Harmonic Ranking for Edge-Weighted Oblivious Matching

Abstract: We study edge-weighted oblivious bipartite matching. The weight of every potential edge is known, but its existence is revealed only when the edge is probed, and a successful probe between two free vertices must be accepted immediately. We give an explicit randomized algorithm with certified competitive ratio $0.698$, improving the previous best guarantee of $0.659$ (Huang, Sun, Wu, and Zhao, FOCS 2025). The result is computer-assisted and verified by a reproducible exact-integer computation. The same algorithm has a $0.698$-competitive online implementation for the vertex-weighted random-arrival model, improving the previous $0.696$ unweighted guarantee of Mahdian and Yan (STOC 2011) and the $0.686$ vertex-weighted guarantee of Peng and Tang (EC 2025). Our algorithm, Harmonic Ranking, is a role-symmetric generalization of \textsc{Ranking}. It assigns an independent random rank $x_z$ to each vertex and probes a potential edge $uv$ in decreasing order of [ w_{uv}\frac{h(x_u)h(x_v)}{h(x_u)+h(x_v)}. ] This harmonic priority arises from a budget-balanced gain split and a mutual-proposal interpretation. The analysis lifts two cutoff curves into indicators, reducing the exponential-size factor-revealing problem to a polynomial-size directed minimum-cut instance. A maximum-flow computation with rounded-down integer capacities gives a rigorous certificate. Independently, we observe that the finite-grid unweighted relaxation of our factor-revealing program coincides exactly with a Mahdian--Yan program.

• The paper introduces Harmonic Ranking, a role-symmetric probing algorithm that achieves a certified 0.698015475248 competitive ratio, improving the prior 0.659 guarantee.
• Its harmonic gain-sharing rule converts an exponentially large cutoff optimization into a polynomial-size min-cut problem, verified through a reproducible exact-integer flow certificate.
• The same algorithm yields a 0.698-competitive method for vertex-weighted bipartite matching with random arrivals, while identifying open questions about optimal score rules and hierarchy gaps.

The problem and the result

The paper studies edge-weighted oblivious bipartite matching: the algorithm knows the bipartition $(L,R)$ and all potential edge weights, but not which edges are realized; probing a pair reveals its existence, and a successful probe between two free vertices must be accepted irrevocably. The competitive ratio is the largest $\Gamma$ such that $\mathbb{E}[ALG]\ge \Gamma\cdot OPT$ on every instance. Deterministic greedy is $1/2$-competitive and optimal among deterministic algorithms. Prior randomized work achieved $1-1/e$ via a one-sided rule (Tang–Wu–Zhang) and $0.659$ via a two-sided product rule (Huang–Sun–Wu–Zhao). This paper introduces Harmonic Ranking, an explicit randomized algorithm with certified ratio

$\mathbb{E}[ALG(G)] \ge 0.698015475248\; OPT(G),$

improving the previous best guarantee by nearly four percentage points. The guarantee is computer-assisted but backed by a reproducible exact-integer maximum-flow computation. A second contribution is an online consequence: the same algorithm is $0.698$-competitive for vertex-weighted bipartite matching under random arrivals, improving both the $0.696$ unweighted bound of Mahdian–Yan and the $0.686$ vertex-weighted bound of Peng–Tang.

Harmonic Ranking from gain sharing

The design principle is that the gain-sharing rule should determine the probing order. In the randomized primal–dual framework, each selected edge's weight is credited to its endpoints ($\Gamma$0); if every edge satisfies $\Gamma$1, scaled expectations are dual feasible and weak duality yields $\Gamma$2. Edge weights create an asymmetry—there is no canonical "item" and "buyer"—which the paper resolves through Mutual Proposals: each free vertex proposes along the incident available pair maximizing its prospective share $\Gamma$3, where $\Gamma$4 is complementary ($\Gamma$5), and a pair is probed exactly when both endpoints propose to each other.

For this to be well defined, proposals must never form directed cycles of length $\Gamma$6. The central structural theorem characterizes long-cycle-free rules completely: among positive complementary rank-based rules, cycle-freeness holds if and only if $\Gamma$7 for some positive function $\Gamma$8, unique up to scaling. The proof is short and clean: any four-cycle with odds-ratio product $\Gamma$9 admits weights realizing a strict proposal cycle, forcing multiplicativity of the odds ratio, which yields the harmonic form. Under this score-balanced rule, multiplying each endpoint's prospective gain by its own score produces the common priority

$\mathbb{E}[ALG]\ge \Gamma\cdot OPT$0

and Mutual Proposals is pathwise equivalent to scanning pairs in decreasing $\mathbb{E}[ALG]\ge \Gamma\cdot OPT$1—this scan is Harmonic Ranking. The characterization explains the algorithm's form but is not used in the competitive analysis. Tie-breaking must satisfy a local monotonicity condition (lowering a rank cannot move nonincident pairs relative to incident ones); a broad class of rules, including those needed for the online implementation, is admissible.

Cutoffs without monotonicity

The analysis adapts the cutoff machinery of Ranking analyses. Fixing all ranks except one endpoint's rank, a one-sided perturbation lemma shows that the event "$\mathbb{E}[ALG]\ge \Gamma\cdot OPT$2 matched strictly earlier than $\mathbb{E}[ALG]\ge \Gamma\cdot OPT$3" is upward closed in $\mathbb{E}[ALG]\ge \Gamma\cdot OPT$4, yielding cutoff curves $\mathbb{E}[ALG]\ge \Gamma\cdot OPT$5 and $\mathbb{E}[ALG]\ge \Gamma\cdot OPT$6 such that $\mathbb{E}[ALG]\ge \Gamma\cdot OPT$7 implies $\mathbb{E}[ALG]\ge \Gamma\cdot OPT$8 strictly earlier, and symmetrically for $\mathbb{E}[ALG]\ge \Gamma\cdot OPT$9. An insertion lemma compares greedy on $1/2$0 with greedy on $1/2$1, showing discrepant acceptances form an alternating path along which each $1/2$2-vertex is matched earlier in the full run.

A key structural departure from prior work: neither cutoff curve need be monotone in the edge-weighted setting. The paper exhibits a size-minimal four-vertex instance (with $1/2$3) in which both cutoffs decrease from $1/2$4 to $1/2$5 before increasing along the diagonal. This contrasts with vertex-weighted random arrivals, where $1/2$6 is provably nondecreasing, and unweighted Ranking, where both are. Consequently the robust classification is coarser: only the binary partition into "$1/2$7 no later than $1/2$8" versus "$1/2$9 strictly earlier" preserves downward closure per section. This loss of structure is precisely what makes the edge-weighted factor-revealing problem harder, and it motivates the indicator lifting below.

From two-curve game to min-cut

Conditioning on all ranks except $1-1/e$0, the threshold-gain lemma gives pointwise lower bounds on $1-1/e$1 and $1-1/e$2 in terms of the indicators $1-1/e$3, $1-1/e$4, and $1-1/e$5. Integrating over the unit square yields a two-curve variational functional $1-1/e$6, and the fixed-score guarantee $1-1/e$7 lower-bounds the competitive ratio, where $1-1/e$8 contains all measurable cutoff pairs whose indicators are almost surely disjoint—a relaxation, since not every pair in $1-1/e$9 need be realizable by an instance.

Direct grid discretization would enumerate $0.659$0 threshold pairs. The enabling idea is a lifting: represent each one-dimensional curve $0.659$1 by its two-dimensional indicator $0.659$2. On a uniform grid with step score $0.659$3, admissible label pairs satisfy three local implications (exclusivity, row-cutoff structure, column-cutoff structure), which become infinite-capacity arcs in a directed network with $0.659$4 nodes and $0.659$5 arcs. Discrete integration by parts expresses all objective capacities as affine functions of the increments $0.659$6, so the minimum cut equals the cellwise minimum of $0.659$7 over all admissible threshold pairs:

$0.659$8

A sampling argument shows $0.659$9, converting any finite flow value into an algorithmic guarantee. Notably, no cross-coordinate arcs encoding cutoff monotonicity are added—consistent with the nonmonotonicity example—and the outer dependence of capacities on the heights remains nonlinear, so numerical search is used only to select a candidate score, after which verification is exact.

Exact-integer certificate

The final score $\mathbb{E}[ALG(G)] \ge 0.698015475248\; OPT(G),$0 is a $\mathbb{E}[ALG(G)] \ge 0.698015475248\; OPT(G),$1-step piecewise-constant function with positive integer heights (normalized to $\mathbb{E}[ALG(G)] \ge 0.698015475248\; OPT(G),$2), generated by rounding a smooth three-parameter profile $\mathbb{E}[ALG(G)] \ge 0.698015475248\; OPT(G),$3 with $\mathbb{E}[ALG(G)] \ge 0.698015475248\; OPT(G),$4, $\mathbb{E}[ALG(G)] \ge 0.698015475248\; OPT(G),$5, $\mathbb{E}[ALG(G)] \ge 0.698015475248\; OPT(G),$6. Capacities are rounded down at scale $\mathbb{E}[ALG(G)] \ge 0.698015475248\; OPT(G),$7; since rounded capacities never exceed exact ones, any feasible flow transfers to the exact network. The verifier recomputes all $\mathbb{E}[ALG(G)] \ge 0.698015475248\; OPT(G),$8 stored $\mathbb{E}[ALG(G)] \ge 0.698015475248\; OPT(G),$9 capacities from exact rational identities and exhibits a feasible flow and source-reachable cut, both of integer value $0.698$0, certifying

$0.698$1

Max-flow/min-cut duality makes this a rigorous proof, not a numerical estimate; the artifact is publicly reproducible. Combined with the fixed-score guarantee, this completes the proof of the main theorem.

Online consequence and the variational hierarchy

In the vertex-weighted random-arrival model, Online Harmonic Ranking samples offline ranks and i.i.d. arrival timestamps, matching each arrival to the available neighbor maximizing $0.698$2. Since $0.698$3 is a common positive factor across neighbors, this maximizes the global harmonic priority; a stable sort by arrival time groups the global scan into arrival blocks without changing the matching, because any inverted pair is vertex-disjoint. The executions are pathwise equal, giving the $0.698$4-competitive online guarantee (assuming $0.698$5 is known).

The paper also organizes a hierarchy of factor-revealing problems indexed by model: edge-weighted (score-balanced rules, arbitrary cutoffs), vertex-weighted (complementary monotone rules, $0.698$6 nondecreasing), and unweighted (both cutoffs nondecreasing), with values satisfying $0.698$7. At the unweighted endpoint, the finite-grid relaxation coincides exactly—at every grid resolution—with the strongly factor-revealing LP of Mahdian and Yan. The proof reflects both coordinates to reduce the whole-cell game to a two-path game, removes transpose overlaps by a decrementing repair argument, then dualizes the transpose-balance constraints into price tables, contracts the fixed-price LP onto binary doubled covers via total unimodularity, and finally balances aggregate matrices by predecessor transport guided by an interval flow computing the penalty $0.698$8. This shows the complementary primal–dual formulation loses no strength relative to the classical program, though it does not assert optimizers lie in the score-balanced subclass.

Limitations and open questions

Several caveats are stated plainly. The relaxation $0.698$9 may include unrealizable cutoff pairs, so the $0.696$0 figure is a certified lower bound on the true factor-revealing value, not necessarily tight. The guarantee is computer-assisted: correctness rests on the reproducible artifact rather than a hand-verifiable closed-form score. The claimed equality between the general-complementary and vertex-weighted programs is explicitly marked as a numerical observation, not a proved identity. The paper leaves open whether the hierarchy inequalities are strict; numerics suggest $0.696$1 possibly with equality between the relaxed edge-weighted and vertex-weighted values, and distinct vertex-weighted versus unweighted values at every tested grid size, but separation in the continuum limit is unresolved. Whether the optimal rule in the larger class $0.696$2 is score-balanced is also open—the authors conjecture it is not, which would locate the edge-/vertex-weighted distinction in the interaction between cutoff geometry and the admissible rule class rather than in monotonicity alone.

Conclusion

The paper delivers a role-symmetric generalization of Ranking whose probing order is derived from a budget-balanced gain split, proves a $0.696$3 competitive ratio for edge-weighted oblivious matching via a polynomial-size min-cut reformulation of an exponentially large factor-revealing game, verifies the constant by exact integer computation, and transfers the guarantee to vertex-weighted online matching under random arrivals. Its methodological contribution—an indicator lifting that turns threshold-curve optimization into maximum flow, together with an exact correspondence to the Mahdian–Yan program—provides a template likely applicable to further refinements of the remaining gap toward the upper end of the hierarchy near $0.696$4.