How many matchings cover the nodes of a graph?

Given an undirected graph, are there $k$ matchings whose union covers all of its nodes, that is, a matching-$k$-cover? A first, easy polynomial solution from matroid union is possible, as already observed by Wang, Song and Yuan (Mathematical Programming, 2014). However, it was not satisfactory neither from the algorithmic viewpoint nor for proving graphic theorems, since the corresponding matroid ignores the edges of the graph. We prove here, simply and algorithmically: all nodes of a graph can be covered with $k\ge 2$ matchings if and only if for every stable set $S$ we have $|S|\le k\cdot|N(S)|$. When $k=1$, an exception occurs: this condition is not enough to guarantee the existence of a matching-$1$-cover, that is, the existence of a perfect matching, in this case Tutte's famous matching theorem (J. London Math. Soc., 1947) provides the right `good' characterization. The condition above then guarantees only that a perfect $2$-matching exists, as known from another theorem of Tutte (Proc. Amer. Math. Soc., 1953). Some results are then deduced as consequences with surprisingly simple proofs, using only the level of difficulty of bipartite matchings. We give some generalizations, as well as a solution for minimization if the edge-weights are non-negative, while the edge-cardinality maximization of matching-$2$-covers turns out to be already NP-hard. We have arrived at this problem as the line graph special case of a model arising for manufacturing integrated circuits with the technology called `Directed Self Assembly'.Comment: 10 page

Optimal Recombination in Genetic Algorithms

This paper surveys results on complexity of the optimal recombination problem (ORP), which consists in finding the best possible offspring as a result of a recombination operator in a genetic algorithm, given two parent solutions. We consider efficient reductions of the ORPs, allowing to establish polynomial solvability or NP-hardness of the ORPs, as well as direct proofs of hardness results

Finding a Maximum Restricted tt-Matching via Boolean Edge-CSP

The problem of finding a maximum $2$-matching without short cycles has received significant attention due to its relevance to the Hamilton cycle problem. This problem is generalized to finding a maximum $t$-matching which excludes specified complete $t$-partite subgraphs, where $t$ is a fixed positive integer. The polynomial solvability of this generalized problem remains an open question. In this paper, we present polynomial-time algorithms for the following two cases of this problem: in the first case the forbidden complete $t$-partite subgraphs are edge-disjoint; and in the second case the maximum degree of the input graph is at most $2t-1$. Our result for the first case extends the previous work of Nam (1994) showing the polynomial solvability of the problem of finding a maximum $2$-matching without cycles of length four, where the cycles of length four are vertex-disjoint. The second result expands upon the works of B\'{e}rczi and V\'{e}gh (2010) and Kobayashi and Yin (2012), which focused on graphs with maximum degree at most $t+1$. Our algorithms are obtained from exploiting the discrete structure of restricted $t$-matchings and employing an algorithm for the Boolean edge-CSP.Comment: 20 pages, 2 figure

Complexity of Discrete Energy Minimization Problems

Discrete energy minimization is widely-used in computer vision and machine learning for problems such as MAP inference in graphical models. The problem, in general, is notoriously intractable, and finding the global optimal solution is known to be NP-hard. However, is it possible to approximate this problem with a reasonable ratio bound on the solution quality in polynomial time? We show in this paper that the answer is no. Specifically, we show that general energy minimization, even in the 2-label pairwise case, and planar energy minimization with three or more labels are exp-APX-complete. This finding rules out the existence of any approximation algorithm with a sub-exponential approximation ratio in the input size for these two problems, including constant factor approximations. Moreover, we collect and review the computational complexity of several subclass problems and arrange them on a complexity scale consisting of three major complexity classes -- PO, APX, and exp-APX, corresponding to problems that are solvable, approximable, and inapproximable in polynomial time. Problems in the first two complexity classes can serve as alternative tractable formulations to the inapproximable ones. This paper can help vision researchers to select an appropriate model for an application or guide them in designing new algorithms.Comment: ECCV'16 accepte