Positive Body Chasing: Dynamic LP Optimization

• Positive body chasing is an online optimization framework that maintains feasible solutions to mixed packing–covering linear programs while minimizing cumulative ℓ1 movement cost.
• It employs memoryless, entropic KL-projections to update solutions based solely on the current violated constraint, achieving competitive recourse bounds.
• The framework supports dynamic algorithms for set cover, load balancing, matching, and MST through dynamic rounding, ensuring logarithmic recourse even with resource augmentation.

Positive body chasing is an online optimization framework concerned with the dynamic maintenance of feasible solutions to mixed packing-covering linear programs, minimizing cumulative adjustment cost under $\ell_1$-movement in the nonnegative orthant. This captures the fully-dynamic, low-recourse regime of many central problems in dynamic algorithms, including set cover, load balancing, hyperedge orientation, minimum spanning tree (MST), and matching (Bhattacharya et al., 2023).

1. Formal Problem Setting

At each time step $t=1,\ldots,T$ in the online sequence, a new “positive body” is revealed:

$$K_t = \{x \in \mathbb{R}_+^n \mid C^t x \geq 1,\; P^t x \leq 1\}$$

Here, $C^t$ and $P^t$ are nonnegative matrices: $C^t x \geq 1$ encodes covering constraints, $P^t x \leq 1$ encodes packing constraints. Points $x^0=0$, $x^1,\ldots,x^T$ with $x^t \in K_t$ must be maintained while minimizing the total recourse (i.e., movement):

$t=1,\ldots,T$0

Because $t=1,\ldots,T$1 is nondecreasing along covering updates and nonincreasing for packing, the one-sided “upward” movement $t=1,\ldots,T$2 approximates the $t=1,\ldots,T$3-cost within a factor of 2.

Throughout, $t=1,\ldots,T$4 denotes the number of nonzeros in the $t=1,\ldots,T$5th row of $t=1,\ldots,T$6, and $t=1,\ldots,T$7 is the maximum row sparsity. A resource augmentation parameter $t=1,\ldots,T$8 may be incorporated to consider constraints $t=1,\ldots,T$9.

2. Memoryless Online Algorithm and KL-Projections

The main result is the construction of an entirely memoryless, $$K_t = \{x \in \mathbb{R}_+^n \mid C^t x \geq 1,\; P^t x \leq 1\}$$0-competitive online algorithm for positive body chasing in $$K_t = \{x \in \mathbb{R}_+^n \mid C^t x \geq 1,\; P^t x \leq 1\}$$1 (Bhattacharya et al., 2023). The algorithmic procedure is as follows:

• Covering Constraint: If a new covering constraint $$K_t = \{x \in \mathbb{R}_+^n \mid C^t x \geq 1,\; P^t x \leq 1\}$$2 is violated by $$K_t = \{x \in \mathbb{R}_+^n \mid C^t x \geq 1,\; P^t x \leq 1\}$$3, the algorithm computes $$K_t = \{x \in \mathbb{R}_+^n \mid C^t x \geq 1,\; P^t x \leq 1\}$$4 as the unique minimizer of the weighted Kullback-Leibler (KL) divergence from $$K_t = \{x \in \mathbb{R}_+^n \mid C^t x \geq 1,\; P^t x \leq 1\}$$5, subject to the constraint and a small “shift” for well-definedness:

$$K_t = \{x \in \mathbb{R}_+^n \mid C^t x \geq 1,\; P^t x \leq 1\}$$6

$$K_t = \{x \in \mathbb{R}_+^n \mid C^t x \geq 1,\; P^t x \leq 1\}$$7

subject to $$K_t = \{x \in \mathbb{R}_+^n \mid C^t x \geq 1,\; P^t x \leq 1\}$$8, then set $$K_t = \{x \in \mathbb{R}_+^n \mid C^t x \geq 1,\; P^t x \leq 1\}$$9.

• Packing Constraint: If a new packing constraint $C^t$0 is violated, $C^t$1 projects $C^t$2 onto the halfspace in KL:

$C^t$3

• Each update relies solely on $C^t$4 and the currently violated constraint—no additional memory or history is used.

The following pseudocode captures the covering case:

$x^0=0$8

3. Theoretical Analysis and Competitive Guarantees

The theoretical analysis proceeds by coupling the online movement cost to a dual linear program (LP) approximating the offline optimum recourse:

• Offline Reference LP: Let $C^t$5 denote any offline feasible path. The primal LP is:

$C^t$6

subject to covering/packing constraints for $C^t$7 and increments $C^t$8.

The dual involves variables $C^t$9 (coverings), $P^t$0 (packings), and “potential” variables $P^t$1.

• Key Lemmas and Bounds:
  • Each covering-step movement is at most $P^t$2.
  • Packing steps subtract only a small quantity, i.e., $P^t$3.
  • Cumulatively, total movement is bounded by $P^t$4 (off-line optimum).
• Lower Bounds: For general convex bodies in $P^t$5, an $P^t$6 lower bound on the competitive ratio is known (Friedman–Linial). Positive bodies permit tighter $P^t$7 bounds—enabled by the sign-structure of normals and the ability to use entropic projections in $P^t$8. No $P^t$9-competitive algorithm is possible even for positive bodies without resource augmentation.

4. Dynamic Rounding and Algorithmic Applications

Fractional solutions generated by positive body chasing are rounded dynamically to obtain integral solutions for dynamic combinatorial problems. This is achieved by threshold or randomized sampling:

• Set Cover: Fractional LP enforces $C^t x \geq 1$0 for each element $C^t x \geq 1$1. Rounding either deterministically (picking $C^t x \geq 1$2 at multiples of $C^t x \geq 1$3) or via randomized sampling (probability $C^t x \geq 1$4) yields $C^t x \geq 1$5 offline recourse and $C^t x \geq 1$6 approximation.
• Load Balancing/Hyperedge Orientation: Jobs are fractionally assigned; use of low-recourse assignment routines ensures poly$C^t x \geq 1$7 recourse and either constant or refined $C^t x \geq 1$8 approximation ratios.
• Matching: Each edge is sampled $C^t x \geq 1$9 times to yield an integral matching after maintaining a short-augmenting-path matching with $P^t x \leq 1$0 absolute recourse per update. Overall, this results in $P^t x \leq 1$1 competitive recourse.
• Minimum Spanning Tree (MST): Cut-LP fractional solutions are rounded by sampling edges with probabilities proportional to $P^t x \leq 1$2 to form sparse subgraphs, on which an actual MST is maintained with $P^t x \leq 1$3 recourse per update; this leads to $P^t x \leq 1$4 total recourse.

Integral algorithms derived in this way achieve “competitive” recourse—at most $P^t x \leq 1$5 times the minimal recourse of any (even offline) algorithm maintaining an integral solution of similar quality, a standard strictly stronger than bounding recourse by absolute measures such as $P^t x \leq 1$6 or $P^t x \leq 1$7 per update.

5. Formulas and Notation Overview

The essential mathematical components:

Concept	Formula/Definition	Context
Positive Body	$P^t x \leq 1$8	Feasible region at time $P^t x \leq 1$9
$x^0=0$0-Recourse	$x^0=0$1	Objective cost function
Weighted KL Divergence	$x^0=0$2	Used in information-projection
Cover-Fix Move	$x^0=0$3 s.t. $x^0=0$4	Update for covering constraints

This framework unifies dynamic problems that can be cast as maintaining solutions to fractional mixed packing–covering LPs, achieves recourse guarantees that bypass the convex-body chasing $x^0=0$5 barrier, and produces the first fully dynamic algorithms with provably competitive recourse for several fundamental problems (Bhattacharya et al., 2023).

6. Significance and Limitations

The positive body chasing paradigm establishes a new regime for dynamic algorithms by exploiting the sign-structure of normals in positive bodies, and demonstrates that entropic, memoryless information projections are sufficient for near-optimal competitiveness in broad mixed packing-covering LPs. The competitive recourse achieved is logarithmic in the sparsity parameter and inverse resource augmentation, exponentially improving over known lower bounds for general convex bodies.

A matching lower bound asserts that the logarithmic dependence is essential; no algorithm can achieve sub-logarithmic (in $x^0=0$6 or $x^0=0$7) competitiveness for positive bodies without resource augmentation.

This framework’s recourse guarantees are “fully dynamic” and competitive on every update sequence relative to any (possibly offline) algorithm, thus distinguishing it sharply from the absolute recourse benchmarks in prior dynamic algorithms literature (Bhattacharya et al., 2023).