Interweaved-Cliques (IC) Design

• Interweaved-Cliques (IC) Design are combinatorial and graph-theoretic structures that generalize cliques and cycles to optimize partitioning and coding schemes.
• They enable efficient algorithms for minimizing broadcast rates and load balancing by leveraging unique I-path properties and interlocked outer cycles.
• IC designs have broad applications in index coding, distributed computing, and extremal combinatorics, offering scalable solutions for resource allocation and network optimization.

An Interweaved-Cliques (IC) Design refers to a family of combinatorial and graph-theoretic structures underlying optimal partitioning, covering, or coding schemes in areas such as index coding, distributed computing, and extremal combinatorics. The fundamental idea is to generalize classical cliques and cycles—maximally interconnected subgraphs—to overlapping but highly structured formations that capture and exploit deeper symmetries, path-uniqueness, and partitioning gains. IC designs admit highly efficient algorithms or constructions for tasks such as minimizing broadcast rate, communication load, or coloring number, and reveal tight order-optimality in a variety of settings.

1. Graphical Formulation of Interweaved-Cliques

The IC design in its classical form arises in index coding, where it models the side-information flow among receivers as a directed graph $G=(V,E)$. A subset of vertices $V_I$ (the inner set) is selected, and the IC-structure is built on the existence of I-paths—directed paths connecting pairs of inner nodes with all intermediate nodes among the non-inner set $V_{NI}=V\setminus V_I$—subject to the following constraints:

• No I-cycles: No directed cycle passes through exactly one inner vertex.
• Unique I-paths: For any $i\neq j\in V_I$, there is exactly one I-path from $i$ to $j$.
• The union of rooted trees $T_i$ at each $i\in V_I$ with leaves $V_I\setminus\{i\}$ covers $G$.
• The subgraph induced by $V_{NI}$ is acyclic; in several important generalizations, certain types of cycles (outer cycles) are permitted but must obey stringent interlocking/intersection conditions (Thapa et al., 2016, Sasi et al., 2018, Sasi et al., 2018, Vaddi et al., 2019).

This graph-theoretic construct embeds both cycles (directed cycles as the degenerate case) and cliques (full interconnection via unique I-paths) as special cases. The notion can be extended to Overlapping Interlinked-Cycle (OIC) structures by recursively introducing and overlapping several layers of inner sets arranged in a polytree, enabling further generality and expressive power (Vaddi et al., 2019).

2. IC Structures with Interlocked Outer Cycles

A key generalization, highly relevant in applied coding and scheduling, involves IC structures with interlocked outer cycles (Sasi et al., 2018, Sasi et al., 2018):

• Outer cycles are cycles contained entirely within $V_{NI}$.
• The family of outer cycles is said to be interlocked if intersections between cycles are always pathwise (not arbitrary), with a Central Cycle Condition ensuring all cycles share structure via a central one.
• The critical combinatorial parameter is $t$, the maximum number of pairwise vertex-disjoint outer cycles. This parameter governs the optimal length of index codes (or, by analogy, the size of other resource-optimal coverings).

Table 1 summarizes key combinatorial objects:

Structure Type	Key Parameters	Governs
IC structure (classical)	$K$ inner, $N$ total nodes	Uniqueness of I-paths, absence of I-cycles
IC + interlocked outer cycles (generalized)	$K$, $N$, $t$ outer cycles	Index code length, partitioning gain
OIC structure	Polytree of semi-inner sets	Capacity, MAIS achievement

The outer cycle arrangement leads to new code/partitioning constructions where the optimal broadcast rate is $L^* = N-K+2-t$ (Sasi et al., 2018).

3. Principles and Methods for IC Design Construction

A canonical methodology for constructing IC designs, particularly in distributed computing and data allocation, follows a deterministic, algebraically structured partitioning:

• Decompose $n$ files into $k$ families, each of size $s$.
• Partition the full set of $d$-tuples into groups indexed by combinatorial types (e.g., multisets of families).
• Assign full-support $d$-tuples and allocate incomplete-support tuples as equitably as possible using combinatorial splitting rules.
• If $n$ is not a multiple of $k$, discard a minimal set of files and reincorporate associated tasks by preserving the partitioning principle (Maheri et al., 9 Jan 2026).
• When the number of groups $N$ exceeds the initial design, further split groups to exactly match $N$, ensuring near-optimal load balancing across groups.

For index coding, the coding algorithm matches these combinatorial splits: inner-message XORs, partially coded non-inner messages along unique I-paths, and residual handling of cycles according to their interlocking and centrality properties (Sasi et al., 2018, Sasi et al., 2018).

4. Optimality, Cost Metrics, and Converse Bounds

IC designs are characterized by provable order-optimality results across their defining cost metrics:

• Communication cost ($\pi_{\mathbf{X}}$): Maximum number of files accessed per group, scaling as $O(n/N^{1/d})$.
• Computation cost ($\delta_{\mathbf{X}}$): Maximum load imbalance per group, maintained at a constant factor when the decomposition is reasonably dense and the partitioning is random or balanced.
• Index code length: For side-information graphs with interlocked cycles, the code length is tightly bounded by $L^* = N-K+2-t$ and achieves the MAIS lower bound under the IC/OIC regime (Vaddi et al., 2019, Sasi et al., 2018).

Universal lower bounds are derived by convexity and vertex-cover arguments. For hypergraphs of $d$-ary dependencies, any allocation must satisfy $\pi_{\mathbf{X}} \geq \phi^{1/d}\cdot n/N^{1/d}$ for any $|\mathbf{X}| = \phi\binom{n}{d}$ (Maheri et al., 9 Jan 2026).

Order-optimality is achieved deterministically, and, importantly, the "blind" property holds: file assignments are fixed independently of $\mathbf{X}$, so allocations remain optimal across multiple changing computational tasks without file reshuffling.

5. Extremal Combinatorics and Nearly Disjoint IC Designs

IC-like structures also play a central role in extremal combinatorics, notably in hypergraph coloring and the construction of "near-designs"—families of cliques that are nearly disjoint (intersections bounded by $t$):

• Polynomially many $r$-cliques, each pair intersecting in at most $t$ vertices, can be constructed algebraically or via random greedy methods (Mubayi et al., 2023).
• Such designs force large chromatic numbers ($\chi(H) \sim r^2$ for $t=1$) despite having only polynomially many cliques and small pairwise intersections.
• Key methods include affine or projective plane line packings (Vandermonde-based) and careful control of intersection profiles via probabilistic techniques.

These results indicate that highly overlapped but structured clique packings induce significant partitioning or coloring hardness, parallel to the resource savings in IC index coding or distributed allocation.

6. Algorithmic Implications and Complexity Considerations

Algorithmic detection of IC or OIC structures is generally NP-hard due to the requirement to search for maximal cliques, cycles, or polytree overlap patterns (Vaddi et al., 2019). Practical strategies employ:

• Heuristics for large clique/cycle detection as seeds.
• Greedy extension by enforcing unique overlaps and structured union via OIC polytree conditions.
• Pruning and verification to comply with IC/OIC defining constraints.

In random intersection graphs, efficient algorithms for robustly recovering overlapping cliques exploit Sum-of-Squares relaxations, bipartite balancedness certificates, and pseudo-distribution-based refutations. Such approaches are effective when clique sizes satisfy $k \gg \sqrt{n\log n}$, and succeed where classical spectral or combinatorial heuristics fail because of large degree fluctuations intrinsic to highly-overlapping clique systems (Göbel et al., 25 Nov 2025). This suggests the computational tractability of IC-like partition recovery is highly sensitive to both overlap degree and noise regime.

7. Extensions, Special Cases, and Applications

Special cases of IC design recover established coding and combinatorial models:

• The clique and cycle, with maximal or minimally overlapping inner sets, are the trivial representatives of the broader IC class.
• When all outer cycles share a common vertex ($t=1$), classic index coding gains are recovered; when cycles overlap less, more extensive partitioning gains are achieved by increasing $t$.
• OIC structures generalize the notion further via recursive overlap hierarchies, tightly matching MAIS-based lower bounds for index code capacity (Vaddi et al., 2019).

Applications of IC design span distributed computing (optimal data/task allocation), network broadcast, hypergraph coloring, and robust community detection in random graphs. The universality property—robustness of allocation or code under arbitrary function decomposition or changing subfunction sets—makes IC design a focal point for the construction of scalable and adaptive resource allocation protocols (Maheri et al., 9 Jan 2026).