Second Price Matching with Complete Allocation and Degree Constraints

TL;DR

This work studies Second Price Matching (2PM) and its perfect-matching variant (2PPM) under degree constraints on bipartite graphs with binary bids. It introduces auxiliary graph constructions and augmentation procedures that, combined with Tutte-Berge-type reasoning, yield a $9/10$-approximation for $(3,2)$-regular inputs and exact polynomial-time algorithms for $(d,2)$-regular graphs with $d\ge 4$ for both 2PM and 2PPM. For the variant where all goods must be matched, the paper proves a tight $(1-1/e)$-hardness in the general case and provides a submodular-maximization formulation that attains the same approximation ratio, with exact solvability on the studied regular graphs. The results highlight a stark contrast between general APX-hardness and tractable structure under regular degree constraints, and they connect augmentation-based matching procedures to classical results like the Tutte-Berge formula. Overall, the paper advances understanding of provable guarantees for second-price combinatorial auctions under structural graph restrictions and clarifies the computational limits of these settings.

Abstract

We study the Second Price Matching problem, introduced by Azar, Birnbaum, Karlin, and Nguyen in 2009. In this problem, a bipartite graph (bidders and goods) is given, and the profit of a matching is the number of matches containing a second unmatched bidder. Maximizing profit is known to be APX-hard and the current best approximation guarantee is $1/2$. APX-hardness even holds when all degrees are bounded by a constant. In this paper, we investigate the approximability of the problem under regular degree constraints. Our main result is an improved approximation guarantee of $9/10$ for Second Price Matching in $(3,2)$-regular graphs and an exact polynomial-time algorithm for $(d,2)$-regular graphs if $d\geq 4$. Our algorithm and its analysis are based on structural results in non-bipartite matching, in particular the Tutte-Berge formula coupled with novel combinatorial augmentation methods. We also introduce a variant of Second Price Matching where all goods have to be matched, which models the setting of expiring goods. We prove that this problem is hard to approximate within a factor better than $(1-1/e)$ and show that the problem can be approximated to a tight $(1-1/e)$ factor by maximizing a submodular function subject to a matroid constraint. We then show that our algorithm also solves this problem exactly on regular degree constrained graphs as above.

Figures (12)

• Figure 1: Instance of 2PPM where all edges correspond to weight $1$ bids on the goods of $A$. The perfect matching taken corresponds to the set of red edges, the unmatched nodes are shaded in blue, and the profit gained by the second highest bidder per good is the set of blue edges. As the matching is perfect over the $6$ nodes of $A$, and there exists $6$ blue edges of second price bids, the profit gained here is $6$ which is maximum possible.
• Figure 2: The construction of the auxiliary graph $G^\prime$
• Figure 3: A tight example of 3-regular graph $G$ (with multiple edges) with $\nu(G)=2|V(G)|/5$. One of the maximum matchings in $G$ is colored in red.
• Figure 4: An example of the construction of the auxiliary graph $G^{\prime\prime}$ of a $(4,2)$-biregular graph $G$ with $B^\prime=\{b_6\}$. The set $A\setminus N(B^\prime)$ is colored in blue and the set $N(B^\prime)$ is colored in red.
• Figure 5: An instance $\mathcal{I}$ of Max $k$-Cover formulated as a bipartite graph: the part on top represents the set of elements $\mathcal{U}$ and the part at bottom represents the set of subsets $\mathcal{S}$.

Theorems & Definitions (37)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 3
• Theorem 4
• proof
• Claim 1
• proof : Proof of Claim.
• Claim 2
• proof : Proof of Claim.