Bipartite Graph Approximation

• Bipartite graph approximation is a method that exploits the two-set partition of graphs to simplify and efficiently approximate NP-hard problems.
• It utilizes low-dimensional cube representations and randomized algorithms, such as RANDUNIT, to improve solutions using LP duality and spectral properties.
• Researchers apply these techniques in distributed computing, geometric data analysis, and combinatorial optimization to achieve scalable, high-precision approximations.

Bipartite graph approximation refers broadly to algorithmic and representational strategies that exploit or approximate bipartite structure to efficiently solve or approximate computationally intractable problems in graphs. This area encompasses both the design of algorithms that leverage the special properties of bipartite graphs (particularly their partitioned structure, LP duality, and spectral properties), as well as methods to approximate general graphs by bipartite ones or to find low-dimensional geometric representations that reveal combinatorial simplifications. Bipartite graph approximation is central in optimization, distributed computing, randomized and quantum algorithms, and geometric data analysis.

1. Structural Representation: Cubicity and Dimensionality

Cubicity is a geometric parameter of a graph, denoted cub(G), defined as the smallest integer $k$ for which the graph is the intersection graph of axis-parallel unit cubes in $\mathbb{R}^k$. For bipartite graphs $G=(A \cup B,E)$, this notion gains significance because many NP-complete problems become efficiently solvable or better approximable if a low-dimensional cube representation is available. The key result is that for $|A|=n_1 \leq |B|=n_2$, and setting $\Delta' = \min\{\Delta_A, \Delta_B\}$ (where $\Delta_A, \Delta_B$ are the maximal degrees in $A$ and $B$),

$$\mathrm{cub}(G) \leq 2(\Delta' + 2) \lceil \ln n_2 \rceil,$$

which refines previous general bounds that depend on the overall maximum degree and total number of vertices. The paper presents a randomized algorithm (RANDUNIT) that constructs such a representation efficiently by generating permutations and forming a sequence of unit interval graphs, with dimension and time complexity scaling as $O(\Delta'(m+n) \lceil \ln n_2 \rceil)$ and overall ambient dimension $3(\Delta'+2)\lceil \ln n_2\rceil$ (0810.2697). This dimensionality reduction underpins efficient exact or approximate algorithms for hard problems such as maximum clique, coloring, and independent set when restricted to bipartite or nearly bipartite graphs.

2. Approximation Algorithms for Combinatorial Problems

Many NP-hard graph problems admit better (sometimes optimal) approximation ratios or even exact solutions when the input is bipartite or can be well-approximated by bipartite graphs.

• Vertex Cover and Max k-Vertex Cover: The natural LP relaxation for minimum vertex cover has integral extreme points on bipartite graphs, allowing tight (factor 1) solutions. Recent work shows that pure combinatorial algorithms achieve a 0.7-approximation ratio for the max $k$-vertex cover problem, improving over greedy ($\approx 0.632$) and simple $2/3$ algorithms (1502.07930). The approach relies on partitioning the problem between the two color classes, constructing several candidate solutions, and employing detailed parameter elimination analyses.
• Oblivious and Weighted Matching: For maximum matching and its variants, randomized (Ranking-type) algorithms achieve the optimal $1 - 1/e$ ratio for oblivious bipartite matching, even with adversarially correlated edge existence, via randomized primal-dual analysis (2002.06037). For vertex-weighted matchings, a $2/3$-approximation can be found in nearly linear time in practice by restricting searches to short augmenting paths and combining solutions via the Mendelsohn-Dulmage theorem (1804.08016).
• Maximum k-Dependent Set: A notable combinatorial (1 + k/(k+2))-approximation was recently achieved for the maximum $k$-dependent set problem on bipartite and König–Egerváry graphs, strictly improving over previous iterative rounding methods. The algorithm repeatedly removes maximum matchings and computes a maximum independent set in the residual graph, with a running time $O(k m \sqrt{n})$ (2110.02487).
• Exact Matching under Constraints: For the Exact Matching problem (finding a perfect matching with exactly $k$ red edges), a deterministic polynomial-time algorithm with one-sided guarantee $\frac{k}{3} \leq k' \leq k$ is given, leveraging cycle decompositions and careful edge selection (2307.02205).

3. Counting and Sampling in Bipartite Graphs

Approximate counting of combinatorial objects in bipartite graphs leverages both structural characteristics and powerful algorithmic frameworks:

• Independent Set Counting: For $d$-regular bipartite expanders, fully polynomial-time approximation schemes (FPTAS) have been developed based on Sapozhenko’s graph container method combined with polymer models and cluster expansion. These methods partition independent sets into structured “containers” and handle deviations via controlled subgraph exploration, with runtime polynomial in $n$ for expanders and subexponential for general $d$-regular bipartite graphs (2109.03744). The container/polymer techniques extend to weighted independent sets (hard-core model) as well, with provable bounds for fugacities above certain thresholds.
• Biclique Counting: To count $(p,q)$-bicliques in large bipartite graphs, a scalable approximate method uses the $(p,q)$-broom (a special spanning tree substructure of the biclique), countable via dynamic programming aided by graph coloring for order-uniqueness. A rejection sampling algorithm then lifts broom counts to biclique estimates, with unbiasedness and error bounds proved analytically; the result offers substantial empirical speed and accuracy compared to prior exact and approximate methods (2505.10471).
• Polymer Model and Imbalance Conditions: On unbalanced bipartite graphs, the cluster expansion and polymer model provide an FPTAS for the partition function of the hardcore model (i.e., independent set counting) when degree/fugacity imbalance satisfies $$64 \lambda_L \lambda_R \leq (1+\lambda_L)^{\Delta_L}$$. The exponential decay of correlations is established for the relevant class, yielding strong control over sample and counting estimates (1906.01666).

4. Distributed Approximation and Computational Barriers

Distributed bipartite graph approximation emphasizes both algorithmic tractability and lower bounds tied to locality:

• Minimum Vertex Cover (CONGEST Model): Efficient distributed algorithms can compute $(1+\varepsilon)$-approximate vertex covers in bipartite graphs in polylogarithmic time using clustering, matching-vertex cover duality, and localized BFS techniques; for general graphs, such efficient algorithms are not known (2011.10014).
• Fundamental Barriers: No o(log n)-time approximation scheme exists for minimum vertex cover in bipartite graphs with maximum degree 3 in the distributed (LOCAL) model. This barrier, established via reductions to local cut minimization on expanders and novel gadget constructions, demonstrates a strong contrast with maximum matching (where constant-time local approximation is possible) (1205.4605).

5. Geometric and Spectral Approximations in Applications

• Point Cloud Denoising: In geometry processing, point clouds are naturally modeled as $k$-nearest neighbor graphs, typically not bipartite. Approximating these with bipartite graphs enables linearization of nonlinear relations (e.g., between coordinates and normals), facilitating convex (quadratic or $\ell_1$-based) optimization with robust regularization terms such as GTV or reweighted graph Laplacian on surface normals. Bipartite approximation is achieved by minimizing the KL divergence between original and bipartite graph edge distributions, partitioning the nodes for alternate optimization and fast convergence. Experimental results confirm superior denoising quality and computational efficiency (1804.10831, 1812.07711).

6. Specialized Problems and Recent Advances

• Graphic TSP in Bipartite Graphs: Improved approximation ratios have been achieved for graphic TSP (Traveling Salesman Problem on unweighted graphs) in cubic and $k$–regular bipartite graphs by designing cycle covers with few cycles, made possible by structural decompositions into perfect matchings and gadgets that control cycle length. The best known is a $9/7$-approximation factor in the cubic case (1311.3640).
• Approximating the Quantum Max-Cut: For the quantum Max-Cut problem, classical polynomial-time algorithms now obtain improved approximation ratios (up to $0.8162$) in bipartite graphs using constructive combinations of SDP relaxations and matching polytope containment, outperforming previous best-known algorithms (2504.11120).
• Bipartization in Geometric Graphs: Substantial progress was made in bipartizing (odd cycle transversal) problems in disk and pseudo-disk graphs, breaking the longstanding $3$-approximation barrier by a multi-pronged strategy involving structural reductions, triangle packing, randomization, and exploiting planarity and degeneracy; the approach generalizes to a broader class of graphs and deletion problems (2407.09356).

7. Theoretical and Broader Implications

Bipartite graph approximation techniques are closely tied to fundamental themes in combinatorics, optimization, and computational complexity, including:

• Improved algorithms for canonical problems (vertex cover, matching, biclique counting) through partition exploitation, substructure sparsification, and duality.
• Integrality gaps and SDP limitations in random and structured bipartite graphs, informing both the power and the limits of convex relaxation methods (1510.05137).
• Connections between geometric graph representations (such as low cubicity) and algorithmic tractability, establishing routes to dimension-reduction leveraging bipartite structure (0810.2697).
• Distributed algorithms’ efficiency and provable barriers, emphasizing non-locality as a constraint for certain LP-dual pairings (1205.4605, 2011.10014).
• Scalable, high-accuracy methods for large-scale bipartite graphs in real applications, including motif counting, recommender systems, machine learning, and geometric data analysis (2505.10471, 1812.07711).

These results collectively highlight both the centrality of bipartite approximation strategies in discrete mathematics and their versatile applicability across theoretical and applied domains.