Shortest Reconfiguration of Perfect Matchings via Alternating Cycles

Motivated by adjacency in perfect matching polytopes, we study the shortest reconfiguration problem of perfect matchings via alternating cycles. Namely, we want to find a shortest sequence of perfect matchings which transforms one given perfect matching to another given perfect matching such that the symmetric difference of each pair of consecutive perfect matchings is a single cycle. The problem is equivalent to the combinatorial shortest path problem in perfect matching polytopes. We prove that the problem is NP-hard even when a given graph is planar or bipartite, but it can be solved in polynomial time when the graph is outerplanar

Faster exponential-time algorithms in graphs of bounded average degree

We first show that the Traveling Salesman Problem in an n-vertex graph with average degree bounded by d can be solved in O*(2^{(1-\eps_d)n}) time and exponential space for a constant \eps_d depending only on d, where the O*-notation suppresses factors polynomial in the input size. Thus, we generalize the recent results of Bjorklund et al. [TALG 2012] on graphs of bounded degree. Then, we move to the problem of counting perfect matchings in a graph. We first present a simple algorithm for counting perfect matchings in an n-vertex graph in O*(2^{n/2}) time and polynomial space; our algorithm matches the complexity bounds of the algorithm of Bjorklund [SODA 2012], but relies on inclusion-exclusion principle instead of algebraic transformations. Building upon this result, we show that the number of perfect matchings in an n-vertex graph with average degree bounded by d can be computed in O*(2^{(1-\eps_{2d})n/2}) time and exponential space, where \eps_{2d} is the constant obtained by us for the Traveling Salesman Problem in graphs of average degree at most 2d. Moreover we obtain a simple algorithm that counts the number of perfect matchings in an n-vertex bipartite graph of average degree at most d in O*(2^{(1-1/(3.55d))n/2}) time, improving and simplifying the recent result of Izumi and Wadayama [FOCS 2012].Comment: 10 page

Counting Euler Tours in Undirected Bounded Treewidth Graphs

We show that counting Euler tours in undirected bounded tree-width graphs is tractable even in parallel - by proving a $\#SAC^1$ upper bound. This is in stark contrast to #P-completeness of the same problem in general graphs. Our main technical contribution is to show how (an instance of) dynamic programming on bounded \emph{clique-width} graphs can be performed efficiently in parallel. Thus we show that the sequential result of Espelage, Gurski and Wanke for efficiently computing Hamiltonian paths in bounded clique-width graphs can be adapted in the parallel setting to count the number of Hamiltonian paths which in turn is a tool for counting the number of Euler tours in bounded tree-width graphs. Our technique also yields parallel algorithms for counting longest paths and bipartite perfect matchings in bounded-clique width graphs. While establishing that counting Euler tours in bounded tree-width graphs can be computed by non-uniform monotone arithmetic circuits of polynomial degree (which characterize $\#SAC^1$) is relatively easy, establishing a uniform $\#SAC^1$ bound needs a careful use of polynomial interpolation.Comment: 17 pages; There was an error in the proof of the GapL upper bound claimed in the previous version which has been subsequently remove

From the Quantum Approximate Optimization Algorithm to a Quantum Alternating Operator Ansatz

The next few years will be exciting as prototype universal quantum processors emerge, enabling implementation of a wider variety of algorithms. Of particular interest are quantum heuristics, which require experimentation on quantum hardware for their evaluation, and which have the potential to significantly expand the breadth of quantum computing applications. A leading candidate is Farhi et al.'s Quantum Approximate Optimization Algorithm, which alternates between applying a cost-function-based Hamiltonian and a mixing Hamiltonian. Here, we extend this framework to allow alternation between more general families of operators. The essence of this extension, the Quantum Alternating Operator Ansatz, is the consideration of general parametrized families of unitaries rather than only those corresponding to the time-evolution under a fixed local Hamiltonian for a time specified by the parameter. This ansatz supports the representation of a larger, and potentially more useful, set of states than the original formulation, with potential long-term impact on a broad array of application areas. For cases that call for mixing only within a desired subspace, refocusing on unitaries rather than Hamiltonians enables more efficiently implementable mixers than was possible in the original framework. Such mixers are particularly useful for optimization problems with hard constraints that must always be satisfied, defining a feasible subspace, and soft constraints whose violation we wish to minimize. More efficient implementation enables earlier experimental exploration of an alternating operator approach to a wide variety of approximate optimization, exact optimization, and sampling problems. Here, we introduce the Quantum Alternating Operator Ansatz, lay out design criteria for mixing operators, detail mappings for eight problems, and provide brief descriptions of mappings for diverse problems.Comment: 51 pages, 2 figures. Revised to match journal pape

K1,3K_{1,3}-covering red and blue points in the plane

We say that a finite set of red and blue points in the plane in general position can be $K_{1,3}$-covered if the set can be partitioned into subsets of size $4$, with $3$ points of one color and $1$ point of the other color, in such a way that, if at each subset the fourth point is connected by straight-line segments to the same-colored points, then the resulting set of all segments has no crossings. We consider the following problem: Given a set $R$ of $r$ red points and a set $B$ of $b$ blue points in the plane in general position, how many points of $R\cup B$ can be $K_{1,3}$-covered? and we prove the following results: (1) If $r=3g+h$ and $b=3h+g$, for some non-negative integers $g$ and $h$, then there are point sets $R\cup B$, like $\{1,3\}$-equitable sets (i.e., $r=3b$ or $b=3r$) and linearly separable sets, that can be $K_{1,3}$-covered. (2) If $r=3g+h$, $b=3h+g$ and the points in $R\cup B$ are in convex position, then at least $r+b-4$ points can be $K_{1,3}$-covered, and this bound is tight. (3) There are arbitrarily large point sets $R\cup B$ in general position, with $r=b+1$, such that at most $r+b-5$ points can be $K_{1,3}$-covered. (4) If $b\le r\le 3b$, then at least $\frac{8}{9}(r+b-8)$ points of $R\cup B$ can be $K_{1,3}$-covered. For $r>3b$, there are too many red points and at least $r-3b$ of them will remain uncovered in any $K_{1,3}$-covering. Furthermore, in all the cases we provide efficient algorithms to compute the corresponding coverings.Comment: 29 pages, 10 figures, 1 tabl