The Multiple-Choice Knapsack problem is a generalization of the Knapsack problem in which items are partitioned into disjoint classes and exactly one item must be selected from each class. Besides the standard dynamic program running in time 𝒪(nc), where n is the total number of items and c is the knapsack capacity, no other pseudopolynomial-time algorithm is currently known. We present an 𝒪(n+w_max^4)-time algorithm, where w_max = max_i (w_{i n_i}-w_{i 1}) is the maximum weight range within a class.

Abstract. In the Bin Packing problem one is given [Formula: see text] items with weights [Formula: see text] and [Formula: see text] bins with capacities [Formula: see text]. The goal is to partition the items into sets [Formula: see text] such that [Formula: see text] for every bin [Formula: see text], where [Formula: see text] denotes [Formula: see text]. Björklund, Husfeldt, and Koivisto [ SIAM J. Comput., 39 (2009), pp. 546–563] presented an [Formula: see text] time algorithm for Bin Packing (the [Formula: see text] notation omits factors polynomial in the input size). In this paper, we show that for every [Formula: see text] there exists a constant [Formula: see text] such that an instance of Bin Packing with [Formula: see text] bins can be solved in [Formula: see text] randomized time. Before our work, such improved algorithms were not known even for [Formula: see text]. A key step in our approach is the following new result in Littlewood–Offord theory on the additive combinatorics of subset sums: For every [Formula: see text] there exists an [Formula: see text] such that if [Formula: see text] for some [Formula: see text], then [Formula: see text].

We consider the problem of computing shortest paths in weighted unit-disk graphs in constant dimension d. Although the single-source and all-pairs variants of this problem are well-studied in the plane case, no non-trivial exact distance oracles for unit-disk graphs have been known to date, even for d = 2. The classical result of Sedgewick and Vitter [Algorithmica '86] shows that for weighted unit-disk graphs in the plane the A^* search has average-case performance superior to that of a standard shortest path algorithm, e.g., Dijkstra’s algorithm. Specifically, if the n corresponding points of a weighted unit-disk graph G are picked from a unit square uniformly at random, and the connectivity radius is r ∈ (0,1), A^* finds a shortest path in G in O(n) expected time when r = Ω(√{log n/n}), even though G has Θ((nr)²) edges in expectation. In other words, the work done by the algorithm is in expectation proportional to the number of vertices and not the number of edges. In this paper, we break this natural barrier and show even stronger sublinear time results. We propose a new heuristic approach to computing point-to-point exact shortest paths in unit-disk graphs. We analyze the average-case behavior of our heuristic using the same random graph model as used by Sedgewick and Vitter and prove it superior to A^*. Specifically, we show that, if we are able to report the set of all k points of G from an arbitrary rectangular region of the plane in O(k + t(n)) time, then a shortest path between arbitrary two points of such a random graph on the plane can be found in O(1/r² + t(n)) expected time. In particular, the state-of-the-art range reporting data structures imply a sublinear expected bound for all r = Ω(√{log n/n}) and O(√n) expected bound for r = Ω(n^{-1/4}) after only near-linear preprocessing of the point set. Our approach naturally generalizes to higher dimensions d ≥ 3 and yields sublinear expected bounds for all d = O(1) and sufficiently large r.

In the Bin Packing problem one is given n items with weights w1, …, wn and m bins with capacities c1, …, cm. The goal is to find a partition of the items into sets S1, …, Sm such that w(Sj) ≤ cj for every bin j, where w(X) denotes Σi∊xwi. Björklund, Husfeldt and Koivisto (SICOMP 2009) presented an time algorithm for Bin Packing. In this paper, we show that for every m ∊ ℕ there exists a constant σm > 0 such that an instance of Bin Packing with m bins can be solved in randomized time. Before our work, such improved algorithms were not known even for m equals 4. A key step in our approach is the following new result in Littlewood-Offord theory on the additive combinatorics of subset sums: For every δ > 0 there exists an ∊ > 0 such that if |{X ⊆ {1, …, n} : w(X) = v}| ≥ 2(1–∊)n for some v then |{w(X) : X ⊆ {1, …, n}}| ≤ 2δn.

In the Equal-Subset-Sum problem, we are given a set $S$ of $n$ integers and the problem is to decide if there exist two disjoint nonempty subsets $A,B \subseteq S$, whose elements sum up to the same value. The problem is NP-complete. The state-of-the-art algorithm runs in $O^{*}(3^{n/2}) \le O^{*}(1.7321^n)$ time and is based on the meet-in-the-middle technique. In this paper, we improve upon this algorithm and give $O^{*}(1.7088^n)$ worst case Monte Carlo algorithm. This answers the open problem from Woeginger's inspirational survey. Additionally, we analyse the polynomial space algorithm for Equal-Subset-Sum. A naive polynomial space algorithm for Equal-Subset-Sum runs in $O^{*}(3^n)$ time. With read-only access to the exponentially many random bits, we show a randomized algorithm running in $O^{*}(2.6817^n)$ time and polynomial space.

For some two-player games (e.g. Go), no accurate and inexpensive heuristic is known for evaluating leaves of a search tree. For other games (e.g. chess), a heuristic is known (sum of piece values). For other games (e.g. Hex), only a local heuristic — one that compares children reliably, but non-siblings poorly — is known (cell voltage drop in the Shannon/Anshelevich electric circuit model). In this paper we introduce a search algorithm for a two-player perfect information game with a reasonable local heuristic. Sibling Conspiracy Number Search (SCNS) is an anytime best-first version of Conspiracy Number Search based not on evaluation of leaf states of the search tree, but — for each node — on relative evaluation scores of all children of that node. SCNS refines CNS search value intervals, converging to Proof Number Search. SCNS is a good framework for a game player. We tested SCNS in the domain of Hex, with promising results. We implemented an 11-by-11 SCNS Hex bot, DeepHex. We competed DeepHex against current Hex bot champion MoHex, a Monte-Carlo Tree Search player, and previous Hex bot champion Wolve, an Alpha-Beta Search player. DeepHex widely outperforms Wolve at all time levels, and narrowly outperforms MoHex once time reaches 4min/move.

For connection games such as Hex or Y or Havannah, finding guaranteed cell-to-cell connection strategies can be a computational bottleneck. In automated players and solvers, sets of such virtual connections are often found with Anshelevich's H-search algorithm: initialize trivial connections, and then repeatedly apply an AND-rule (for combining connections in series) and an OR-rule (for combining connections in parallel). We present FastVC Search, a new algorithm for finding such connections. FastVC Search is more effective than H-search when finding a representative set of connections quickly is more important than finding a larger set of connections slowly. We tested FastVC Search in an alpha-beta player Wolve, a Monte Carlo tree search player MoHex, and a proof number search implementation called Solver. It does not strengthen Wolve, but it significantly strengthens MoHex and Solver.