152 research outputs found
Maximum Skew-Symmetric Flows and Matchings
The maximum integer skew-symmetric flow problem (MSFP) generalizes both the
maximum flow and maximum matching problems. It was introduced by Tutte in terms
of self-conjugate flows in antisymmetrical digraphs. He showed that for these
objects there are natural analogs of classical theoretical results on usual
network flows, such as the flow decomposition, augmenting path, and max-flow
min-cut theorems. We give unified and shorter proofs for those theoretical
results.
We then extend to MSFP the shortest augmenting path method of Edmonds and
Karp and the blocking flow method of Dinits, obtaining algorithms with similar
time bounds in general case. Moreover, in the cases of unit arc capacities and
unit ``node capacities'' the blocking skew-symmetric flow algorithm has time
bounds similar to those established in Even and Tarjan (1975) and Karzanov
(1973) for Dinits' algorithm. In particular, this implies an algorithm for
finding a maximum matching in a nonbipartite graph in time,
which matches the time bound for the algorithm of Micali and Vazirani. Finally,
extending a clique compression technique of Feder and Motwani to particular
skew-symmetric graphs, we speed up the implied maximum matching algorithm to
run in time, improving the best known bound
for dense nonbipartite graphs.
Also other theoretical and algorithmic results on skew-symmetric flows and
their applications are presented.Comment: 35 pages, 3 figures, to appear in Mathematical Programming, minor
stylistic corrections and shortenings to the original versio
Hard cases of the multifacility location problem
AbstractLet μ be a rational-valued metric on a finite set T. We consider (a version of) the multifacility location problem: given a finite set V⊇T and a function c:V2→Z+, attach each element x∈V−T to an element γ(x)∈T minimizing ∑c(xy)μ(γ(x)γ(y)):xy∈V2, letting γ(t)≔t for each t∈T. Large classes of metrics μ have been known for which the problem is solvable in polynomial time. On the other hand, Dalhaus et al. (SIAM J. Comput. 23 (4) (1994) 864) showed that if T={t1,t2,t3} and μ(titj)=1 for all i≠j, then the problem (turning into the minimum 3-terminal cut problem) becomes strongly NP-hard. Extending that result and its generalization in (European J. Combin. 19 (1998) 71), we prove that for μ fixed, the problem is strongly NP-hard if the metric μ is nonmodular or if the underlying graph of μ is nonorientable (in a certain sense)
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