LP-Type Problems: Structure & Applications

• LP-type problems are abstract finite-constraint optimization problems defined by a totally ordered value function that obeys monotonicity and locality.
• They enable efficient algorithmic strategies like Clarkson-style sampling in distributed, streaming, and parallel models for applications in linear programming, geometry, and machine learning.
• Small certificate subsets, or bases, and bounded combinatorial dimensions are key to controlling complexity and facilitating provable efficiency in these optimization frameworks.

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 (Hinnenthal et al., 2019, Assadi et al., 2019).

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: $$\forall\,F\subseteq G\subseteq H\text{ with }w(F)=w(G),\ \forall\,h\in H,\quad w(G)<w(G\cup\{h\})\Longrightarrow w(F)<w(F\cup\{h\}).$$ Monotonicity expresses that adding constraints never improves the optimum, while locality states that if a constraint violates a larger set $G$, then it already violates every minimal set $F$ attaining the same value as $G$ (Hinnenthal et al., 2019).

Other expositions use the notation $(S,f)$ and add a feasibility condition: $f(\emptyset)=\min_{A\subseteq S} f(A),$ with $f(A)<\infty$ whenever $w:2^H\to \mathbb R\cup\{\pm\infty\}$0 is feasible. In the same notation, monotonicity is

$w:2^H\to \mathbb R\cup\{\pm\infty\}$1

and locality is written as

$w:2^H\to \mathbb R\cup\{\pm\infty\}$2

A further presentation emphasizes a basis-certificate form of locality: $w:2^H\to \mathbb R\cup\{\pm\infty\}$3 Across these formulations, LP-type structure is characterized by order-theoretic monotonicity and a certificate property ensuring that violations can be witnessed by small subsets (Assadi et al., 2019, Suter et al., 2020).

2. Bases, combinatorial dimension, and violators

A subset $w:2^H\to \mathbb R\cup\{\pm\infty\}$4 is called a basis, or extremal set, if

$w:2^H\to \mathbb R\cup\{\pm\infty\}$5

and every proper subset $w:2^H\to \mathbb R\cup\{\pm\infty\}$6 satisfies $w:2^H\to \mathbb R\cup\{\pm\infty\}$7. In the alternative notation, $w:2^H\to \mathbb R\cup\{\pm\infty\}$8 is a basis if $w:2^H\to \mathbb R\cup\{\pm\infty\}$9 and every proper subset $$\forall\,F\subseteq G\subseteq H:\qquad w(F)\le w(G)\le w(H).$$0 has $$\forall\,F\subseteq G\subseteq H:\qquad w(F)\le w(G)\le w(H).$$1. More generally, $$\forall\,F\subseteq G\subseteq H:\qquad w(F)\le w(G)\le w(H).$$2 may be a basis for a subinstance $$\forall\,F\subseteq G\subseteq H:\qquad w(F)\le w(G)\le w(H).$$3 when $$\forall\,F\subseteq G\subseteq H:\qquad w(F)\le w(G)\le w(H).$$4 and no proper subset preserves that value (Hinnenthal et al., 2019, Assadi et al., 2019).

The combinatorial dimension, denoted $$\forall\,F\subseteq G\subseteq H:\qquad w(F)\le w(G)\le w(H).$$5, $$\forall\,F\subseteq G\subseteq H:\qquad w(F)\le w(G)\le w(H).$$6, or $$\forall\,F\subseteq G\subseteq H:\qquad w(F)\le w(G)\le w(H).$$7 in the cited works, is the maximum size of any basis, either over the full instance or over all subinstances. This parameter is the central structural invariant: if it is constant, the corresponding LP-type problem admits fast randomized algorithms in gossip, streaming, coordinator, MPC, and multi-pass sketching models. The same algorithmic literature also tracks the VC dimension $$\forall\,F\subseteq G\subseteq H:\qquad w(F)\le w(G)\le w(H).$$8 of the violator range system

$$\forall\,F\subseteq G\subseteq H:\qquad w(F)\le w(G)\le w(H).$$9

or assumes $$\forall\,F\subseteq G\subseteq H\text{ with }w(F)=w(G),\ \forall\,h\in H,\quad w(G)<w(G\cup\{h\})\Longrightarrow w(F)<w(F\cup\{h\}).$$0, so that sampling results such as weighted $$\forall\,F\subseteq G\subseteq H\text{ with }w(F)=w(G),\ \forall\,h\in H,\quad w(G)<w(G\cup\{h\})\Longrightarrow w(F)<w(F\cup\{h\}).$$1-nets apply (Çekirge et al., 15 Jul 2025, Assadi et al., 2019).

A violator is any constraint whose insertion strictly increases the current value. In algorithmic notation, if $$\forall\,F\subseteq G\subseteq H\text{ with }w(F)=w(G),\ \forall\,h\in H,\quad w(G)<w(G\cup\{h\})\Longrightarrow w(F)<w(F\cup\{h\}).$$2 is a current basis, the violator set is

$$\forall\,F\subseteq G\subseteq H\text{ with }w(F)=w(G),\ \forall\,h\in H,\quad w(G)<w(G\cup\{h\})\Longrightarrow w(F)<w(F\cup\{h\}).$$3

and in the distributed gossip setting a node tests whether

$$\forall\,F\subseteq G\subseteq H\text{ with }w(F)=w(G),\ \forall\,h\in H,\quad w(G)<w(G\cup\{h\})\Longrightarrow w(F)<w(F\cup\{h\}).$$4

for each locally held constraint $$\forall\,F\subseteq G\subseteq H\text{ with }w(F)=w(G),\ \forall\,h\in H,\quad w(G)<w(G\cup\{h\})\Longrightarrow w(F)<w(F\cup\{h\}).$$5 (Assadi et al., 2019, Hinnenthal et al., 2019).

Problem family	Value notion	Dimension statement in the cited works
Linear programming in $$\forall\,F\subseteq G\subseteq H\text{ with }w(F)=w(G),\ \forall\,h\in H,\quad w(G)<w(G\cup\{h\})\Longrightarrow w(F)<w(F\cup\{h\}).$$6 variables	Optimum objective over the feasible polytope	$$\forall\,F\subseteq G\subseteq H\text{ with }w(F)=w(G),\ \forall\,h\in H,\quad w(G)<w(G\cup\{h\})\Longrightarrow w(F)<w(F\cup\{h\}).$$7 in one formulation; $$\forall\,F\subseteq G\subseteq H\text{ with }w(F)=w(G),\ \forall\,h\in H,\quad w(G)<w(G\cup\{h\})\Longrightarrow w(F)<w(F\cup\{h\}).$$8 in another
Smallest enclosing ball in $$\forall\,F\subseteq G\subseteq H\text{ with }w(F)=w(G),\ \forall\,h\in H,\quad w(G)<w(G\cup\{h\})\Longrightarrow w(F)<w(F\cup\{h\}).$$9	Radius of the minimum enclosing ball	$G$0
Hard-margin linear SVM	Optimum margin $G$1	$G$2
Core vector machines	Reduction of SVM to MEB in feature space	$G$3

The differing LP dimension conventions reflect differing presentations in the cited works; the common point is that low-dimensional instances have basis size bounded solely by the ambient dimension parameter rather than by the number of constraints (Hinnenthal et al., 2019, Assadi et al., 2019).

3. Clarkson-style sampling and multiplicity growth

A recurrent algorithmic pattern is Clarkson-style random sampling. In one distributed formulation, each node $G$4 maintains a local multiset $G$5, samples a random multiset $G$6 of size $G$7 by pull operations, computes a basis $G$8, identifies local violators $G$9, propagates violators by push operations when $F$0 is small enough, filters received constraints independently to keep $F$1 manageable, and triggers gossip-based termination when $F$2 and the sampled set attains the current global estimate. Whenever $F$3 and sampling succeeds, a constant fraction of the total multiset is expected to violate $F$4; when those violators double their multiplicities, every optimal basis element’s multiplicity grows by at least a factor $F$5 every $F$6 successful iterations. After $F$7 iterations, the multiplicity of some basis element would exceed the bounded total multiset size unless a correct basis has already been found (Hinnenthal et al., 2019).

A related meta-algorithm for streaming and distributed models maintains a nonnegative weight $F$8 on each constraint. It repeatedly samples a weighted $F$9-net $G$0 of size

$G$1

computes a basis $G$2 of size at most $G$3, forms the violator set

$G$4

and checks whether

$G$5

If so, the algorithm boosts the weights of violated constraints by a factor $G$6; otherwise it discards the sample and repeats. Termination occurs when $G$7, at which point $G$8 is a basis for $G$9. The complexity analysis hinges on the number of successful weight-boosting rounds, which is $(S,f)$0, together with the ability to implement each weighted-sampling step efficiently in the target computational model (Assadi et al., 2019).

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 $(S,f)$1 anonymous nodes operating in synchronous rounds with $(S,f)$2-bit messages. In each round, a node may push information to, or pull information from, a peer chosen uniformly at random, and may do $(S,f)$3 total pushes and pulls per round. For LP-type problems of constant combinatorial dimension $(S,f)$4 and input size $(S,f)$5, the distributed algorithm achieves, with high probability, round complexity

$(S,f)$6

and work per node per round

$(S,f)$7

Random filtering plus Chernoff bounds keeps local multisets from exploding, and a push-pull termination subroutine ensures that once one node finds $(S,f)$8, all nodes learn it in $(S,f)$9 additional rounds with $f(\emptyset)=\min_{A\subseteq S} f(A),$0 work (Hinnenthal et al., 2019).

The same bounded-dimension perspective yields near-tight trade-offs in several big-data models. For an LP-type problem of dimension $f(\emptyset)=\min_{A\subseteq S} f(A),$1, and for any integer $f(\emptyset)=\min_{A\subseteq S} f(A),$2 and $f(\emptyset)=\min_{A\subseteq S} f(A),$3, one can compute $f(\emptyset)=\min_{A\subseteq S} f(A),$4 with high probability using the following informal bounds:

• Streaming: $f(\emptyset)=\min_{A\subseteq S} f(A),$5 passes and $f(\emptyset)=\min_{A\subseteq S} f(A),$6 space.
• Coordinator: $f(\emptyset)=\min_{A\subseteq S} f(A),$7 rounds and $f(\emptyset)=\min_{A\subseteq S} f(A),$8 total bits of communication.
• MPC: $f(\emptyset)=\min_{A\subseteq S} f(A),$9 rounds with per-machine load $f(A)<\infty$0. Low $f(A)<\infty$1 permits $f(A)<\infty$2-net samples of size $f(A)<\infty$3 for suitable $f(A)<\infty$4, which is what drives the reduction in space, communication, and rounds (Assadi et al., 2019).

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 (Assadi et al., 2019). At the same time, the framework has limits: for hitting set and set cover, the worst-case $f(A)<\infty$5 can be as large as $f(A)<\infty$6. Nevertheless, for range spaces of size $f(A)<\infty$7 with optimal cover size $f(A)<\infty$8, the gossip framework yields an $f(A)<\infty$9-round, $w:2^H\to \mathbb R\cup\{\pm\infty\}$00-work algorithm computing an $w:2^H\to \mathbb R\cup\{\pm\infty\}$01-approximate hitting set by sampling $w:2^H\to \mathbb R\cup\{\pm\infty\}$02 elements and reusing the same multiplicity-doubling analysis (Hinnenthal et al., 2019).

5. Multi-pass linear sketches and lower bounds

A 2025 development studies LP-type problems in the multiple linear sketch model, equivalent to an $w:2^H\to \mathbb R\cup\{\pm\infty\}$03-pass streaming algorithm that may choose a fresh linear sketch in each pass. For LP-type problems with combinatorial and VC dimensions $w:2^H\to \mathbb R\cup\{\pm\infty\}$04, and for any

$w:2^H\to \mathbb R\cup\{\pm\infty\}$05

there is a randomized algorithm making

$w:2^H\to \mathbb R\cup\{\pm\infty\}$06

passes and using

$w:2^H\to \mathbb R\cup\{\pm\infty\}$07

words of space, where

$w:2^H\to \mathbb R\cup\{\pm\infty\}$08

Equivalently,

$w:2^H\to \mathbb R\cup\{\pm\infty\}$09

By setting $w:2^H\to \mathbb R\cup\{\pm\infty\}$10, the factor $w:2^H\to \mathbb R\cup\{\pm\infty\}$11 becomes $w:2^H\to \mathbb R\cup\{\pm\infty\}$12, yielding space polynomial in $w:2^H\to \mathbb R\cup\{\pm\infty\}$13 and polylogarithmic in $w:2^H\to \mathbb R\cup\{\pm\infty\}$14 (Çekirge et al., 15 Jul 2025).

The upper-bound proof replaces explicit sampling over the original input with a discrete metric $w:2^H\to \mathbb R\cup\{\pm\infty\}$15-net $w:2^H\to \mathbb R\cup\{\pm\infty\}$16 of size

$w:2^H\to \mathbb R\cup\{\pm\infty\}$17

snaps each point to a unique net point, and then simulates weighted Clarkson rounds over $w:2^H\to \mathbb R\cup\{\pm\infty\}$18. Weights on net points are multiplied by $w:2^H\to \mathbb R\cup\{\pm\infty\}$19 when they violate the current sample. To implement sampling and weight estimation in sublinear memory, the method uses a bank of $w:2^H\to \mathbb R\cup\{\pm\infty\}$20 samplers and $w:2^H\to \mathbb R\cup\{\pm\infty\}$21 estimators, each requiring $w:2^H\to \mathbb R\cup\{\pm\infty\}$22 space per class. In $w:2^H\to \mathbb R\cup\{\pm\infty\}$23 successful rounds, the weight of an optimal basis becomes dominant, so the current solution is correct on $w:2^H\to \mathbb R\cup\{\pm\infty\}$24, and hence $w:2^H\to \mathbb R\cup\{\pm\infty\}$25-correct on the original point set $w:2^H\to \mathbb R\cup\{\pm\infty\}$26 (Çekirge et al., 15 Jul 2025).

The framework specializes to several geometric LP-type problems. For Minimum Enclosing Ball, where $w:2^H\to \mathbb R\cup\{\pm\infty\}$27, and for linear SVM with $w:2^H\to \mathbb R\cup\{\pm\infty\}$28, one-pass $w:2^H\to \mathbb R\cup\{\pm\infty\}$29-approximation requires $w:2^H\to \mathbb R\cup\{\pm\infty\}$30 space, but the multi-pass sketch yields an $w:2^H\to \mathbb R\cup\{\pm\infty\}$31-pass, $w:2^H\to \mathbb R\cup\{\pm\infty\}$32-space $w:2^H\to \mathbb R\cup\{\pm\infty\}$33-approximation. For bounded $w:2^H\to \mathbb R\cup\{\pm\infty\}$34-variable LP with $w:2^H\to \mathbb R\cup\{\pm\infty\}$35, the result is additive $w:2^H\to \mathbb R\cup\{\pm\infty\}$36-approximation with the same polynomial-space and $w:2^H\to \mathbb R\cup\{\pm\infty\}$37-pass profile. For bounded SDP with trace-one PSD matrices $w:2^H\to \mathbb R\cup\{\pm\infty\}$38, the cited summary states $w:2^H\to \mathbb R\cup\{\pm\infty\}$39 and gives additive $w:2^H\to \mathbb R\cup\{\pm\infty\}$40-approximation in $w:2^H\to \mathbb R\cup\{\pm\infty\}$41 space and $w:2^H\to \mathbb R\cup\{\pm\infty\}$42 passes (Çekirge et al., 15 Jul 2025).

Lower bounds indicate that these gains do not come for free. For exact two-dimensional LP, any randomized $w:2^H\to \mathbb R\cup\{\pm\infty\}$43-pass streaming algorithm with constant success probability must use

$w:2^H\to \mathbb R\cup\{\pm\infty\}$44

space, and any $w:2^H\to \mathbb R\cup\{\pm\infty\}$45-round coordinator algorithm, even with $w:2^H\to \mathbb R\cup\{\pm\infty\}$46 sites, requires

$w:2^H\to \mathbb R\cup\{\pm\infty\}$47

total bits of communication. For approximate MEB and linear SVM in the high-accuracy regime $w:2^H\to \mathbb R\cup\{\pm\infty\}$48, any one-pass streaming algorithm must use

$w:2^H\to \mathbb R\cup\{\pm\infty\}$49

words of space. These lower bounds explain why the pass complexity of the newer sketching algorithms is central rather than incidental (Assadi et al., 2019, Çekirge et al., 15 Jul 2025).

6. Thresholded LP-type problems, monotone Boolean functions, and MaxCon

LP-type structure also admits a thresholded interpretation. Given an LP-type objective

$w:2^H\to \mathbb R\cup\{\pm\infty\}$50

and a threshold $w:2^H\to \mathbb R\cup\{\pm\infty\}$51, define

$w:2^H\to \mathbb R\cup\{\pm\infty\}$52

Because $w:2^H\to \mathbb R\cup\{\pm\infty\}$53 is monotone, $w:2^H\to \mathbb R\cup\{\pm\infty\}$54 is a monotone Boolean function on $w:2^H\to \mathbb R\cup\{\pm\infty\}$55. In robust fitting, one typically sets $w:2^H\to \mathbb R\cup\{\pm\infty\}$56 by rewriting $w:2^H\to \mathbb R\cup\{\pm\infty\}$57 as the negative number of outliers in $w:2^H\to \mathbb R\cup\{\pm\infty\}$58, so that $w:2^H\to \mathbb R\cup\{\pm\infty\}$59 iff $w:2^H\to \mathbb R\cup\{\pm\infty\}$60 is infeasible. The lower sets $w:2^H\to \mathbb R\cup\{\pm\infty\}$61 form an independence system closed under subsets (Suter et al., 2020).

This viewpoint introduces the influence of an element $w:2^H\to \mathbb R\cup\{\pm\infty\}$62: $w:2^H\to \mathbb R\cup\{\pm\infty\}$63 Equivalently, in the Fourier expansion of $w:2^H\to \mathbb R\cup\{\pm\infty\}$64, $w:2^H\to \mathbb R\cup\{\pm\infty\}$65 is proportional to the first-order coefficient on $w:2^H\to \mathbb R\cup\{\pm\infty\}$66. 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 $w:2^H\to \mathbb R\cup\{\pm\infty\}$67 splits into an inlier set of size $w:2^H\to \mathbb R\cup\{\pm\infty\}$68 and outliers of size $w:2^H\to \mathbb R\cup\{\pm\infty\}$69, then every inlier $w:2^H\to \mathbb R\cup\{\pm\infty\}$70 and outlier $w:2^H\to \mathbb R\cup\{\pm\infty\}$71 satisfy

$w:2^H\to \mathbb R\cup\{\pm\infty\}$72

For two disjoint ideal structures of sizes $w:2^H\to \mathbb R\cup\{\pm\infty\}$73, influences are ordered by structure-membership class: $w:2^H\to \mathbb R\cup\{\pm\infty\}$74 More generally, for $w:2^H\to \mathbb R\cup\{\pm\infty\}$75 disjoint ideal structures, the influences are totally ordered by the bit-pattern of membership (Suter et al., 2020).

These results motivate an influence-guided search for Maximum Consensus. Starting from $w:2^H\to \mathbb R\cup\{\pm\infty\}$76, the algorithm repeatedly queries a feasibility oracle for $w:2^H\to \mathbb R\cup\{\pm\infty\}$77; when $w:2^H\to \mathbb R\cup\{\pm\infty\}$78 is infeasible, the oracle returns a small infeasible basis $w:2^H\to \mathbb R\cup\{\pm\infty\}$79 with $w:2^H\to \mathbb R\cup\{\pm\infty\}$80. The algorithm estimates the influences of elements in $w:2^H\to \mathbb R\cup\{\pm\infty\}$81, removes the highest-influence element, and repeats until feasibility is reached. An optional local “up-lift” step then tries to reinsert excluded points whenever feasibility is preserved. Each iteration costs $w:2^H\to \mathbb R\cup\{\pm\infty\}$82 oracle calls to estimate the $w:2^H\to \mathbb R\cup\{\pm\infty\}$83 influences, plus one oracle call for termination; in practice $w:2^H\to \mathbb R\cup\{\pm\infty\}$84. In the line-fitting example summarized in the paper, a synthetic instance with $w:2^H\to \mathbb R\cup\{\pm\infty\}$85 points—$w:2^H\to \mathbb R\cup\{\pm\infty\}$86 near one line and $w:2^H\to \mathbb R\cup\{\pm\infty\}$87 pure outliers—leads the influence algorithm to remove the true outliers first and recover the $w:2^H\to \mathbb R\cup\{\pm\infty\}$88-point inlier set in two steps. For fundamental-matrix estimation on KITTI and AdelaideRMF sequences, the same summary states that influence-guided search nearly matches the optimal consensus found by expensive $w:2^H\to \mathbb R\cup\{\pm\infty\}$89 search, runs orders of magnitude faster, and often outperforms RANSAC-style heuristics in finding the true largest consensus (Suter et al., 2020).

The thresholded viewpoint shows that LP-type problems are not only an abstract optimization class but also a bridge to monotone Boolean functions and violator spaces. A plausible implication is that the same basis-and-violation logic that drives Clarkson-style optimization can also guide feasibility-oriented search procedures in robust estimation, provided the problem admits a meaningful thresholded formulation.