsilon )$$ $(11/e\epsilon )$ in the cost. Second, we consider the special case when each$$O(\frac{1}{\epsilon }\log r)$$ $O(\frac{1}{\u03f5}logr)$ is a obtained from a truncated coverage function and obtain an algorithm that generalizes previous work on partial set cover ($$f_i$$ ${f}_{i}$PartialSC ), 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 highlevel model and the lens of submodularity in addressing this class of covering problems. 
Abstract Let
denote the standard Haar system on [0, 1], indexed by$$(h_I)$$ $\left({h}_{I}\right)$ , the set of dyadic intervals and$$I\in \mathcal {D}$$ $I\in D$ denote the tensor product$$h_I\otimes h_J$$ ${h}_{I}\otimes {h}_{J}$ ,$$(s,t)\mapsto h_I(s) h_J(t)$$ $(s,t)\mapsto {h}_{I}\left(s\right){h}_{J}\left(t\right)$ . We consider a class of twoparameter function spaces which are completions of the linear span$$I,J\in \mathcal {D}$$ $I,J\in D$ of$$\mathcal {V}(\delta ^2)$$ $V\left({\delta}^{2}\right)$ ,$$h_I\otimes h_J$$ ${h}_{I}\otimes {h}_{J}$ . This class contains all the spaces of the form$$I,J\in \mathcal {D}$$ $I,J\in D$X (Y ), whereX andY are either the Lebesgue spaces or the Hardy spaces$$L^p[0,1]$$ ${L}^{p}[0,1]$ ,$$H^p[0,1]$$ ${H}^{p}[0,1]$ . We say that$$1\le p < \infty $$ $1\le p<\infty $ is a Haar multiplier if$$D:X(Y)\rightarrow X(Y)$$ $D:X\left(Y\right)\to X\left(Y\right)$ , where$$D(h_I\otimes h_J) = d_{I,J} h_I\otimes h_J$$ $D({h}_{I}\otimes {h}_{J})={d}_{I,J}{h}_{I}\otimes {h}_{J}$ , and ask which more elementary operators factor through$$d_{I,J}\in \mathbb {R}$$ ${d}_{I,J}\in R$D . A decisive role is played by theCapon projection given by$$\mathcal {C}:\mathcal {V}(\delta ^2)\rightarrow \mathcal {V}(\delta ^2)$$ $C:V\left({\delta}^{2}\right)\to V\left({\delta}^{2}\right)$ if$$\mathcal {C} h_I\otimes h_J = h_I\otimes h_J$$ $C{h}_{I}\otimes {h}_{J}={h}_{I}\otimes {h}_{J}$ , and$$I\le J$$ $\leftI\right\le \leftJ\right$ if$$\mathcal {C} h_I\otimes h_J = 0$$ $C{h}_{I}\otimes {h}_{J}=0$ , as our main result highlights: Given any bounded Haar multiplier$$I > J$$ $\leftI\right>\leftJ\right$ , there exist$$D:X(Y)\rightarrow X(Y)$$ $D:X\left(Y\right)\to X\left(Y\right)$ such that$$\lambda ,\mu \in \mathbb {R}$$ $\lambda ,\mu \in R$ i.e., for all$$\begin{aligned} \lambda \mathcal {C} + \mu ({{\,\textrm{Id}\,}}\mathcal {C})\text { approximately 1projectionally factors through }D, \end{aligned}$$ $\begin{array}{c}\lambda C+\mu (\phantom{\rule{0ex}{0ex}}\text{Id}\phantom{\rule{0ex}{0ex}}C)\phantom{\rule{0ex}{0ex}}\text{approximately 1projectionally factors through}\phantom{\rule{0ex}{0ex}}D,\end{array}$ , there exist bounded operators$$\eta > 0$$ $\eta >0$A ,B so thatAB is the identity operator ,$${{\,\textrm{Id}\,}}$$ $\phantom{\rule{0ex}{0ex}}\text{Id}\phantom{\rule{0ex}{0ex}}$ and$$\Vert A\Vert \cdot \Vert B\Vert = 1$$ $\Vert A\Vert \xb7\Vert B\Vert =1$ . Additionally, if$$\Vert \lambda \mathcal {C} + \mu ({{\,\textrm{Id}\,}}\mathcal {C})  ADB\Vert < \eta $$ $\Vert \lambda C+\mu (\phantom{\rule{0ex}{0ex}}\text{Id}\phantom{\rule{0ex}{0ex}}C)ADB\Vert <\eta $ is unbounded on$$\mathcal {C}$$ $C$X (Y ), then and then$$\lambda = \mu $$ $\lambda =\mu $ either factors through$${{\,\textrm{Id}\,}}$$ $\phantom{\rule{0ex}{0ex}}\text{Id}\phantom{\rule{0ex}{0ex}}$D or .$${{\,\textrm{Id}\,}}D$$ $\phantom{\rule{0ex}{0ex}}\text{Id}\phantom{\rule{0ex}{0ex}}D$ 
Abstract Let us fix a prime
p and a homogeneous system ofm linear equations for$$a_{j,1}x_1+\dots +a_{j,k}x_k=0$$ ${a}_{j,1}{x}_{1}+\cdots +{a}_{j,k}{x}_{k}=0$ with coefficients$$j=1,\dots ,m$$ $j=1,\cdots ,m$ . Suppose that$$a_{j,i}\in \mathbb {F}_p$$ ${a}_{j,i}\in {F}_{p}$ , that$$k\ge 3m$$ $k\ge 3m$ for$$a_{j,1}+\dots +a_{j,k}=0$$ ${a}_{j,1}+\cdots +{a}_{j,k}=0$ and that every$$j=1,\dots ,m$$ $j=1,\cdots ,m$ minor of the$$m\times m$$ $m\times m$ matrix$$m\times k$$ $m\times k$ is nonsingular. Then we prove that for any (large)$$(a_{j,i})_{j,i}$$ ${\left({a}_{j,i}\right)}_{j,i}$n , any subset of size$$A\subseteq \mathbb {F}_p^n$$ $A\subseteq {F}_{p}^{n}$ contains a solution$$A> C\cdot \Gamma ^n$$ $\leftA\right>C\xb7{\Gamma}^{n}$ to the given system of equations such that the vectors$$(x_1,\dots ,x_k)\in A^k$$ $({x}_{1},\cdots ,{x}_{k})\in {A}^{k}$ are all distinct. Here,$$x_1,\dots ,x_k\in A$$ ${x}_{1},\cdots ,{x}_{k}\in A$C and are constants only depending on$$\Gamma $$ $\Gamma $p ,m andk such that . The crucial point here is the condition for the vectors$$\Gamma $\Gamma <p$
in the solution$$x_1,\dots ,x_k$$ ${x}_{1},\cdots ,{x}_{k}$ to be distinct. If we relax this condition and only demand that$$(x_1,\dots ,x_k)\in A^k$$ $({x}_{1},\cdots ,{x}_{k})\in {A}^{k}$ are not all equal, then the statement would follow easily from Tao’s slice rank polynomial method. However, handling the distinctness condition is much harder, and requires a new approach. While all previous combinatorial applications of the slice rank polynomial method have relied on the slice rank of diagonal tensors, we use a slice rank argument for a nondiagonal tensor in combination with combinatorial and probabilistic arguments.$$x_1,\dots ,x_k$$ ${x}_{1},\cdots ,{x}_{k}$ 
Abstract In this article, we study the moduli of irregular surfaces of general type with at worst canonical singularities satisfying
, for any even integer$$K^2 = 4p_g8$$ ${K}^{2}=4{p}_{g}8$ . These surfaces also have unbounded irregularity$$p_g\ge 4$$ ${p}_{g}\ge 4$q . We carry out our study by investigating the deformations of the canonical morphism , where$$\varphi :X\rightarrow {\mathbb {P}}^N$$ $\phi :X\to {P}^{N}$ is a quadruple Galois cover of a smooth surface of minimal degree. These canonical covers are classified in Gallego and Purnaprajna (Trans Am Math Soc 360(10):54895507, 2008) into four distinct families, one of which is the easy case of a product of curves. The main objective of this article is to study the deformations of the other three, non trivial, unbounded families. We show that any deformation of$$\varphi $$ $\phi $ factors through a double cover of a ruled surface and, hence, is never birational. More interestingly, we prove that, with two exceptions, a general deformation of$$\varphi $$ $\phi $ is twotoone onto its image, whose normalization is a ruled surface of appropriate irregularity. We also show that, with the exception of one family, the deformations of$$\varphi $$ $\phi $X are unobstructed even though does not vanish. Consequently,$$H^2(T_X)$$ ${H}^{2}\left({T}_{X}\right)$X belongs to a unique irreducible component of the Gieseker moduli space. These irreducible components are uniruled. As a result of all this, we show the existence of infinitely many moduli spaces, satisfying the strict Beauville inequality , with an irreducible component that has a proper quadruple sublocus where the degree of the canonical morphism jumps up. These components are above the Castelnuovo line, but nonetheless parametrize surfaces with non birational canonical morphisms. The existence of jumping subloci is a contrast with the moduli of surfaces with$$p_g > 2q4$$ ${p}_{g}>2q4$ , studied by Horikawa. Irreducible moduli components with a jumping sublocus also present a similarity and a difference to the moduli of curves of genus$$K^2 = 2p_g 4$$ ${K}^{2}=2{p}_{g}4$ , for, like in the case of curves, the degree of the canonical morphism goes down outside a closed sublocus but, unlike in the case of curves, it is never birational. Finally, our study shows that there are infinitely many moduli spaces with an irreducible component whose general elements have non birational canonical morphism and another irreducible component whose general elements have birational canonical map.$$g\ge 3$$ $g\ge 3$ 
Abstract 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
and other complexity classes do not have small circuits (in the worst case and/or on average) from various circuit classes$\mathsf {Quasi}\text {}\mathsf {NP} = \mathsf {NTIME}[n^{(\log n)^{O(1)}}]$ $\mathrm{Quasi}\mathrm{NP}=\mathrm{NTIME}\left[{n}^{{\left(\mathrm{log}n\right)}^{O\left(1\right)}}\right]$ , by showing that$\mathcal { C}$ $C$ admits nontrivial satisfiability and/or$\mathcal { C}$ $C$# SAT algorithms which beat exhaustive search by a minor amount. In this paper, we present a new strong lower bound consequence of having a nontrivial# SAT algorithm for a circuit class . Say that a symmetric Boolean function${\mathcal C}$ $C$f (x _{1},…,x _{n}) issparse if it outputs 1 onO (1) values of . We show that for every sparse${\sum }_{i} x_{i}$ ${\sum}_{i}{x}_{i}$f , and for all “typical” , faster$\mathcal { C}$ $C$# SAT algorithms for circuits imply lower bounds against the circuit class$\mathcal { C}$ $C$ , which may be$f \circ \mathcal { C}$ $f\circ C$stronger than itself. In particular:$\mathcal { C}$ $C$# SAT algorithms forn ^{k}size circuits running in 2^{n}/$\mathcal { C}$ $C$n ^{k}time (for allk ) implyN E X P does not have circuits of polynomial size.$(f \circ \mathcal { C})$ $(f\circ C)$# SAT algorithms for size$2^{n^{{\varepsilon }}}$ ${2}^{{n}^{\epsilon}}$ circuits running in$\mathcal { C}$ $C$ time (for some$2^{nn^{{\varepsilon }}}$ ${2}^{n{n}^{\epsilon}}$ε > 0) implyQ u a s i N P does not have circuits of polynomial size.$(f \circ \mathcal { C})$ $(f\circ C)$Applying
# SAT algorithms from the literature, one immediate corollary of our results is thatQ u a s i N P does not haveE M A J ∘A C C ^{0}∘T H R circuits of polynomial size, whereE M A J is the “exact majority” function, improving previous lower bounds againstA C C ^{0}[Williams JACM’14] andA C C ^{0}∘T H R [Williams STOC’14], [MurrayWilliams STOC’18]. This is the first nontrivial lower bound against such a circuit class.