--- title: 'LP-Type Problems: Structure & Applications' url: https://www.emergentmind.com/topics/lp-type-problems type: topic --- # LP-Type Problems: Structure & Applications Searching arXiv for relevant papers on LP-type problems and recent developments. LP-type problems are abstract finite-constraint optimization problems in which a value function on subsets is totally ordered and satisfies monotonicity together with a locality principle. In this framework, minimal subsets attaining the global optimum act as certificates, and the size of such certificates controls algorithmic complexity. The class includes (integer) linear programming, geometric problems such as smallest enclosing ball and polytope distance, set problems such as hitting set and set cover, and several machine-learning formulations including hard-margin linear SVM and core vector machines [1904.10706][1903.05617]. ## 1. Axiomatic structure A standard presentation begins with a finite ground set \(H\) of constraints and a function \[ w:2^H\to \mathbb R\cup\{\pm\infty\} \] mapping each subset to a totally ordered value. Two axioms are required. The first is monotonicity: \[ \forall\,F\subseteq G\subseteq H:\qquad w(F)\le w(G)\le w(H). \] The second is locality: \[ \forall\,F\subseteq G\subseteq H\text{ with }w(F)=w(G),\ \forall\,h\in H,\quad w(G)f(B)\}, \] and in the distributed gossip setting a node tests whether \[ w(R_i)f(B)\}, \] and checks whether \[ \sum_{e\in V} w(e)\le \varepsilon \sum_e w(e). \] If so, the algorithm boosts the weights of violated constraints by a factor \(n^{1/r}\); otherwise it discards the sample and repeats. Termination occurs when \(V=\emptyset\), at which point \(B\) is a basis for \(S\). The complexity analysis hinges on the number of successful weight-boosting rounds, which is \(O(\delta r)\), together with the ability to implement each weighted-sampling step efficiently in the target computational model [1903.05617]. The common methodological theme is that small bases permit aggressive subsampling without losing correctness: weighted or multiplicity-based updates make true basis elements increasingly likely to appear in later samples, and locality ensures that violations can be detected relative to a current basis rather than by recomputing the full instance. ## 4. Distributed, streaming, and parallel complexity In the gossip model, the network consists of \(n\) anonymous nodes operating in synchronous rounds with \(O(\log n)\)-bit messages. In each round, a node may push information to, or pull information from, a peer chosen uniformly at random, and may do \(O(\mathrm{polylog}\,n)\) total pushes and pulls per round. For LP-type problems of constant combinatorial dimension \(\delta\) and input size \(|H|=\mathrm{poly}(n)\), the distributed algorithm achieves, with high probability, round complexity \[ O(\delta\log n) \] and work per node per round \[ O(\mathrm{poly}\log n). \] Random filtering plus Chernoff bounds keeps local multisets from exploding, and a push-pull termination subroutine ensures that once one node finds \(w(H)\), all nodes learn it in \(O(\log n)\) additional rounds with \(O(\log n)\) work [1904.10706]. The same bounded-dimension perspective yields near-tight trade-offs in several big-data models. For an LP-type problem of dimension \(\delta\), and for any integer \(r\ge 1\) and \(\varepsilon\in(0,1)\), one can compute \(f(S)\) with high probability using the following informal bounds: - **Streaming:** \(O(\delta r)\) passes and \(O(\delta\lambda n^{1/r})\) space. - **Coordinator:** \(O(\delta r)\) rounds and \(O(\delta^2 n^{1/r}+\delta^2 k)\) total bits of communication. - **MPC:** \(O(\delta/\varepsilon^2)\) rounds with per-machine load \(O(\delta^2 n^\varepsilon)\). Low \(\delta\) permits \(\varepsilon\)-net samples of size \(O(\delta n^\alpha)\) for suitable \(\alpha\), which is what drives the reduction in space, communication, and rounds [1903.05617]. These results extend beyond geometric optimization in the narrow sense. Low dimensional LP-type problems are stated to arise in robust regression, support vector machines, and core vector machines, and their efficient support is motivated as valuable for large-scale machine learning queries in database systems [1903.05617]. At the same time, the framework has limits: for hitting set and set cover, the worst-case \(\delta\) can be as large as \(\Theta(n)\). Nevertheless, for range spaces of size \(s\) with optimal cover size \(d\), the gossip framework yields an \(O(\log n)\)-round, \(O(d\log n+\mathrm{poly}\log n)\)-work algorithm computing an \(O(d\log(ds))\)-approximate hitting set by sampling \(O(d\log(ds))\) elements and reusing the same multiplicity-doubling analysis [1904.10706]. ## 5. Multi-pass linear sketches and lower bounds A 2025 development studies LP-type problems in the multiple linear sketch model, equivalent to an \(r\)-pass streaming algorithm that may choose a fresh linear sketch in each pass. For LP-type problems with combinatorial and VC dimensions \(O(d)\), and for any \[ s\in[1,\ d\log(1/\varepsilon)], \] there is a randomized algorithm making \[ O(ds) \] passes and using \[ S(d,\varepsilon,s)=O\!\bigl(s\,N^{3/s}\bigr)\times \mathrm{poly}(d,\log N) \] words of space, where \[ N=\bigl(\tfrac{\sqrt d}{\varepsilon}\bigr)^d. \] Equivalently, \[ S(d,\varepsilon,s)=O\!\Bigl(s\,(\tfrac{\sqrt d}{\varepsilon})^{3d/s}\Bigr)\cdot \mathrm{poly}\bigl(d,\log(1/\varepsilon)\bigr). \] By setting \(s=d\log(1/\varepsilon)\), the factor \(N^{3/s}\) becomes \(\mathrm{polylog}(1/\varepsilon)\), yielding space polynomial in \(d\) and polylogarithmic in \(1/\varepsilon\) [2507.11484]. The upper-bound proof replaces explicit sampling over the original input with a discrete metric \(\varepsilon\)-net \(Q\subseteq \mathbb R^d\) of size \[ N=O\!\bigl((\tfrac{\sqrt d}{\varepsilon})^d\bigr), \] snaps each point to a unique net point, and then simulates weighted Clarkson rounds over \(Q\). Weights on net points are multiplied by \(N^{1/s}\) when they violate the current sample. To implement sampling and weight estimation in sublinear memory, the method uses a bank of \(\ell_0\) samplers and \(\ell_0\) estimators, each requiring \(\mathrm{polylog}(N)\) space per class. In \(O(\nu s)\) successful rounds, the weight of an optimal basis becomes dominant, so the current solution is correct on \(Q\), and hence \((1+O(\varepsilon))\)-correct on the original point set \(P\) [2507.11484]. The framework specializes to several geometric LP-type problems. For Minimum Enclosing Ball, where \(\nu=\lambda=d+1\), and for linear SVM with \(\nu,\lambda=O(d)\), one-pass \((1+\varepsilon)\)-approximation requires \((1/\varepsilon)^{\Omega(d)}\) space, but the multi-pass sketch yields an \(O(d^2\log(1/\varepsilon))\)-pass, \(\mathrm{poly}(d,\log(1/\varepsilon))\)-space \((1+O(\varepsilon))\)-approximation. For bounded \(d\)-variable LP with \(\nu=d,\lambda=d+1\), the result is additive \(\varepsilon\)-approximation with the same polynomial-space and \(O(d^2\log(1/\varepsilon))\)-pass profile. For bounded SDP with trace-one PSD matrices \(X\in\mathbb R^{d\times d}\), the cited summary states \(\nu,\lambda=O(d^2)\) and gives additive \(\varepsilon\)-approximation in \(\mathrm{poly}(d,\log(1/\varepsilon))\) space and \(O(d^2\log(1/\varepsilon))\) passes [2507.11484]. Lower bounds indicate that these gains do not come for free. For exact two-dimensional LP, any randomized \(r\)-pass streaming algorithm with constant success probability must use \[ \Omega\bigl(r^{-3}n^{1/2r}\bigr) \] space, and any \(r\)-round coordinator algorithm, even with \(k=2\) sites, requires \[ \Omega\bigl(r^{-2}n^{1/2r}\bigr) \] total bits of communication. For approximate MEB and linear SVM in the high-accuracy regime \(d<(1/\varepsilon)^{0.999}\), any one-pass streaming algorithm must use \[ \Omega\bigl((1/\varepsilon)^{\Omega(d)}\bigr) \] words of space. These lower bounds explain why the pass complexity of the newer sketching algorithms is central rather than incidental [1903.05617][2507.11484]. ## 6. Thresholded LP-type problems, monotone Boolean functions, and MaxCon LP-type structure also admits a thresholded interpretation. Given an LP-type objective \[ f:2^H\to \mathbb R\cup\{-\infty\} \] and a threshold \(\tau\), define \[ \varphi_\tau(G)= \begin{cases} 1,& f(G)\ge \tau,\\ 0,& f(G)<\tau. \end{cases} \] Because \(f\) is monotone, \(\varphi_\tau\) is a monotone Boolean function on \(2^H\). In robust fitting, one typically sets \(\tau=0\) by rewriting \(f(G)\) as the negative number of outliers in \(G\), so that \(\varphi_0(G)=1\) iff \(G\) is infeasible. The lower sets \(\{F:\varphi_\tau(F)=0\}\) form an independence system closed under subsets [2005.05490]. This viewpoint introduces the influence of an element \(i\in H\): \[ I_i(\varphi)=\Pr_{X\sim \mathrm{Unif}(2^H)}\bigl[\varphi(X)\neq \varphi(X\oplus\{i\})\bigr]. \] Equivalently, in the Fourier expansion of \(\varphi\), \(I_i(\varphi)\) is proportional to the first-order coefficient on \(\{i\}\). The cited results state that, in thresholded LP-type or MaxCon problems, outliers tend to have high influence, while true inliers have low influence. Under ideal assumptions, if \(H\) splits into an inlier set of size \(k\) and outliers of size \(N-k\), then every inlier \(i\) and outlier \(j\) satisfy \[ I_i(\varphi)