Fractional Biclique Partition Number

• Fractional Biclique Partition Number is the LP relaxation of covering a bipartite graph’s edges with weighted bicliques, equivalent to the fractional binary rank of its adjacency matrix.
• It provides a framework connecting matrix factorization with combinatorial biclique covers, highlighting differences between exact partitions and their LP relaxations.
• Studies on examples like the Domino graph reveal non-multiplicative behavior under Kronecker products, influencing asymptotic performance and complexity bounds.

Searching arXiv for the cited paper and closely related work on biclique partitions, fractional binary rank, and Kronecker products. The fractional biclique partition number is the linear-programming relaxation of the biclique partition number of a bipartite graph. For a bipartite graph $G=(U,V,E)$ with biclique family $\mathcal{B}$, it assigns nonnegative weights to bicliques so that every edge is covered with total weight exactly $1$, and it minimizes the total biclique weight. In the matrix formulation, it coincides with the fractional binary rank of the bipartite adjacency matrix. The treatment in "Engineering Insights into Biclique Partitions and Fractional Binary Ranks of Matrices" (Ghosal et al., 10 Feb 2025) places the notion in the setting of Kronecker powers, establishes its equivalence to fractional binary matrix factorization, develops a column-generation framework for computing it, and shows by explicit counterexample that it is not multiplicative under the Kronecker product.

1. Definition and basic LP formulation

Let $G=(U,V,E)$ be a bipartite graph with $E\subseteq U\times V$. A biclique is a complete bipartite subgraph $B=K_{S,T}$ induced by $S\subseteq U$ and $T\subseteq V$, with edge set $S\times T$ (Ghosal et al., 10 Feb 2025). The biclique partition number $bp(G)$ is the smallest number of bicliques whose edge sets form a partition of $\mathcal{B}$0: every edge must belong to exactly one selected biclique, and no edge may be used twice.

The fractional biclique partition number, denoted here by $\mathcal{B}$1, is the LP relaxation of $\mathcal{B}$2:

$\mathcal{B}$3

subject to

$\mathcal{B}$4

$\mathcal{B}$5

The equality constraints are essential: fractional partitions require exact edge coverage, not merely sufficient coverage (Ghosal et al., 10 Feb 2025).

A closely related quantity is the fractional biclique cover number $\mathcal{B}$6, defined by the LP

$\mathcal{B}$7

subject to

$\mathcal{B}$8

$\mathcal{B}$9

Since the feasible region of $1$0 contains that of $1$1, one has

$1$2

This distinction between partition and cover is structurally important. A recurring misconception is that the fractional partition and fractional cover numbers should behave similarly because both are LP relaxations over the same biclique family. The data in (Ghosal et al., 10 Feb 2025) shows that they can differ substantially, both numerically and in their behavior under Kronecker products.

2. Matrix-theoretic equivalences

Let $1$3 be the bipartite adjacency matrix of $1$4, with $1$5 if and only if $1$6. The paper recalls the classical equivalence

$1$7

where $1$8 is the binary rank: the smallest $1$9 such that there exist binary matrices $G=(U,V,E)$0 and $G=(U,V,E)$1 with $G=(U,V,E)$2 under ordinary arithmetic (Ghosal et al., 10 Feb 2025).

The fractional binary rank, denoted $G=(U,V,E)$3 in the paper, is obtained by allowing a nonnegative diagonal matrix $G=(U,V,E)$4:

$G=(U,V,E)$5

subject to

$G=(U,V,E)$6

with $G=(U,V,E)$7, $G=(U,V,E)$8, and diagonal entries $G=(U,V,E)$9. The paper establishes the equivalence

$E\subseteq U\times V$0

The cover analogue is also identified. The Boolean rank $E\subseteq U\times V$1 equals the biclique cover number $E\subseteq U\times V$2, and its fractional relaxation $E\subseteq U\times V$3 equals $E\subseteq U\times V$4 (Ghosal et al., 10 Feb 2025). These correspondences make the fractional biclique partition number simultaneously a graph covering parameter, a matrix factorization parameter, and an LP relaxation of an exact combinatorial decomposition problem.

The paper also recalls the fooling-set, or isolating-set, lower bound. A fooling set in $E\subseteq U\times V$5 is a set of ones such that no two lie in a common all-ones submatrix; equivalently, it is a maximum induced matching in $E\subseteq U\times V$6. If $E\subseteq U\times V$7 denotes the size of a maximum fooling set, then

$E\subseteq U\times V$8

Thus the fractional biclique cover number is at least as strong as the widely used fooling-set bound (Ghosal et al., 10 Feb 2025).

3. Kronecker products and asymptotic behavior

For binary matrices $E\subseteq U\times V$9 and $B=K_{S,T}$0, the Kronecker product $B=K_{S,T}$1 is the $B=K_{S,T}$2 block matrix whose $B=K_{S,T}$3-block is $B=K_{S,T}$4. If $B=K_{S,T}$5 and $B=K_{S,T}$6 are the corresponding bipartite graphs, then the product graph $B=K_{S,T}$7 has bipartition $B=K_{S,T}$8 and an edge $B=K_{S,T}$9 whenever $S\subseteq U$0 and $S\subseteq U$1; its adjacency matrix is $S\subseteq U$2 (Ghosal et al., 10 Feb 2025).

Writing

$S\subseteq U$3

the asymptotic fractional binary rank is defined as

$S\subseteq U$4

The paper uses geometric normalization via the $S\subseteq U$5th root, not normalization by $S\subseteq U$6. Existence of the limit follows from submultiplicativity and Fekete’s lemma (Ghosal et al., 10 Feb 2025).

The central structural result is non-multiplicativity:

$S\subseteq U$7

for some binary matrices $S\subseteq U$8. The explicit counterexample in the paper is the Domino matrix $S\subseteq U$9, for which

$T\subseteq V$0

Hence the fractional binary rank is not multiplicative with respect to $T\subseteq V$1 (Ghosal et al., 10 Feb 2025).

This has an immediate asymptotic consequence. Since strict submultiplicativity occurs already at the second Kronecker power, the asymptotic quantity $T\subseteq V$2 can be strictly smaller than the one-shot value $T\subseteq V$3. The paper exhibits this separation explicitly for the Domino.

4. The Domino graph as the canonical case study

The Domino graph $T\subseteq V$4 is the bipartite graph with bipartition sizes $T\subseteq V$5 and adjacency matrix

$T\subseteq V$6

Equivalently, with $T\subseteq V$7 and $T\subseteq V$8, its edges are $T\subseteq V$9, $S\times T$0, $S\times T$1, $S\times T$2, $S\times T$3, $S\times T$4, $S\times T$5 (Ghosal et al., 10 Feb 2025).

For the base graph, the paper gives an explicit fractional biclique partition showing

$S\times T$6

and also shows

$S\times T$7

Optimality of $S\times T$8 is certified by a dual witness $S\times T$9 satisfying $bp(G)$0 and $bp(G)$1 (Ghosal et al., 10 Feb 2025).

For the second Kronecker power, the paper constructs an explicit fractional biclique partition of $bp(G)$2 using $bp(G)$3 bicliques of weight $bp(G)$4 each, yielding

$bp(G)$5

A matching dual certificate shows optimality at $bp(G)$6 (Ghosal et al., 10 Feb 2025). This is the explicit obstruction to multiplicativity.

The paper further computes $bp(G)$7 for $bp(G)$8 and reports the corresponding $bp(G)$9th roots:

$\mathcal{B}$00	$\mathcal{B}$01	$\mathcal{B}$02
1	2.5	2.5
2	6	2.449490
3	13.818792	2.399699
4	32.040389	2.379164
5	75.201302	2.372712

Combining these computations with the analytical lower bound described below, the paper reports

$\mathcal{B}$03

This yields a strict gap

$\mathcal{B}$04

demonstrating that asymptotic fractional binary rank can improve over the base fractional rank (Ghosal et al., 10 Feb 2025).

5. Column generation and computational engineering

Direct enumeration of all bicliques is infeasible for Kronecker powers. For $\mathcal{B}$05, the number of edges is $\mathcal{B}$06, the number of maximal bicliques grows as $\mathcal{B}$07, and the total number of bicliques is $\mathcal{B}$08 because one maximal biclique has size $\mathcal{B}$09, yielding $\mathcal{B}$10 sub-bicliques (Ghosal et al., 10 Feb 2025). The paper therefore formulates the computation of $\mathcal{B}$11 through column generation.

The restricted master LP is the partition LP over a current subset $\mathcal{B}$12:

$\mathcal{B}$13

subject to

$\mathcal{B}$14

Its dual is

$\mathcal{B}$15

subject to

$\mathcal{B}$16

For a biclique $\mathcal{B}$17, the reduced cost is

$\mathcal{B}$18

Any biclique with $\mathcal{B}$19, with $\mathcal{B}$20 in the experiments, has negative reduced cost and can improve the primal objective (Ghosal et al., 10 Feb 2025).

Pricing is handled in two stages. First, the inclusion-wise maximal bicliques of the current graph are computed using MBEA [Zhang 2014]. Then, for each maximal biclique $\mathcal{B}$21, the paper solves an ILP for the maximum-sum sub-biclique under the dual weights:

$\mathcal{B}$22

subject to

$\mathcal{B}$23

with $\mathcal{B}$24 and $\mathcal{B}$25. Here $\mathcal{B}$26 selects left vertices, $\mathcal{B}$27 selects right vertices, and $\mathcal{B}$28 indicates whether edge $\mathcal{B}$29 is included, ensuring that the selected edges form a biclique (Ghosal et al., 10 Feb 2025). These pricing problems are solved independently per maximal biclique and in parallel using Gurobi 10.0.

The paper emphasizes engineering choices for initialization and memory control. Initial biclique sets include trivial feasible sets such as star bicliques, all bicliques for small base graphs, and an inductive construction for Kronecker powers: if $\mathcal{B}$30 and $\mathcal{B}$31 are feasible sets for $\mathcal{B}$32 and $\mathcal{B}$33, then

$\mathcal{B}$34

is feasible for $\mathcal{B}$35 (Ghosal et al., 10 Feb 2025). Using only bicliques with positive weight in the optimal solutions for $\mathcal{B}$36 and $\mathcal{B}$37 gives a strong warm start for $\mathcal{B}$38.

To control memory and runtime, the paper prunes biclique columns whose dual constraints are slack in more than $\mathcal{B}$39 successive iterations. Pruned columns may later reappear through pricing. With this heuristic, iteration time for $\mathcal{B}$40 decreased from approximately $\mathcal{B}$41 hours to approximately $\mathcal{B}$42 minutes; the full computation required $\mathcal{B}$43 column-generation iterations. The reported hardware was $\mathcal{B}$44 physical cores ($\mathcal{B}$45 logical threads), an Intel Xeon E5-2680 v3 @2.50 GHz, and $\mathcal{B}$46 GB RAM. Gurobi 10.0 was used with the concurrent optimizer, and barrier plus crossover was fastest. In the final iterations, barrier took approximately $\mathcal{B}$47 minutes, while crossover could take up to approximately $\mathcal{B}$48 hours (Ghosal et al., 10 Feb 2025).

The paper notes that $\mathcal{B}$49 was out of reach with the available computational resources because of memory limits in building the initial master problem. This indicates that the current bottleneck is not merely pricing difficulty but also the size of the warm-start column pool.

6. Bounds, comparisons, and unresolved questions

The lower-bound structure begins with

$\mathcal{B}$50

or in matrix form,

$\mathcal{B}$51

The fooling-set bound refines this to

$\mathcal{B}$52

so the fractional biclique cover number is at least as strong as the isolating-set bound and can improve it strictly. The paper gives the example of the Crown graph $\mathcal{B}$53, for which

$\mathcal{B}$54

(Ghosal et al., 10 Feb 2025).

For Kronecker products, the paper proves a stronger inequality:

$\mathcal{B}$55

The proof sketch is a projection argument: from an optimal fractional partition of $\mathcal{B}$56, one groups bicliques by projection onto $\mathcal{B}$57; the total weight in each block $\mathcal{B}$58 is at least $\mathcal{B}$59; after normalization by $\mathcal{B}$60, one obtains a feasible fractional cover of $\mathcal{B}$61 (Ghosal et al., 10 Feb 2025). Applying the same argument symmetrically yields the maximum bound.

Taking $\mathcal{B}$62 gives

$\mathcal{B}$63

hence

$\mathcal{B}$64

and therefore

$\mathcal{B}$65

Together with submultiplicativity and the integral inequality, the paper records the chain

$\mathcal{B}$66

A crucial contrast is that $\mathcal{B}$67 is multiplicative under $\mathcal{B}$68:

$\mathcal{B}$69

whereas $\mathcal{B}$70 is not (Ghosal et al., 10 Feb 2025). This separates the fractional partition problem from the fractional cover problem at a structural level.

The broader unresolved issue concerns the integral, not fractional, setting. The paper cites the longstanding open problem of whether

$\mathcal{B}$71

for all binary matrices $\mathcal{B}$72, equivalently whether

$\mathcal{B}$73

The results in (Ghosal et al., 10 Feb 2025) do not settle this question, but they show that the fractional relaxation behaves differently from fractional Boolean rank and can exhibit strict asymptotic improvement. The paper further states that biclique covers and partitions are connected to nondeterministic and deterministic communication complexity, and that the asymptotic fractional rank informs rates and limits in product constructions. A plausible implication is that sharper understanding of $\mathcal{B}$74 may affect both matrix factorization theory and product-based complexity bounds.

The Domino case summarizes the present state of knowledge in concrete form:

$\mathcal{B}$75

and

$\mathcal{B}$76

Within the scope of (Ghosal et al., 10 Feb 2025), these facts establish that the fractional biclique partition number is not multiplicative under Kronecker products, that its asymptotic counterpart can be strictly smaller than its base value, and that progress on tighter asymptotic bounds is currently constrained by both combinatorial explosion and large-scale LP engineering.