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Register a new userEnvy-free Matchings in Bipartite Graphs and their Applications to Fair Division
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Added by
Erel Segal-Halevi
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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A matching in a bipartite graph $G:=(X + Y,E)$ is said to be envy-free if no unmatched vertex in $X$ is adjacent to a mathced vertex in $Y$. Every perfect matching is envy-free, but envy-free matchings may exist even when perfect matchings do not. We provide a polynomial-time algorithm for finding an envy-free matching of maximum cardinality. For edge-weighted bipartite graphs, we provide a polynomial-time algorithm for finding a maximum-cardinality envy-free matching of minimum weight. We show how envy-free matchings can be used in various fair division problems with either continuous resources (cakes) or discrete ones. In particular, we show a symmetric algorithm for proportional cake-cutting, an algorithm for $1$-out-of-$(2n-2)$ maximin-share allocation of discrete goods, and an algorithm for $1$-out-of-$lfloor 2n/3rfloor$ maximin-share allocation of discrete bads (chores) among $n$ agents.
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In this paper we further investigate the well-studied problem of finding a perfect matching in a regular bipartite graph. The first non-trivial algorithm, with running time $O(mn)$, dates back to K{o}nigs work in 1916 (here $m=nd$ is the number of edges in the graph, $2n$ is the number of vertices, and $d$ is the degree of each node). The currently most efficient algorithm takes time $O(m)$, and is due to Cole, Ost, and Schirra. We improve this running time to $O(min{m, frac{n^{2.5}ln n}{d}})$; this minimum can never be larger than $O(n^{1.75}sqrt{ln n})$. We obtain this improvement by proving a uniform sampling theorem: if we sample each edge in a $d$-regular bipartite graph independently with a probability $p = O(frac{nln n}{d^2})$ then the resulting graph has a perfect matching with high probability. The proof involves a decomposition of the graph into pieces which are guaranteed to have many perfect matchings but do not have any small cuts. We then establish a correspondence between potential witnesses to non-existence of a matching (after sampling) in any piece and cuts of comparable size in that same piece. Kargers sampling theorem for preserving cuts in a graph can now be adapted to prove our uniform sampling theorem for preserving perfect matchings. Using the $O(msqrt{n})$ algorithm (due to Hopcroft and Karp) for finding maximum matchings in bipartite graphs on the sampled graph then yields the stated running time. We also provide an infinite family of instances to show that our uniform sampling result is tight up to poly-logarithmic factors (in fact, up to $ln^2 n$).
We consider the well-studied problem of finding a perfect matching in $d$-regular bipartite graphs with $2n$ vertices and $m = nd$ edges. While the best-known algorithm for general bipartite graphs (due to Hopcroft and Karp) takes $O(m sqrt{n})$ time, in regular bipartite graphs, a perfect matching is known to be computable in $O(m)$ time. Very recently, the $O(m)$ bound was improved to $O(min{m, frac{n^{2.5}ln n}{d}})$ expected time, an expression that is bounded by $tilde{O}(n^{1.75})$. In this paper, we further improve this result by giving an $O(min{m, frac{n^2ln^3 n}{d}})$ expected time algorithm for finding a perfect matching in regular bipartite graphs; as a function of $n$ alone, the algorithm takes expected time $O((nln n)^{1.5})$. To obtain this result, we design and analyze a two-stage sampling scheme that reduces the problem of finding a perfect matching in a regular bipartite graph to the same problem on a subsampled bipartite graph with $O(nln n)$ edges that has a perfect matching with high probability. The matching is then recovered using the Hopcroft-Karp algorithm. While the standard analysis of Hopcroft-Karp gives us an $tilde{O}(n^{1.5})$ running time, we present a tighter analysis for our special case that results in the stronger $tilde{O}(min{m, frac{n^2}{d} })$ time mentioned earlier. Our proof of correctness of this sampling scheme uses a new correspondence theorem between cuts and Halls theorem ``witnesses for a perfect matching in a bipartite graph that we prove. We believe this theorem may be of independent interest; as another example application, we show that a perfect matching in the support of an $n times n$ doubly stochastic matrix with $m$ non-zero entries can be found in expected time $tilde{O}(m + n^{1.5})$.
In this paper we study envy-free division problems. The classical approach to such problems, used by David Gale, reduces to considering continuous maps of a simplex to itself and finding sufficient conditions for this map to hit the center of the simplex. The mere continuity of the map is not sufficient for reaching such a conclusion. Classically, one makes additional assumptions on the behavior of the map on the boundary of the simplex (for example, in the Knaster--Kuratowski--Mazurkiewicz and the Gale theorem). We follow Erel Segal-Halevi, Frederic Meunier, and Shira Zerbib, and replace the boundary condition by another assumption, which has the meaning in economy as the possibility for a player to prefer an empty part in the segment partition problem. We solve the problem positively when $n$, the number of players that divide the segment, is a prime power, and we provide counterexamples for every $n$ which is not a prime power. We also provide counterexamples relevant to a wider class of fair or envy-free division problems when $n$ is odd and not a prime power. In this arxiv version that appears after the official publication we have corrected the statement and the proof of Lemma 3.4.
We study the fair division of items to agents supposing that agents can form groups. We thus give natural generalizations of popular concepts such as envy-freeness and Pareto efficiency to groups of fixed sizes. Group envy-freeness requires that no group envies another group. Group Pareto efficiency requires that no group can be made better off without another group be made worse off. We study these new group properties from an axiomatic viewpoint. We thus propose new fairness taxonomies that generalize existing taxonomies. We further study ne
We study the computational complexity of two well-known graph transversal problems, namely Subset Feedback Vertex Set and Subset Odd Cycle Transversal, by restricting the input to $H$-free graphs, that is, to graphs that do not contain some fixed graph~$H$ as an induced subgraph. By combining known and new results, we determine the computational complexity of both problems on $H$-free graphs for every graph $H$ except when $H=sP_1+P_4$ for some $sgeq 1$. As part of our approach, we introduce the Subset Vertex Cover problem and prove that it is polynomial-time solvable for $(sP_1+P_4)$-free graphs for every $sgeq 1$.
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